Game theory. The Art of Strategic Thinking (A. Dixit, B. J. Nailbuff)

Abstract

Game theory is rigorous strategic thinking. It is the art of predicting the opponent’s next move, coupled with the knowledge that he is doing the same. The main part of the theory contradicts the usual worldly wisdom and common sense, so its study can form a new view of the structure of the world and the interaction of people. Using examples from cinema, sports, politics, history, the authors show how almost all companies and people are involved in interactions described by game theory. Knowing this subject will make you more successful in business and life.

How should one behave in society?

OUR ANSWER to this question has nothing to do with ethics or etiquette. We are not going to compete with philosophers, preachers or parents. The subject of our reflections is not too exalted, but it has no less influence on life than moral norms and rules of conduct. This book is about strategic behavior. We are all strategists, whether we like it or not. It’s better to be a good strategist than a bad one, and the purpose of this book is to help you develop your skills in finding and applying effective strategies.

Work and life in society are reduced to a continuous stream of decisions. Which career to choose, how to run a company, who to choose as a life partner, how to raise children, and whether to run for president are just some of the life-changing decisions you have to make. All of these situations have one thing in common: you are not in a vacuum. On the contrary, you are surrounded by people who actively make decisions that are somehow related to yours. And this relationship of decisions has a great influence on your thoughts and actions.

As an illustration, consider the difference between the forest.e solutions. and the general. When the forestry decides how to cut the forest, he does not expect resistance from it: his environment is neutral. But when a general tries to defeat an enemy army, he must anticipate and overcome any resistance that might interfere with his plans. Like a general, you should be aware of the fact that your business rivals, future spouse, and even your children are strategic thinkers. Their goals often conflict with your goals, but they can also coincide with them. Your own choice should allow for the possibility of conflict and create the conditions for cooperation. This book will teach you not only to think strategically, but also to turn your thoughts into actions.

Game theory is a branch of the social sciences that studies strategic decision making. Game theory covers games ranging from chess to parenting, from tennis to takeovers, from advertising to arms control. George Mikes, an English humorist of Hungarian origin, once said: “Many inhabitants of the continent believe that life is a game; The British think that cricket is a game.” In our opinion, both are right.

All games require a variety of skills. Basic skills, such as being able to hit the basket in basketball, knowing precedent in the law, or being able to keep a straight face in poker, are one category of skills; the ability to think strategically is different. Strategic thinking builds on core skills and helps you figure out how to get the most out of them. If you know the laws, you must develop a strategy to protect your client. If you know how good your football team members are at running or passing, and how well the opposing team is at defending, you as a coach must make a decision about whether the players should run or pass. Sometimes, as in the case of a nuclear confrontation, thinking strategically also means knowing when it is appropriate to stop playing.

Game theory as a science has not yet fully formed, and strategic thinking in many respects remains an art. Our overriding goal is to make you practitioners of this art, but this requires a serious knowledge of the basic concepts and methods of science called «strategy». For this reason, we have based this book on a combination of the two approaches. Chapter 1 begins with examples of the art of strategy that show how strategic questions arise in a variety of decision-making processes. We introduce readers to effective, less effective, and even outright bad strategies that were used by participants in games that took place in real life. All of these examples reflect a certain conceptual model. In chapters 2-4, we explore the foundations of strategic science with examples, each of which illustrates a principle. We then focus on the concepts and strategies for finding the right course of action in specific situations: how to mix moves when another player can use any of your system actions to their advantage; how to change the course of the game in your favor, as well as how to manipulate information in the process of strategic interaction. We conclude by describing a number of general categories of strategic situations, such as negotiation, auctions, elections, and incentive creation, that will show you all of these principles and strategies in action.

Science and art are fundamentally different: science is learned with a systematic, logical approach, while art can only be mastered through case studies, experience, and hard practice. In describing the foundations of strategic science, you will find principles and general rules: for example, backward reasoning is discussed in Chapter 2, and Nash equilibrium in Chapter 4. On the other hand, the art of strategy, which you may need in a variety of situations, will require you to additional efforts. Each specific situation is characterized by unique features. They must be taken into account, as well as the general principles of the science of strategy. The only way to improve your skills in the art of strategy is to use the method of inductive reasoning, that is, to analyze what steps have been taken in similar situations in the past. We intend to increase your strategic IQ with numerous examples, including tutorials, included in each chapter, as well as examples collected in the final chapter.

You’ll find examples throughout the book ranging from the familiar, fairly simple or funny (usually taken from literature, sports, or movies) to the very ominous, such as a nuclear confrontation. The examples of the first group are just a neat and fun way to introduce you to the basic concepts of game theory. As for the examples of the second group, many readers may think that the horror that the topic of nuclear war inspires makes it impossible to rationally analyze it. But we believe that since the Cold War is long over, the game-theoretic aspects of the arms race and the Cuban crisis can be analyzed in terms of strategic logic, somewhat detached from their emotional component.

Case study analysis is similar to what you encountered in business school classes. Each example describes a set of specific circumstances to which you must apply the principles outlined in the relevant chapter in order to determine the correct strategy for the situation. In some examples, there are open questions, but this is exactly what happens in real life. There are times when there is no unambiguously correct solution, but only flawed ways to overcome the problem. Carefully reviewing the case study before you read the next chapter will help you understand the ideas in this chapter much better than just reading the main text. In order for you to apply what you have learned in practice, the final chapter contains a series of case studies arranged in order of increasing difficulty.

We hope that by the end of this book you will be more effective managers, negotiators, athletes, politicians, or parents. Be warned: not all strategies that will help you achieve your goals will appeal to your opponents. If you want to play fair, tell them about our book.

Part I

Chapter 1 Ten Stories of Strategy

LET’S BEGIN THE BOOK with ten stories about strategy taken from different areas of life. They provide a first idea of ​​what the best course of action is. Many of you have no doubt encountered similar situations in your daily life and found the right solution either through reflection or through trial and error. For some, some of the proposed solutions may come as a surprise, but these examples are not presented here to surprise you. Our goal is to show that such situations are common, that they all boil down to a certain set of interrelated questions, and that a systematic analysis of these situations can bear fruit.

In the following chapters, we offer guidelines for creating effective strategies based on these concepts. Treat these stories like an appetizer before the main course: they are only meant to whet your appetite, not to fill you up.

Story 1. Number Guessing Game

Believe it or not, we invite you to play the same game with us. We chose a number between 1 and 100; your task is to guess this number. If you name it correctly on the first try, we will pay you $100.

Of course, we are not going to pay you $100 at all: it would be too expensive, especially since we are going to help you guess the number. But when you play, we want you to think that we will actually pay you this money; and we will play with you on the same terms.

The probability of guessing the number on the first try is quite low: one in a hundred. In order to increase your chances of winning, we will give you five attempts, and after each attempt we will say which number you called — more or less. Of course, the faster you name the correct number, the greater the reward. If you guess the number on the second try, you will receive $80. On the third attempt, your reward is reduced to $60, on the fourth it is reduced to $40, and on the fifth it will be $20. If you need more than five attempts, it means that the game is over and you will not get anything.

Ready to play? We are ready too. Most likely, you are asking yourself the question: how can you play with a book? It’s really difficult, but still possible. If you wish, you can go to the site artofstrategy.info and play interactively. And here we can assume how you will play the game and make the appropriate response moves.

Your first guess is 50? This is the most common guess and, unfortunately for you, too large a number.

Perhaps your second attempt is 25? Having given the number 50 as the first number, most people choose 25 as the second. It’s a pity, but this number is too small. In the next step, most people say the number 37. Unfortunately, 37 is also too low. How about 42? Again, too little.

Let’s pause, look at the situation from the outside and analyze it. You have the fifth attempt left — the last chance to win money from us. You know that the desired number is greater than 42 and less than 50. You have seven choices: 43, 44, 45, 46, 47, 48, and 49. Which of these numbers will you choose?

Up until now, you’ve been trying to guess a number by choosing the average from the remaining interval. This is an ideal strategy for a game in which a number has been chosen at random — a special term is used to designate such a search strategy: «entropy minimization». You get the most information possible from every guess you make, so you can get closer to the number you’re looking for in the shortest amount of time. Microsoft CEO Stephen Ballmer is said to have used the game as a test during job interviews. In Ballmer’s understanding, the correct answer should be: 50, 25, 37, 42, …. He was primarily interested in whether the candidate for the vacant position was able to solve this problem in the most logical and efficient way.

We offer a different solution. In Ballmer’s problem, the number was chosen arbitrarily, so the engineer’s strategy of «divide the population by two and win» was quite appropriate. Getting the maximum information from each guess minimizes the expected number of guesses, and therefore allows you to win the most money. However, in our case, the number was not chosen randomly. Remember how we told you from the start that we were going to play like we were really going to pay you money? No one will reimburse us for the amounts that you hypothetically have to pay, which means that it is better to save them than to give them to you. Therefore, we have deliberately chosen a number that will be difficult for you to calculate. Think for yourself: would it be reasonable for us to guess the number 50? It would cost us a fortune!

The main lesson of game theory is that you need to put yourself in the place of another player. We put ourselves in your shoes and assumed that you would call the number 50 first, then 25, then 37 and 42. Understanding how you would play significantly reduced the likelihood that you would guess our number, and thereby reduce the amount of money which we would have to pay.

By explaining this before the end of the game, we gave you a head start. Now you understand what kind of game you are actually playing. What will be your final guess for which you can get $20? What number do you choose?

Is it number 49?

Congratulations! Yourself, not you. You are trapped again! We thought of the number 48. In fact, all this reasoning about choosing a number that is difficult to find by choosing the average number from the interval was aimed precisely at misleading you. We wanted you to choose the number 49, thereby securing our number 48. Remember: our job is not to give you our money.

In order to beat us in this game, you had to be at least one step ahead of us. You should have been thinking like this: «They want us to choose 49, so I’ll choose 48.» Of course, if we were to assume that you are that smart, we would choose the number 47 or even 49.

The point of our game with you is not to show you how cunning we are, but to clearly illustrate what makes any situation a game: you must take into account the goals and strategies of other players. When you guess a number chosen at random, no one tries to hide this number from you. Therefore, you can apply an engineering approach by choosing the average value from the interval and thereby get the best result. But if you are playing a real game, you need to analyze how the other player will act and how their decisions will affect your strategy.

Story 2. Victory at the cost of defeat

We must admit to readers that we watched the reality show Survivor (“Survivor”; the Russian equivalent is “The Last Hero”. Approx. Per.). There’s no way we could have won on that island. If we hadn’t given up because of the pangs of hunger, the other participants in the game would probably have got rid of us for being «smarts». However, we were very interested in trying to predict how the game would end. It came as no surprise to us that short, stocky nudist Richard Hatch outwitted, beat and outlasted everyone else by becoming the first winner of this CBS reality show and winning a $1 million prize. He had a special gift: the ability to act strategically without looking like a strategist.

The most cunning tactical move Hatch made in the last episode. Only three players remained in the game. Richard’s rivals were 72-year-old former Navy SEAL Rudy Bosch and 23-year-old river guide Kelly Wigglesworth. The last test was to stand on a support and hold on to the idol of immunity in the center of the circle. The participant of the game, who stood the longest, went to the final. It is also important that the winner decides which of the two losers will go with him to the final.

At first glance, it might seem that physical endurance played a major role in this test. But let’s analyze the situation more carefully. All three players understood that Rudy was the most likely winner. The only thing Richard could hope for was to reach the final with Kelly.

There were only two possibilities to achieve this. First, Kelly will win this challenge and choose Richard. The second is that Richard will win and choose Kelly. Richard could count on Kelly to choose him. She realized that Rudy was very popular with the participants in the game and that her only chance of winning was to reach the final with Richard.

It seemed that the situation was developing in such a way that whoever won in the last test, Kelly or Richard, each of them would choose the other as a rival. Consequently, Richard could try to stay in the game — at least until Rudy is off the hook. The only problem was that Richard and Rudy had long formed an alliance. If Richard won the challenge and didn’t choose Rudy, it would set Rudy and all of his friends against Richard. And that could have cost Richard the win. One of the characteristic features of the Survivor show is that the winner is determined by the participants who have dropped out of the game. Therefore, each participant in the game must behave very carefully with his opponents.

From Richard’s point of view, the final test could follow one of three scenarios.

• Rudy wins. He chooses Richard, but Rudy has a better chance of winning.

• Kelly wins. She is smart enough to understand that her only chance of winning is to get rid of Rudy and fight Richard in the final.

• Richard wins. If he chooses Rudy, he will defeat him in the final. If he chooses Kelly, she can defeat him as Richard will lose the support of Rudy and his many friends.

After comparing all possible scenarios, Richard came to the conclusion that it is best for him to lose in this test. He needs Rudy out of the game, but it’s best if Kelly does all the dirty work for him. The smartest move would be to bet on Kelly to win this challenge. She has already won three of the previous four; in addition, since Kelly, as a guide, led excursions in nature, she was in the best physical shape.

In this book, we often make digressions, which we call «problems for training thinking.» In such digressions, more complex elements of the game are considered, which we have passed over in silence in the main text. For example, in the example above, Richard could have waited a bit to see who would be out first. If Kelly had been off the hook first, it would have been more profitable for Richard to beat Rudy and pick Kelly than to let Rudy win and fight him in the final. Besides, Richard might have thought that Kelly was resourceful enough to calculate the same options and also be the first to exit the game. In the following chapters, you will learn how to use a systematic approach to find ways to win the game. Our ultimate goal is to help you change the way you think about strategic situations and realize that you won’t always have time to analyze every possible scenario.

This version of the development of events gave Richard a pleasant bonus: there was no need to stand on a support under the hot sun. At the very beginning of the challenge, show host Jeff Probst offered an orange slice to anyone who decides to stop further fighting. Richard got off the pole and got an orange.

After 4 hours and 11 minutes, Rudy made an unsuccessful attempt to change position, broke away from the idol of immunity and lost the challenge. Kelly chose Richard to compete in the final. Rudy decided the outcome of the vote by casting his vote in favor of Richard, and Richard Hatch was the first winner of the reality show Survivor.

Looking at the situation in retrospect, this calculation seems simple enough. But Richard was able to foresee all possible scenarios before they happened (Richard would do well if he also thought about the consequences of not paying taxes on the $ 1 million he won. On May 16, 2006, he was sentenced to 51 months in prison conviction for tax evasion). In Chapter 2, you’ll find a number of tools that will help you predict the game and even give you the chance to try your hand at the next season of reality TV.

Story 3. Lucky hand

Do athletes really have a “happy hand”? Sometimes it seems that Yao Ming (Yao Ming is a Chinese basketball player. Note ed.) is simply unable not to hit the ball in the basket or that Sachin Tendulkar (Sachin Tendulkar is an Indian cricketer. Note ed.) cannot but win a hundred in cricket. Sports commentators who observe such long periods of uninterrupted success for some athletes claim that they have a «lucky hand.» However, psychology professors Thomas Gilovich, Robert Vallone, and Amos Tversky believe that this conclusion does not correspond to the real state of affairs. 17.). They argue that if you toss a coin long enough, sooner or later it will come up heads or tails many times in a row. According to these psychologists, commentators, who sometimes have nothing to talk about, simply choose periods of successful play from a long playing season. They come in exactly the same way as after a whole series of tossing a coin falls on one side several times in a row. These psychologists offer a more accurate, evidence-based test using the basketball game as an example. They count all the times a certain player hits the basket, and then calculate the percentage of those episodes when the next throw is also successful. The same calculations are made for those cases when a hit is followed by a miss. You can talk about a «lucky hand» only when, after hitting the basket, a hit follows more often than a miss.

Psychologists conducted this test among the players of the Philadelphia 76ers basketball team. The results obtained disproved the «happy hand» theory. When a player made a successful throw, the next time he most often missed; when he missed, the next throw was more likely to be successful. This pattern was observed even in Andrew Tawney, who had a reputation as a player capable of making a series of successful throws. Does this mean that we should be talking here about a hand that operates on the principle of a stroboscope, as in a flashing beacon in which the light turns on and off?

Game theory offers another explanation. The statistics suggest that players lack the ability to make long streaks of hits, but they do not disprove the possibility that players with a «lucky hand» can actually «warm up» the game in one way or another. The difference between a series of successful rolls and a «lucky hand» arises due to the interaction between the offensive strategy and the defensive strategy. Let’s say Andrew Tawney really does have a «lucky hand.» Of course, in this case, the players of the opposing team will begin to push him away from the ball, which can reduce the percentage of the ball hitting the basket.

And that is not all. When opposing defenders focus on Tawney, one of his teammates is off guard and his chances of hitting the basket are increased. In other words, Tawney’s «happy hand» provides an increase in team performance, although Tawney’s own individual performance may decrease. Thus, in order to check whether the players of the team have “happy hands”, it is necessary to analyze the periods of successful play of the team as a whole.

A similar phenomenon is observed in many team sports. In American football, a brilliant running back helps his teammates make forward passes, and a talented wide receiver gets the ball forward as the opposition is forced to “patronize” star players. In the 1986 FIFA World Cup final, Argentina’s star Diego Maradona didn’t score a goal, but his passes through the West German defenders netted Argentina two goals. The value of a star player cannot be judged solely by his personal performance; his contribution to improving the performance of other members of the team plays a crucial role, and statistics on the number of assists allow this contribution to be assessed. In hockey, the individual performance of the players is determined equally by the number of both assists and goals scored.

The player can even help himself when one of his “lucky hands” supports the other. Cleveland Cavaliers star LeBron James eats and writes with his left hand, but prefers to shoot for the basket with his right hand (although he still shoots more accurately with his left hand). Defenders know that LeBron is right-handed, so they try to protect their basket from shots with their right hands. But they can’t focus on just that, as LeBron’s left-handed throws are too effective to leave them defenseless.

What happens if, between seasons, LeBron works on improving his left-handed shots? The opposing team’s defenders will react to this by paying more attention to covering his shots from the left. As a result, LeBron will have more opportunities to shoot with his right hand. In this example, the left hand not only knows what the right hand is doing, but also helps it.

Building on this theme in Chapter 5, we show that if the left hand is stronger, it can be used even less frequently than the right. Many of you must have learned this from your own experience while playing tennis. If your backhand is weaker than your righthand, your opponents will know and play against your left hand more often, but with this practice your backhand will improve. Once your left and right strikes are equally effective, opponents will no longer be able to take advantage of your left arm weakness. You will begin to use the right hand more often — this is the true advantage of practicing the left hand.

Story 4. To be or not to be a leader

After the first four heats in the finals of the America’s Cup (America’s Cup is a world-famous sailing regatta. The Cup is named after the schooner America, which won the prestigious sailing race in 1851. Approx. Per.), the yacht Liberty with skipper Dennis Conner led 3-1 in a series of heats to determine the best of the seven yachts. On the morning of the fifth swim, cases of champagne were delivered to the pier where the Liberty yacht was supposed to moor. On the spectator yacht, the Liberty crew’s wives, dressed in red, white and blue tank tops and shorts, were already looking forward to being photographed as their husbands once again won the 132-year-old United States Cup. America2. But this was not destined to happen.

At the start, Liberty gained 37 seconds of lead when Australia II made a false start and therefore returned to the starting line. Australian yacht skipper John Bertrand attempted to make up for lost time by veering off course to port in the hope of a lucky wind change. Dennis Conner decided to leave the Liberty on the right of course. But Bertrand’s risky step paid off. The direction of the wind shifted five degrees in favor of Australia II; in the end, this yacht won the race, beating the Liberty by 1 minute 47 seconds. Conner was criticized for not following the same course that Australia II had taken, this proved to be a major strategic mistake. After two more heats, Australia II won the entire series.

The sailing regatta provides an opportunity to analyze an interesting reverse version of the follow-the-leader strategy. As a rule, the leading sailboat copies the strategy of the ship following it. When a lagging sailboat changes course, the leader does the same. The leader copies the actions of the laggard, even if his strategy is clearly ineffective. Why? Because in sailing — like in ballroom dancing — only victory matters. If you are already in first place, the surest way to stay first is to imitate the actions of those who are following you. the ship shifts course to the right and the other to the left, the leader will have to decide which actions to repeat (and whether to do it at all)​).

Equity analysts and economic forecasters also often follow a strategy of imitation. Leading forecasters have an interest in following the majority and making predictions that do not contradict those of other analysts: this reduces the likelihood that people will change their minds about the abilities of these specialists. On the other hand, beginners often use risky strategies; they predict either a sharp rise or a precipitous decline in the economy. In most cases, their conclusions turn out to be erroneous, and everyone forgets about them. However, from time to time some of them make correct predictions and become famous because of it.

Competition in industry and technology offers new evidence for this view. In the personal computer market, Dell is known less for innovation than for its ability to bring unified technologies to the mass market. At one time, most of the new ideas were born in Apple, Sun and other start-up companies. Risky innovation is their main and perhaps only chance to gain market share. This situation is developing not only in the market of high-tech products. Procter & Gamble, the «Dell» of the diaper market, began using Kimberly-Clark’s invention of Velcro on their diapers and thus regained market dominance.

There are two ways to move second. You either start copying the leader’s actions as soon as he reveals his approach (as in a sailing regatta), or you wait until it becomes clear that success or failure will bring this approach (as in the computer field). In business, waiting is the better way to go because, unlike in sports, competition in business is not a winner-take-all competition. Therefore, market leaders do not follow companies that have unexpectedly achieved success until they are convinced of the advantages of their course.

Story 5

When the Catholic Church demanded that Martin Luther renounce his speeches against the power of popes and councils, he did not renounce his convictions: “I cannot and do not want to renounce anything, for it is wrong and unrighteous to do anything against conscience.” He did not compromise: “Here I stand, and I cannot do otherwise.” New York: Abingdon-Cokesbury, 18). Luther’s intransigence was based on the conviction that his position was sent down to him by God. Where there was talk of righteousness, there was no room for compromise. Luther’s resilience had long-term consequences: his opposition to the papacy marked the beginning of the Protestant Reformation and significant changes in the medieval Catholic Church.

Intransigence helped Charles de Gaulle to become a very influential player in the arena of international relations. His biographer Don Cook said: «De Gaulle managed to achieve power with nothing but unshakable moral principles, intellect, personal qualities and a sense of his own destiny.» However, the main feature of Charles de Gaulle was his intransigence. During World War II, as the self-proclaimed leader of a defeated and occupied country, he himself organized negotiations with Roosevelt and Churchill. In the 60s, his presidential «No!» several times prevented France from joining the European Economic Community (EEC).

How did Charles de Gaulle’s inflexibility give him a strong negotiating position? When he took a truly unshakable position, the other participants in the negotiations had only two options — to accept it or reject it. For example, it was de Gaulle who opposed England’s entry into the European Economic Community — in 1963, and then in 1968. Other countries were forced to either accept his veto or destroy the EEC. De Gaulle carefully weighed his position in order to achieve its acceptance. However, this often led to an unfair distribution of benefits in favor of France. De Gaulle’s intransigence made it impossible for the other side to put forward a counterproposal acceptable to it.

In practice, this is easier said than done for two reasons. The first is due to the fact that in most cases during negotiations, conditions are discussed that are not related to momentary benefits. Feeling that you are taking a too self-interested position may lead to the reluctance of other negotiators to do business with you in the future. Or they will take a tougher position in the next negotiations, trying to avoid possible losses. On a personal level, an unfair victory can ruin business or even personal relationships with other people. In fact, Charles de Gaulle’s biographer David Schonbrun accused him of chauvinism: “In relations between people, he who loves no one himself is rarely loved; he who does not know how to make friends has no friends. That is why de Gaulle’s rejection of friendship harmed France.» What seems like a compromise now may turn out to be a more effective strategy in the future.

The second problem is to acquire the necessary level of steadfastness. Martin Luther and Charles de Gaulle have developed toughness through personal qualities, but it comes at a cost. An unbending character cannot be turned on and off according to the situation. Intransigence weakens the opponent and forces him to make concessions, but with the same success leads to the fact that small losses grow into serious problems.

Ferdinand de Lesseps was a mediocre engineer, but possessed incredible foresight and determination. He became famous thanks to the construction of the Suez Canal in conditions in which it seemed impossible. Lesseps did not recognize the impossible and for this reason was able to achieve his goal. Subsequently, he tried to apply the same method in the construction of the Panama Canal, but it ended in disaster. The sands of the Nile obeyed his will, tropical malaria did not. The problem of Lesseps took place in the fact that his tough character did not allow him to admit defeat even when the battle was lost.

The bed of the Suez Canal is located at sea level. Laying this canal was comparatively easy, as it ran through low country in the desert. The bed of the Panama Canal was in much more difficult terrain, with highlands, lakes, and dense jungle. Lesseps’ attempt to dig a channel at sea level ended in failure. Subsequently, the US Army Corps of Engineers coped with this task by applying a different method — a system of locks, which included lakes located along the route of the canal.

How to achieve selective inflexibility? There is no perfect solution here, but there are ways in which you can set a course and stick to it (this is the topic of Chapter 7).

Story 6. Strategic weight loss

Cindy Naxon-Schechter wanted to lose weight. She knew exactly what it took: eat less and move more. She knew all about the food pyramid and the hidden calories in drinks. However, nothing helped her lose weight. After the birth of her second child, she gained almost 20 kilograms, and she could not get rid of them.

That’s why Cindy accepted ABC’s offer to help her lose weight. On December 9, 2005, she came to a photography studio located on the West Side of Manhattan. There she was asked to wear a bikini. Cindy hadn’t worn a bikini since she was nine, and now wasn’t exactly the right time to do it again.

The setting in the photo studio was reminiscent of what unfolds backstage during the pre-production of Sports Illustrated magazine with photos of supermodels in bathing suits. There were lights and cameras everywhere, and Cindy wore only a lime-green bikini. The producers prudently installed a hidden heater in the room so that Cindy would not freeze. Click. Smile. Click. Smile. What the hell was she thinking? And click again.

If everything goes as Cindy hopes, no one will see these pictures of her. She made a deal with ABC Primetime that if Cindy lost seven kilos in two months, those photos would be destroyed. But at the same time, no one will help her. No trainer, no instructor, no special diets. She already knew what to do. All that was needed was additional motivation and a reason to start today, not tomorrow.

Now Cindy has additional motivation. If she does not lose the promised weight, the ABC channel will show her photos and videos in prime time. Cindy has already signed a permit for this, waiving any claims against the channel.

Losing seven kilograms in two months is an achievable task, but not so easy. A series of holiday parties and Christmas dinners loomed ahead. Cindy couldn’t risk putting off her task until the New Year. She should have started right away.

Cindy was well aware of all the dangers of being overweight: an increased risk of diabetes, acute cardiovascular disease, and even death. But it still wasn’t enough to scare her into action. What Cindy feared the most was that her ex-boyfriend would see her in a bikini on national TV. It was hardly worth counting on the fact that he would not watch this show. If Cindy herself doesn’t tell him, her best friend will.

Lori Edwards was not satisfied with either her appearance or her well-being. It didn’t help that she worked as a bartender surrounded by hot young guys in their twenties. Neither the Weight Watchers diet (“Guardians of Weight”), nor South Beach (“South Coast”), nor Slim-Fast (“We lose weight quickly”) helped — it’s impossible to list everything. Laurie was heading in the wrong direction, so she needed help to change her course. When Laurie told her friends about the show, they thought it was the dumbest thing she’d ever done. The photographs captured a look on her face, as if she was wondering, “What am I doing?”

Ray also wanted to lose weight. He had recently married and was only in his twenties, but he looked to be in his forties. When he walked onto the set in his bathing suit, the photos weren’t very attractive. Click. Smile. Click.

Ray himself would not have taken such a risk. However, his wife wanted him to lose weight and was willing to help him. She offered to diet with him. And then she took an even bolder step. Ray’s wife also wore a bikini. She wasn’t as plump as Ray, but she wouldn’t have dared to wear a bikini either.

Ray’s wife and Cindy found themselves in different situations. Ray’s wife didn’t make a deal like Cindy did. She didn’t have to weigh herself. She didn’t even need to lose weight. And her photos would have been shown only if Ray had not managed to lose weight.

For Ray, this meant that his stakes were raised even higher. He could either lose weight or lose his wife.

All these people, two women and one married couple, bared their souls in front of the cameras, and not only. Why did they go for it? They weren’t the type to show off. The producers of ABC carefully checked them all. None of the five contestants on the show wanted these photos to appear on TV; they were all sure that this would not happen.

These people were playing against their future selves. The present self wants the future self to diet and exercise. The future self wants ice cream and TV. In most cases, the future self takes over, because it is the future self that has to do the real thing. The whole secret is to change the incentives that drive the future self, then its behavior will change.

In Greek mythology, Odysseus wanted to hear the Sirens sing. He knew that if he allowed his future self to listen to this song, his future self would steer the ship towards the rocks. So he tied his hands—literally. Odysseus ordered his men to tie his hands to the mast and cover his ears with wax. When dieting, this approach is known as the empty fridge strategy.

Cindy, Laurie and Ray went even further. They put themselves in a position from which only strict adherence to a diet could help them get out. Perhaps you think it’s always good to have more options. But if you think strategically, reducing the number of possible events can bring more benefits. Thomas Schelling writes in his book about how the Athenian commander Xenophon fought with his back to a deep gorge. He deliberately chose such a position so that his soldiers would not have the opportunity to retreat. In that battle, their backs were stiff, but they won.

Similarly, Cortés sank his ships after arriving in Mexico. This decision was supported by his entire detachment. With a huge numerical advantage of the enemy, 600 soldiers of Cortes decided that they would either defeat the Aztecs or die. The Aztecs could retreat deep into their territory, and there was no way for the soldiers of Cortes to escape or retreat. By making the defeat even more disastrous than it would have been on its own, Cortes increased his chances of victory — and won (Cortes was also helped by the fact that the Aztecs mistook him for their god Quetzalcoatl, «the feathered serpent»).

What helped Cortes and Xenophon helped Cindy, Laurie and Ray. Two months later, just in time for Valentine’s Day, Cindy lost almost eight kilograms. Ray dropped ten kilograms and reduced his waist by two holes in his belt. The threat of publicizing the photos was the motive that made them start, but when they really started working on themselves, they did it only for themselves. Lori dropped the required seven kilograms in the first month. But she continued to work and over the next month she lost another six kilograms. By shedding 14 kilos (which was XNUMX percent of her weight), Laurie was able to wear clothes two sizes smaller. Her friends no longer think being on an ABC show is a stupid idea.

By now, it should not surprise you that one of the authors of this book was involved in the creation of this show (The show Life: The Game («Life is a game») aired on March 16, 2006. A continuation of this show, in which the threat was countered with positive reinforcement, aired on December 20, 2006). Perhaps we should have called this book «Strategic Weight Loss» — it would have sold very well. Alas, we called the book differently. But we will return to this type of strategic action in Chapter 6.

Story 7. Buffett’s dilemma

In an article on campaign finance reform, the «Oracle of Omaha» Warren Buffett proposed limiting private contributions between $1000 and $5000 and banning all other contributions. No contributions from corporations, unions; no soft money. Sounds great, except that it’s not feasible.

Campaign finance reform is facing such resistance because the legislators who must approve it stand to lose the most if it is passed. It is the benefits of fundraising that keep them busy.

From 1992 to 2000, Dan Rostenkowski became the only sitting congressman to lose his re-election bid. The chances of success for an incumbent member of Congress at the time were 604 out of 605, or 99,8 percent. When Rostenkowski lost, he was charged with 17 counts of bribery, obstruction of justice and embezzlement of public funds.

How can you force people to do something that is contrary to their interests? This means putting them in a situation known as a prisoner’s dilemma («Prisoner’s dilemma» is a more common version of this term, but we use the plural («prisoner’s dilemma»), because if two or three prisoners are not involved in the situation, then there is no dilemmas). Here’s what Warren Buffett has to say about it:

Imagine that some eccentric billionaire (but not me!) makes the following proposal: if the bill is defeated, this eccentric billionaire will donate a billion dollars (soft money makes everything possible) in any way possible to the political party that will cast the most votes. for passing the bill. With this diabolical application of game theory, the bill will get through Congress without spending a dime (suggesting that he isn’t all that eccentric).

Imagine that you are a Democratic legislator and consider your options. If you think the Republicans support the bill and you yourself oppose it, then if successful, you will provide the Republicans with $1 billion, thereby putting in their hands the resources that will keep them dominant for the next ten years. Therefore, it makes no sense for you to oppose the bill if it is supported by the Republicans. On the other hand, if Republicans oppose this bill and you support it, you stand a chance of making $1 billion.

Thus, whatever the Republicans do, the Democrats must support the bill. Of course, the same logic applies to Republicans. They should support the bill regardless of the position of the Democrats. As a result, both parties will support the bill, and our billionaire will get what he wants for nothing. Buffett notes that the mere fact that such a plan is effective «would confirm the absurdity of claims that money does not affect congressional votes.»

This situation is referred to as a prisoner’s dilemma, as both sides are forced to take actions that are contrary to their mutual interests (Active participants in such a game lose, but a third party can benefit from it. Campaign finance reform would displease incumbent politicians, but all of us She would only benefit). In the classic Prisoner’s Dilemma, the police interrogate two suspects separately. Each of them is offered to confess and is promised a much harsher sentence if he does not have time to confess first. Thus, each of the prisoners considers it more profitable for himself to confess to the crime, although it would be better for both if they were silent.

Truman Capote gives a good example of the prisoner’s dilemma in his book In Cold Blood. Richard «Dick» Hickok and Perry Edward Smith are arrested for the wanton murder of the Clutter family. There were no witnesses, but the prison informant gave the police their names. During the interrogation, the police turned them against each other. Capote allows us to get into Perry’s mind:

…This, like the fictional ‘witness’, was just said to make him twitch. This cannot be. Or they mean… Oh, if only he could talk to Dick! But they were kept apart; Dick was in a cell on another floor. … And Dick? Maybe they did the same trick with him. Dick was a smart, persuasive actor, but there was no «core» in him, he panicked too easily. … «And before you left this house, you killed everyone who was there.» It may very well be that they sing the same song to each of the ex-convicts in Kansas. They must have interrogated hundreds of people and accused many of them of this murder; Dick and I are just two more of the same…

And Dick, too, was awake in his cell below, and, as he later recalled, just as eager to talk to Perry—to find out what the freak had told them.

As a result, first Dick confessed, and then Perry (Although they both hoped that the confession would commute their sentence, in this case this did not happen: both criminals were sentenced to death). That is the nature of this game.

The problem of collective action is one of the variants of the prisoner’s dilemma, although in this case there are not two «prisoners», but many more. In a children’s story, the mice decided that their lives would be much safer if the cat had a bell around its neck. There is only one problem: who would risk their life to hang a bell around a cat’s neck?

Both mice and people face this problem. How do dictators who do not enjoy the love of the people manage to control huge masses of people for a long time? Why can one bully terrorize the whole school? In both cases, concerted action by the masses has a good chance of success.

However, the process of communication and coordination required to carry out such actions is fraught with great difficulties, and dictators, knowing the strength of the masses, take special measures to make this process even more difficult. When people are forced to act apart and only hope that someday the right moment will come, the question arises: “Who will be the first?” Such a leader will pay a heavy price — a broken nose or even life. And his reward will be posthumous glory or the gratitude of his descendants. There are people in this world who are driven by duty or honor, but most believe that such a price exceeds the possible benefits.

Khrushchev first spoke out condemning the Stalinist purges at the XNUMXth Congress of the Communist Party of the Soviet Union. After his dramatic speech, someone from those present asked what he himself was doing at that time. Khrushchev asked the questioner to stand up and give his name. The hall was silent. Khrushchev said: «That’s what I did.»

Each person acts in their own interests, which leads to adverse consequences for the entire group. Apparently, the prisoner’s dilemma is the most difficult one in game theory. We will return to this topic in Chapter 3 to discuss what can be done in this case. But we must warn you from the very beginning: we do not claim that the outcome of the game will suit the players. Many economists (including ourselves) tirelessly talk about the benefits of a free market. This conclusion is based on the assumption that the system of free pricing determines the behavior of the individual. In most cases of strategic interaction, there is no invisible hand of the market to guide the actions of the baker, butcher, or any other person. Consequently, there is no reason to expect that the outcome of the game will be favorable for its individual participants or for society as a whole. It’s not enough to play well — you have to be sure you’re playing the right game.

Story 8. Mixing moves

Apparently, Takashi Hashiyama has trouble making decisions. The two largest auction houses, Sotheby’s and Christie’s, have made very attractive offers to auction his company’s $18 million art collection. Instead of picking one auction house out of the two, Hashiyama suggested that both of them play a game of rock, paper, scissors to determine the winner. Yes, yes, rock, paper, scissors. The rock breaks the scissors, the scissors cut the paper, and the paper wraps around the rock.

Christie’s chose scissors, Sotheby’s chose paper. Scissors cut paper, so Christie’s got the job and earned a $3 million commission. Given the stakes were so high, could game theory help in this situation?

At first glance, it may seem that in such a game it is impossible to predict the actions of its participants. If Sotheby’s had known that scissors would be chosen at Christie’s, then they would have chosen stone. It doesn’t matter what you choose: there will always be something that will be stronger. Therefore, it is very important that your opponent cannot predict your move.

In preparation for the game, Christie’s turned to local experts for advice, namely the children of their employees, who regularly play the game. Eleven-year-old Alice said: «Everyone knows to always start with scissors.» Flora, Alice’s twin sister, added her opinion: “Rock is too obvious, but scissors beat paper. Since they are beginners, it is best for them to start with scissors.”

Sotheby’s took a different approach. They believed that this was a matter of chance, so there is no place for strategy. The choice of paper was no worse than the other options.

What is interesting in this example is that each side is right in its own way. If Sotheby’s had chosen its strategy arbitrarily (assuming stone, scissors and paper have an equal chance), then whatever Christie’s preferred would have been equally good. Each variation has a one-third chance of winning, one-third of losing, and one-third of drawing.

But Christie’s did not choose a strategy randomly. Therefore, Sotheby’s would do well to think about what advice Christie’s might get and do whatever it takes to beat them. If everyone really knows to start with scissors, then Sotheby’s better start with Bart Simpson’s favorite choice — good old rock.

In a way, both players in this game are half wrong. Since Sotheby’s had no strategy, Christie’s efforts were meaningless. But it is with these Christie’s efforts in mind that Sotheby’s should be thinking strategically.

When it comes to a single session of the game, making a random choice is not so difficult. But if the game is repeated, a more skillful approach must be found. Mixing moves is not limited to predictable change of strategies. Your opponent, having observed you, can just as easily exploit any pattern in your game as it can repeat the same strategy over and over again. Unpredictability is the most important thing in mixing moves.

As it turns out, most people behave predictably. You can test it out for yourself by playing the game of rock paper scissors online: computer programs are able to recognize patterns and that’s why they beat you. In an attempt to confuse everything, players too often use the same strategies in turn. This leads to the unexpected success of the «avalanche» strategy: rock, rock, rock.

Also, people tend to make decisions based on what the other side did last time. If both Sotheby’s and Christie’s started the game with scissors, there would be a draw and the game would have to start over. Given what Flora had said, Sotheby’s would have expected Christie’s to rock (to beat their scissors). So Sotheby’s would have chosen paper, while Christie’s would have preferred scissors. Of course, such a template approach cannot be correct either. If he had been correct, Sotheby’s could have played with a stone and won.

Imagine what would happen if there were some well-known formula for selecting objects for tax audits. Before filing your tax return, you could use this formula to determine if the IRS will audit you. If verification is possible, but you see a way to «adjust» your income statement until the formula no longer predicts verification, chances are you would. If verification is unavoidable, you would prefer to tell the truth. The predictability of the actions of the tax service would end up with the fact that the tax audit would be carried out in the wrong place. Everyone who comes under scrutiny would know what awaits them and would rather act honestly. On the other hand, those people who managed to escape the test would answer only to their own conscience. If the tax office randomly selects items to be checked, everyone is at risk of being subjected to such a check, and this creates an additional incentive for honesty.

The importance of randomized strategy proved to be one of the first discoveries of game theory. This is a fairly simple and intuitive idea, but it needs clarification to be successfully used in practice. It is not enough for a tennis player to know that he will have to beat off the opponent’s serves from the left and right. He should have some idea of ​​how often he will have to return a right hand — 30 or 64 percent of the time; he also needs to understand that the answer to this question depends on the relative strength of the blows on both sides. In Chapter 5, we will discuss methods for finding the answer to this question.

We would like to conclude this section with the following remark. In the game of rock, paper, scissors, it was not Sotheby’s that lost the most, but Mr. Hashiyama. His decision to use the game gave the two auction houses an equal opportunity to earn commissions. Rather than let the two rivals agree on a commission split, he could have his own auction. Both companies were willing, even very interested, to take over the sale of his art collection for a 12 percent commission. The auction house that offered the lower bid would be the winner. Am I hearing 11 percent? One…two…

The standard fee is 20 percent of the first $800 and 000 percent of the remainder. Four paintings by Takashi Hashiyama sold for $12 million; total commissions were $17,8 million.

Story 9

In the movie Guys and Dolls, gambler Skye Masterson talks about the valuable advice he once received from his father:

One day you will meet a guy on your way who will show you a brand new, unopened deck of cards. Then he will offer you a bet that he will draw the jack of spades from the deck, and if he wins, he will blow a stream of cider in your ear. But son, don’t take that bet on any account or you’ll end up with cider in your ear — and that’s as true as standing here in front of me.

In one of the episodes of the film, Nathan Detroit offered Sky Masterson such a bet: what Mindy sells more — strudel or cottage cheese pies. Nathan knew the correct answer (strudels) and was willing to bet Sky on the cheesecake.

It should be noted that Sky never learned his father’s lesson. A minute later, he makes a bet that Nathan doesn’t know what color his butterfly is. Sky can’t win under any circumstances. If Nathan knows what color his butterfly is, he will bet and win. As it turned out, Nathan did not know the color and therefore did not accept the bet. Of course, that was the point of the game. Sky bet that Nathan would not accept his offer.

At first glance, this example may seem extreme. Of course, no one would make such a stupid bet. Or would you? Look at the market for futures contracts on the Chicago Board Options Exchange. When a market player offers to sell you a futures contract, he will only make money if you lose it.

Buying shares is different from dealing in futures contracts. When you buy shares, you provide the company with capital, which ensures its more rapid growth. Therefore, in this case, both parties can win — both you and the company.

If you are a farmer and grow soybeans for sale, then a futures contract will be your insurance against the risk of adverse price fluctuations in the future. Similarly, if you are selling soy milk and need to buy soy, such a contract is also insurance for you, not a gamble.

However, the volume of trading in futures contracts on the exchange suggests that most of the people who buy and sell them are traders, not farmers or manufacturers. For them, this deal is a zero-sum game. When both parties to a deal agree to make it, each of them believes that they will make money. Therefore, one of the parties to the transaction is mistaken. This is the nature of the zero-sum game: only one side wins.

Therein lies the paradox. How can both parties to the transaction believe that they will outsmart each other? One of them is definitely wrong. Why do you think that someone else is wrong, and not you? Let’s say you don’t have insider information. If someone is willing to sell you a futures contract, any money you make will be lost to the other side. Why do you think you are smarter? Don’t forget: the other party to a deal is only willing to make it with you because they think they’re smarter than you.

In poker, players face this paradox when it comes to raising the stakes. If a player only bets when they have good cards, the other players in the game will soon figure it out. In response to raising the stakes, they will give up the fight, so this player will not be able to hit a big pot. If there are players who will go for a raise, this means that they have even better cards, so our poor player is in for a big loss. Other players will only play against a strong card if they think you are bluffing. In order to convince them of this, you should bet often enough, while occasionally bluffing. The result is an interesting dilemma. You would like other players to fold when you bluff, this would give you the opportunity to win with bad cards. But then you can’t win by taking a big pot. To convince other players to raise, you need to be caught bluffing.

The more experienced the players, the harder it is to convince them to bet big against you. Take as an example an episode of a big-stakes game between Erik Lindgren and Daniel Negreanu, two of the top poker players.

…Negreanu, sensing a bad card, raised the bets by two thousand dollars. “I have already bet two hundred and seventy thousand, so I have two thousand left,” Negreanu said. But Eric looks at my chips and says, «How much do you have left?» — and goes all-in, putting everything he had on the line. According to the rules of the tournament, Negreanu had only 90 seconds to decide whether he should make the same bet. In doing so, he risked losing all his money if Lindgren bluffed, or he would have to fold and lose a fair amount of the money he had already deposited in the bank.

«I didn’t think he could be that stupid,» Negreanu said. But he wasn’t stupid either. He was just one step ahead. He knew that I knew that he wouldn’t do such a stupid thing, so by doing this so-called stupid thing, he actually did a great job.»

Of course, you should not bet against such poker champions — but when is it worth taking the risk? Groucho Mark (Julius Henry «Groucho» Marks — American actor, comedian. Approx. ed.) once said the famous phrase that he would never join a club that would agree to accept him. For the same reason, it may not be worth accepting bets that other people offer you. You should be concerned even if you won the auction: the fact that you bid the highest amount means that other bidders do not consider the item as valuable as you. There is a special term for someone who wins an auction and then finds out they overpaid: «winner’s curse».

Every action tells us something about what the person who did it knows, and you should use that information, along with what you know yourself, as a guide to action. In Chapter 10, we’ll cover how to avoid the winner’s curse by participating in auctions.

There are game rules that will put you on an equal footing with other players. One way to deal with one-sided information is to invite the less informed participant in the transaction to choose which position to take. If Nathan Detroit had agreed in advance to make a bet, whichever side Sky chose, even the information he had would not have helped him. In the stock and currency markets, as well as in other financial markets, people can take either side of a transaction. On some exchanges, including the London Stock Exchange, when you ask for a stock quote, the market maker is required to tell you both the buy and sell price before they know which side of the trade you are on. Without such preventive measures, market makers could profit from confidential information, and third-party investors would be so afraid of being cheated that it could lead to a collapse of the entire market. The purchase price and the sale price are not equal to each other; the difference between them is denoted by the term «spread between the ask price and the buy price». In liquid markets, such a spread is very small, which means that any order to buy or sell contains not much information. We will return to the role of information in Chapter 8.

Story 10. Game theory can be dangerous to health

Late one evening after a conference in Jerusalem, two American economists (including one of the authors of this book) took a taxi and asked the driver to take them to a hotel. The driver immediately realized that they were American tourists and refused to turn on the meter. Instead, he declared that he loved the Americans and promised to take less money from us than what would have been on the meter. Of course, we were somewhat skeptical about his promise. Why on earth would this stranger offer us a lower price than the meter if we are willing to pay what we are owed? How do we even know if the driver charged too high?

On the other hand, we did not promise the driver to pay more than the meter. We decided to use our knowledge of game theory. If we start haggling right away and our negotiations reach an impasse, we’ll have to look for another taxi. But if we postpone these negotiations until we arrive at the hotel, our position will be much stronger. And it was very difficult to find a taxi.

And here we are. The driver demanded 2500 Israeli shekels ($2,75). Who knows if this amount was fair? Since bargaining is common in Israel, Barry objected to this amount and offered 2200 shekels. The driver was furious. He said that it was impossible to get from there to here for such a sum. Before we could continue the conversation, he locked all the doors and headed back at breakneck speed, ignoring traffic lights and pedestrians. Have we been kidnapped and taken to Beirut? No. Our driver returned to the place where we got into his taxi, pretty gu.e. drove us out of the cabin and shouted after us: «Let’s see how far you can go now for 2200 shekels.»

We found another taxi and for exactly 2200 shekels we got to the hotel.

Of course, the time we spent was not worth three hundred shekels. On the other hand, the story itself cost money. It shows how dangerous it is to negotiate with someone who has not yet read our book. In short, pride and irrationality cannot be ignored. Sometimes it’s better to let yourself be fooled if it doesn’t cost you much.

There is another lesson to be learned from this story. In fact, we acted in that situation is not very far-sighted. Imagine how much stronger our bargaining position with the driver would be if we started negotiating the price after we got out of the car. (Of course, when you hire a taxi, you need to follow the reverse logic. If you tell the driver where you need to go before you even get in the car, this may end up with your taxi driver immediately looking for another client. Get in a taxi first , and then say where you need to go.)

A few years after this story was published, we received the following letter:

Dear professors!

Of course, you do not know my name, but I think you will remember my story. I studied in Jerusalem, and in the evenings worked as a taxi driver. Now I work as a consultant. I came across your book by chance when it was published in Hebrew. You may be wondering that I also tell this story to my clients. Yes, all this really happened late at night in Jerusalem. As for the rest, as far as I remember, everything was somewhat different.

Due to studying and working in the evenings, I had almost no time left to spend with my fiancée. Therefore, I found this way out: I decided to take her with me on trips around the city, putting me in the front seat. Although she was silent then, it was a big mistake on your part not to mention her in your story.

My meter broke, but you didn’t believe me. And I’m too tired to argue. When we arrived, I asked you for a very fair price — 2500 shekels. I even hoped that you would round this amount up to 3000 shekels. You rich Americans could afford to pay a 50 cent tip.

I couldn’t believe you were trying to fool me. Your refusal to pay a fair price has dishonored me in the eyes of my fiancee. Although I was poor, I did not need your handouts.

You Americans think we should enjoy every crumb you throw at us. But listen: we have to teach you a lesson in the game called life. My wife and I have been together for twenty years, and we still laugh at the stupid Americans who spent half an hour driving back and forth in a taxi just to save a mere twenty cents.

Yours sincerely,

(name not provided)

In truth, we have not received such a letter. We just wanted to illustrate one important lesson of game theory: you need to understand the position of the other player. You have to analyze what he knows, what motivates him and even what he thinks of you. George Bernard Shaw famously said about the golden rule that you should not treat others the way you would like them to treat you, because they may have very different tastes. Strategic thinking means that you must make every effort to understand the position and relationships between other players in the game, including the position of those players who prefer to remain silent.

We have come to the last important point. You may feel like you’re playing one game when it’s just part of a larger game. There is always a bigger game.

The most important features

The above examples introduced us to the principles of making strategic decisions. What follows is a summary of these principles in the form of lessons learned from our stories.

Think 4:8 when you’re trying to figure out what the other player is up to.

This is the principle of positive thinking that Tommy Newberry talks about in his book The 4:8 Principle: The Secret to a Joy-Filled Life and also in the book Think 4 :8:4 Days to a Joy-Filled Life for Teens To think according to the principle of 8:40 means to think as the holy apostle Paul teaches in his epistle to the Philippians: virtue and praise, consider these things.” Note. per.

Think about Richard Hatch’s ability to figure out all the next steps in order to understand what should be done. The story of the «happy hand» teaches that in strategy, as in physics, «for every action there is a reaction.» We do not live and operate in a vacuum. Therefore, we cannot expect that when we change our behavior, everything else will remain unchanged. Charles de Gaulle’s success in negotiations means that «a stuck wheel gets lube» (You may have heard something similar about a «squeaky wheel»; a stuck wheel needs more lubrication. Of course, sometimes such a wheel is replaced with a new one). However, being unyielding is not easy, especially if you have to be even more adamant than a stubborn opponent. This stubborn rival may well be your future self, especially when it comes to dieting. If you are fighting or dieting, cutting off your escape routes, this will help you strengthen your resolve.

The stories of the prisoner’s dilemma and the mice dreaming of having a bell hanging around the cat’s neck show how difficult it is to achieve a result when it requires coordination and self-sacrifice. In technological races, as much as in sailing, the laggards try to use more innovative strategies, and the leaders often imitate the actions of those following them.

The rock-paper-scissors example highlights the strategic advantage of being unpredictable. This course of action has the added benefit of making life a little more interesting. Our taxi driver makes it clear that the other participants in the game are also people, not machines. Pride, irritation and other emotions can influence their decisions. Putting yourself in the place of another person means seeing the situation from his point of view, and not from your point of view.

We could give many more examples and learn from them, but this is not the best way to methodically study strategy games. To do this, it is better to approach this topic in a different way. We choose a certain principle (for example, strategic commitment, cooperation, or mixing of strategies) and analyze examples illustrating this principle until you fully understand its essence. Then we will give you the opportunity to apply this principle in the case study, which is given at the end of the corresponding chapter.

Case Study: Choosing the Correct Answer from Multiple Choices

We believe that everything that happens in life is a game, even what at first glance does not seem like a game. Let’s take as an example the following question from the GMAT (Graduate Management Admissions Test) — a test for applicants to a master’s degree in management.

Unfortunately, copyright does not allow us to bring the question here, but this should not stop us. Which of the following answers is correct:

a) 4π sq. inches

b) 8π sq. inches

c) 16 sq. inches

d) 16π ​​sq. inches

e) 32π sq. inches

We understand that you are at a disadvantage because you do not know the question itself. And yet we believe that by applying game theory, you can find the right answer.

Example Analysis

In this series of answers, answer c falls out of the general pattern. Because it’s so different from the rest of the answers, chances are it’s the wrong answer. Given that all of the answers use the square inch as the unit of measurement, we can assume that the answer should be a full square, such as 4π or 16π.

That’s a good start and shows you have good test-taking skills, but we haven’t even started applying game theory yet. Think about the game that the person who is asking these questions is playing. What are its goals?

This person needs people who understand the problem to find the right answer, and people who do not understand the problem cannot solve it. Therefore, incorrect answers must be carefully selected in terms of their attractiveness to those who do not know the correct answer. For example, if the question «How many feet are in one mile?» among the possible options there are answers “giraffe” or even “16π”, it is unlikely that any of the test takers will choose them.

Now let’s analyze the situation from the other side. Let’s say 16 square inches is the correct answer. What question must be asked so that the correct answer is 16 square inches, but the person taking the test would think that the correct answer is 32π square inches? There are not many such questions. People don’t tend to use the number pi just for fun. “Have you seen my new car? She travels 10pi miles on one gallon of gasoline.» Therefore, we can safely exclude 16 square inches from the correct answers.

Now let’s analyze the perfect squares: 4π and 16π. Let’s say the correct answer is 16π square inches. Perhaps the question was put like this: what is the area of ​​a circle with a radius of 4 inches? The formula for calculating the area of ​​a circle looks like this: πr2. However, someone who does not remember the formula for the area of ​​a circle might confuse it with the formula for the circumference of a circle, 2πr. (Yes, we know that the circumference is measured in inches, not square inches, but a person capable of making such a mistake is unlikely to understand this.)

Note that if r = 4, then 2πr is equal to 8π — because of this, a person may choose the wrong answer b. In addition, this person may confuse things further and use the formula 2πr2, which will force him to choose 32π, or e, as the correct answer. This person can drop pi and choose 16 or c, or forget about squaring the radius and choose the πr formula for area, which will lead them to think of choosing 4pi or a. In general, if the correct answer is 16π, we can make a strong case for how the person taking the test might choose each of the remaining wrong answers.

But what if the correct answer is 4π (then r = 2)? Remember that people often make the mistake of confusing circle with area. If a student uses the wrong 2πr formula, they will get 4π, albeit with the wrong unit. From the point of view of the person writing the test, there is nothing worse than allowing the test taker to get the correct answer based on erroneous reasoning. Therefore, a 4π answer would be the worst correct answer, as it would allow too many incompetent people to choose the correct answer and pass the test.

At this point, you can stop further analysis. We are convinced that the correct answer is 16π. By thinking about the goals of the test writer, we can calculate the correct answer without even knowing the question itself.

However, we do not recommend that you take the tests without first reading the questions. We assume that if you are smart enough to understand all this logic, then you probably know the formula for the area of ​​a circle. But anything happens in life. You may well encounter a situation where among the answers to test questions there is something completely unfamiliar to you or the relevant material was not covered in the course that you studied. In such cases, analyzing the game called «take tests» will help you find the right answer.

Chapter 2

Your move, Charlie Brown

In the Peanuts comic (“Babes”), this episode is repeated: Lucy holds the ball on the ground and invites Charlie Brown to run up and hit him. At the very last moment, Lucy removes the ball, Charlie Brown swings his foot in the air, falls on his back, and Lucy gloats.

Anyone would tell Charlie not to play with Lucy. Even if Lucy hadn’t done this trick on him last year (and the year before, and the year before), Charlie already knew her personality, so he could have predicted her actions.

When Charlie decides whether or not to accept Lucy’s offer, her actions are in the future. But the fact that she will perform these actions in the future does not mean that Charlie should consider them unpredictable. He should have known that of the two options (let him hit the ball and watch it fall), Lucy would likely prefer the latter. Therefore, Charlie must assume that when the time comes, she will move the ball aside. Logically speaking, the likelihood that Lucy will let Charlie hit the ball is actually very small. To rely on such a possibility is to allow hope to triumph over experience, as Dr. Johnson said about remarriage. Charlie must dismiss this possibility as unimportant and foresee that by accepting Lucy’s offer, he will inevitably fall on his back. This means that he should reject the offer.

Two types of strategic interaction

The hallmark of any strategic game is the interdependence of the players’ decisions. Participants in such a game can follow two ways of interacting. The first way is through sequential interaction, as in the case of Charlie Brown. When it’s Charlie’s turn to make a decision, he must look into the future and analyze how his current actions will affect Lucy’s, as well as his own actions in the future.

The second way is parallel interaction, as in the Prisoner’s Dilemma story in Chapter 1. In this case, the players act simultaneously, without knowing anything about the current actions of the other players. Nevertheless, each participant in the game must be aware that there are other active players who also understand that they are not alone in this game, and so on. Therefore, each participant in the game needs to put himself in the place of all the other players and try to calculate the result of the game. His own optimal move is an integral element of such calculations.

When playing a strategy game, you must determine what type of interaction is present in it — parallel or sequential. Some games, such as soccer, have elements of both types of interaction. This means that you must tailor your strategy to the specific situation. This chapter has given a general description of the ideas and principles that will help you play sequential games; Parallel-move games are the subject of Chapter 3. We start with fairly simple examples, like the story of Charlie Brown. This is done with a purpose: such stories are of little value in themselves, and the right strategies can be easily found through ordinary intuition, which makes it possible to clearly identify the main ideas. The following chapters provide more realistic and complex case studies.

First rule of strategy

The general principle of all games with sequential moves is that each player must analyze the future responses of other players and use this information to find his optimal move in the present. This idea is so important that it needs to be formulated as a basic rule of strategic behavior:

RULE #1: Look ahead and think backwards.

Analyze what your initial decisions might lead to, and based on this information, calculate the best move.

In the story with Charlie Brown, it was easy enough for anyone (except for Charlie Brown himself) to do this. Of the two possible options, he chose the one on which — also of the two possible ones — Lucy also stopped. Most strategic situations involve a longer sequence of decisions, with multiple choices at each decision point. The process of making the right decisions in such games can be represented as a tree diagram. Let me explain to you how to use such «trees».

Decision tree and game tree

In the context of the first rule of the strategic game (look ahead and reason backwards), a sequence of decisions is possible even when decisions are made by one person who is not involved in a strategic game with other people. Let’s take as an example a fragment of a poem by Robert Frost (Fragment from Robert Frost’s poem «The Road Not Taken» (translated by V. Toropov — «The Unchosen Road») about the road in the autumn forest:

And if it becomes unbearable to live,

I will remember the old choice willy-nilly:

Fork in two roads — I chose the one

Where you bypass travelers a mile away.

Everything else does not matter.

Let’s represent this situation in the form of the following scheme:

But the choice doesn’t have to end there. Each road can have its own forks, which means that the road map will be more complex. Here is an example from our own life.

Travelers looking to get from Princeton to New York have a choice. At the first point of decision-making, they need to determine the mode of transportation: by bus, train or car. Those who prefer a car can choose the Verrazano-Narrows Bridge, the Holland Tunnel, the Lincoln Tunnel or the George Washington Bridge. Those who take the train must decide whether they should transfer to the PATH train (PATH train — a train that runs along the PATH (Port Authority Trans-Hudson) high-speed underground railway connecting Manhattan with the cities of Hoboken, Jersey City, Harrison and Newark. Note lane) or go further to Pennsylvania Station. Once in New York City, travelers who chose the train or bus will have to think about how to get to their destination next: walk, take the subway (using the usual local or high-speed line), bus or taxi. The optimal choice depends on many factors, such as price, speed, possible traffic jams, the final destination in New York, or even simply not wanting to breathe the air of a toll freeway in New Jersey.

The roadmap, which presents the options at each branching point of the path, resembles a tree with branches appearing one after the other, which is why such a diagram is called a tree. The correct way to use such a map, or tree, is not to take the route whose first branch seems to be the best for you — for example, because you prefer to travel by car rather than by train, all other things being equal, and then drive over the bridge Verrazano Narrows when you get there. You must analyze your future actions and use the information received in order to make the right choice at an earlier stage. For example, if you want to get to the business district of the city, it is better to travel by high-speed train, not by car, since the PATH provides a direct connection between Newark and downtown New York.

With the help of such a tree, one can show possible scenarios for the development of events in a strategy game, but in this case one more element will be added. At least two people participate in any game. Therefore, different players can make decisions at different branching points of the tree. A person who makes a choice at one or another branching point needs to analyze not only his own future decisions, but also the decisions of other players. He must anticipate what they will do, try to put himself in their place and think as they would think in the appropriate situation. To emphasize the difference, we will use two terms for trees: a «game tree» is a tree that represents the sequence of decisions in a strategy game; A decision tree is a sequence of decisions made by one person.

Charlie Brown in football and in business

The Charlie Brown story that opened this chapter is extremely simple, but it will help you get familiar with the concept of the game tree. The game begins at the moment when Lucy offers Charlie to hit the ball, and Charlie must decide whether he should accept this offer. If Charlie refuses, the game is over. If he accepts Lucy’s offer, she will have to choose between letting Charlie hit the ball or putting the ball away. We can show this by adding another fork in this road map.

As already mentioned, Charlie must be prepared for Lucy to choose the top branch. Therefore, he needs to remove the bottom branch from Lucy’s choices. If he himself chooses the upper branch, this will certainly lead to an unpleasant fall. Thus, for Charlie, the best way out is to choose the lower branch. In the figure, these choices are indicated by bold lines with arrows.

Do you think this is too simple an example? Here is his version in business. Imagine the following scenario. As an adult, Charlie spends his holidays in the recently reformed former Marxist country of Freedonia. Charlie talks to a local businessman named Fredo, who tells him what great profit opportunities he would create if he had enough capital. And then he proposes to Charlie: “Invest $100 in me, and in a year I will turn it into $000, which we will split equally. Therefore, in a year you will more than double your capital.” The opportunity that Fredo is talking about is indeed very tempting, and Charlie would be willing to sign a corresponding contract in accordance with the laws of Freedonia. But how reliable are these laws? If Fredo runs away with all that money at the end of the year, will Charlie, already in the United States, be able to sue him for breach of contract in the Freedonia court? This court can either make a biased decision in favor of a citizen of its country, or act too slowly, or it can be bribed by Fredo. All this means that Charlie has to play a game with Fredo, the tree of which is shown in the following figure. (Note that if Fredo fulfills the contract, he will pay Charlie $500; from this amount, the initial investment of $000 must be deducted. Therefore, Charlie’s profit will be $250).

What do you think Fredo will do? In the absence of clear and good reasons to believe his promises, Charlie must anticipate that Fredo will run away with all the money (just as little Charlie should have thought at one time that Lucy would put the ball aside). The trees of the two games are actually identical in many ways. However, how many times have such “charlies” failed to draw the right conclusions in such situations?

What reason could Charlie have for believing in Fredo’s promises? Perhaps it applies to many other businesses that need funding from the United States or that need to export their products to the US. In such a case, Charlie could strike back at Fredo by ruining his reputation in the United States or having his merchandise seized. Therefore, this game may be part of a larger game, perhaps even a long-term interaction that guarantees Fredo’s honesty. However, if the transaction is one-time (as in the example shown above), the logic of the reverse reasoning is obvious.

We would like to emphasize three important points on the example of this game. First, different games can be expressed in terms of identical or very similar mathematical forms (trees or tables like those in the following chapters). Using a formal mathematical representation allows you to draw parallels and transfer knowledge about the game from one situation to another. This is an important function of theory in any subject area: it makes it possible to identify essential common elements in seemingly different situations, as well as to adhere to a unified, and therefore simpler approach to their analysis. Many tend to subconscious rejection of the theory as such. But we think this is wrong. Of course, every theory has its limitations. Specific conditions or events can significantly supplement or change the recipes that the theory provides. However, if the theory is completely abandoned, you can lose a valuable starting point for reflection, and this will seriously complicate the solution of the problem. We must try to make game theory our assistant, and not an obstacle in the matter of strategic thinking.

Second moment. Fredo must understand that the strategically minded Charlie will take his offer with a grain of salt and will not invest money, thereby depriving him of the opportunity to earn $ 250. Therefore, Fredo has a strong incentive to make his promise credible. As a sole proprietor, Fredo has little to no influence over the country’s weak legal system, which means he can’t dispel such investor doubts. What other methods are available to him? We will analyze the issue of policy credibility and how to achieve it in Chapters 000 and 6.

The third and probably the most important point concerns the results that the participants in the game will come to, depending on which option they choose. It is not always the case that if one player gets more, the other will certainly get less. A situation in which Charlie decides to invest and Fredo to fulfill the contract is more beneficial for both of them than a situation in which Charlie does not invest. Unlike competitions or contests, there are no winners and losers in such games; in terms of game theory, these games need not be zero-sum games. They can end in win or loss for both sides. In fact, in most games in business, politics, and social interaction, there is both a community of interest (for example, when Charlie and Fredo can both make a profit if Fredo finds a way to convince his partner that he will fulfill the contract) and a conflict of interest (in particularly if Fredo decides to cash in on Charlie and run away with all the money after Charlie makes an investment). It is the combination of commonality and conflict of interest that makes the analysis of such games so interesting and relevant.

More complex trees

We decided to take an example of a more complex game tree from the realm of politics. One of the cartoons on American politics shows that the US Congress is not averse to allocating funds from the federal budget for local needs, and the presidents of the country are trying in every possible way to reduce the bloated budgets that Congress passes. Of course, presidents have their own preferences for such spending, so they would only want to cut spending that they don’t like. To do this, they need to have the right to selective veto, which would give them the opportunity to reduce spending on certain items. In the State of the Union report presented to the US Congress in January 1987, Ronald Reagan spoke very eloquently about this: “Give us the tool that 43 governors have — the right to veto line by line so that we can exclude senseless projects and privileges granted for political reasons: all those articles which by themselves would not have survived.

At first glance, the right to veto certain articles of the bill will only strengthen the power of the president and under no circumstances will lead to negative consequences. Nevertheless, there are situations in which the president would be better off without such a tool. The fact is that the presence of the president’s right of selective veto will affect the strategy for passing laws that Congress adheres to. A simple game will show exactly how this can happen.

With regard to this topic, the essence of the situation that developed in 1987 was as follows. Let’s assume that two items of expenditure are presented for consideration: the modernization of cities («M») and the missile defense system («P»). The Congress preferred the former, while the President preferred the latter. Nevertheless, both Congress and the president chose to include both of these articles in the bill rather than maintain the status quo. The table shows the assessment of possible scenarios for the development of events in the case of two players (4 is the highest assessment, 1 is the lowest).

The game tree, provided that the president does not have the right to selective veto, is shown in the figure. The President will sign a bill that would include both Sections M and P, or that would have only Section P, but would veto the bill if only Section M was included. Knowing this, Congress chooses to include both articles. We again show the choices made at each node in the tree by marking the corresponding choices with thick arrow lines. Note that we do this at all points where the president is supposed to have to make a choice, although some of the choices have been called into question by previous decisions of Congress. The fact is that the choice of the Congress largely depends on the analysis of the possible steps of the president in the event that the Congress would prefer another option. In order to illustrate this logic, it is necessary to map the president’s actions in all possible situations.

An analysis of this game allows us to draw the following conclusion: in this case, both sides will consider it necessary to choose the second preferred option (score 3).

Now suppose the president has a selective veto. In this case, the game tree will look like this:

At the same time, Congress makes the following forecast: if both articles are passed, the president will veto article M and leave only article P. Therefore, it is best for Congress to either pass only M and see the president veto this article, or not to pass both articles. . Perhaps Congress will favor the first scenario if it can make political gains from the president’s veto. On the other hand, the president, too, can benefit politically from such maintenance of budgetary discipline. Let’s assume that both options balance each other, which means that Congress has no particular preference for one or the other choice. However, both options give each side the opportunity to get a result that occupies only third place (score 2). Consequently, even the president may find himself in a difficult position due to the presence of additional freedom of choice.

In many US states, governors have selective veto power. Does this mean that such states have lower levels of budget spending and budget deficits than those states whose governors do not have selective veto power? Syracuse University professor Douglas Holtz-Eakin (former director of the Congressional Budget Office) performed a statistical analysis showing no such differences (see The Line Item Veto and Public Sector Budgets, Journal of Public Economics 36 (1988): 269–292). ).

This game illustrates an important conceptual point. In a situation where decisions are made by one person, having more freedom of action will not bring any harm. But in games, the extra freedom of action can be harmful, as it can influence the behavior of other participants in the game. Moreover, tying your own hands is sometimes beneficial. This “commitment advantage” is discussed in more detail in Chapters 6 and 7.

We applied the reverse reasoning method in a very simple game (Charlie Brown) and then used it in a more complex game (selective veto). The basic principle remains the same regardless of the difficulty level of the game. However, if we are talking about games in which each player has several choices at each decision point, the game tree can very quickly become so complex that it will be difficult to build or use. So, for example, in chess, 20 branches come from the root vertex: a chess player playing with white pieces can either move each of his pawns one or two squares, or move his knight in one of two directions. For each of these moves, the black chess player can respond with 20 moves. Therefore, already at this level we have 400 different paths. The number of branches coming from nodes at the next levels is even greater. The complete solution of a chess game by building a game tree is beyond the power of even the most powerful computer of all that exists today or can be created in the next few decades. Therefore, in such cases, it is necessary to apply other methods, such as the method of partial analysis. Next, we will talk about how chess players solved this problem.

Between these two extremes are games of medium difficulty in areas such as business, politics and everyday life. There are two approaches to such games. The first is the use of computer programs to build trees and calculate solutions15. On the other hand, many games of medium complexity are solved by logical analysis of the game tree without building the tree itself. Let’s illustrate this approach with the example of a game in one of the TV shows, in which each participant tries to «outplay, outsmart and last longer» all the other participants.

Strategies for Survivor game participants

The CBS reality show Survivor has a lot of interesting strategies. In one episode of Survivor: Thailand, two teams (or two tribes) played a game that was a perfect illustration of the application of the principle of “look ahead and reason backwards” in theory and practice16. Twenty-one flags were placed on the playing field between the tribes; members of each tribe had to remove these flags in turn. When it was the turn of one of the tribes, his representative could remove 1, 2 or 3 flags. Removing 0 flags (in other words, passing your turn) was not allowed, as well as removing four or more flags at a time. The winning team was the one that took the last flag if it was left alone, or all if there were two or three left17. The losing tribe had to expel its own representative, which weakened the tribe’s position in future trials. In fact, the loss in this case played a decisive role, and in the end, a member of another tribe won the main prize — a million dollars. Thus, the ability to choose the right strategy for this game was quite valuable.

The participants of the show were divided into two tribes — Suk Jai and Chuay Gan; the Suk Jai tribe made the first move. This tribe started by removing 2 flags, leaving 19 flags on the field. Before you read on, pause for a moment and consider: How many checkboxes would you choose to remove in their place?

Write down the answer to this question somewhere and keep reading. In order to understand how this game should be played, and to compare the correct strategy with how both tribes actually played, pay attention to two significant points. First, before the start of the game, both tribes were given a few minutes to discuss this challenge. One member of the Chuai Gan tribe, African-American Ted Rogers, who was a software engineer, remarked, «At the end of the game, we should leave them with four flags.» This is true: left with 4 flags, the Suk Jai tribe must remove 1, 2 or 3 flags. In this case, the Chuai Gan tribe will only have to wait for their turn, remove 3, 2 or 1 flag, respectively — and win. The Chuai Gan tribe understood correctly and used this opportunity: when there were 6 flags left, they removed two of them.

But there is another significant moment. On the previous turn, when the Suk Jai tribe took 3 flags out of the remaining 9, one of the representatives of this tribe, Shi Ann, who knew how to articulate thoughts clearly and was rightfully proud of her analytical skills, suddenly realized: “If Chuai Gan takes two flags, we are finished.” . This meant that the Suk Jai had just made a wrong move. What should be done next in this situation?

Shi Ann or one of her tribesmen should have thought the same way as Ted Rogers did and try to leave another tribe with four flags, but applying that logic to that tribe’s next turn. How to leave another tribe with 4 flags on its next turn? Leaving it with 8 flags on the previous one! When this tribe takes 1, 2, or 3 flags out of eight, you will take 3, 2, or 1 flag, leaving them with four flags, as planned. Therefore, the Suk Jai tribe would have to switch places with the Chuai Gan tribe and take only 1 out of 9 flags. Shi Ann’s analytical mind began to work actively, but one turn late! Apparently, Ted Rogers’ analytical skills were even better. But is it?

Why did the Suk Jai tribe end up with 9 flags on the previous turn? Because in Chuai Gan they removed 2 flags out of 11 from the field on their previous move. Ted Rogers should have taken his reasoning one step further. The Chuai Gan tribe needed to take 3 flags, leaving Suk Jai with 8 flags — a losing position.

The same reasoning can be continued in reverse order. In order to leave another tribe with 8 flags, you should leave it with 12 flags on the previous turn; to do this, you must leave it with 16 flags a move earlier and 20 flags a move before this move. Thus, the Suk Jai tribe should have started the game by removing only 1 flag from the playing field, and not 2, as happened in reality. Such a move would provide this tribe with an inevitable victory, leaving the Chuai Gan tribe with 20, 16, … 4 flags on the next moves.

Is the player who makes the first move guaranteed to win all games? No. If this flag game had started with 20 flags instead of 21, the player who moved second would have won. Also, in some games (like tic-tac-toe with a 3 × 3 board), each player can choose a strategy that will ensure a draw.

And now let’s remember the first move of the Chuai Gan tribe. They had 19 flags. If Chuai Gan followed their own logic, they should have taken 3 flags, leaving Suk Jai with 16 flags and thus dooming this tribe to inevitable defeat. Starting from any stage of the game at which the opponent made a wrong move, the team that made the next move could seize the initiative and win. But the Chuai Gan tribe also did not play perfectly.

The fate of two key representatives of these tribes is also interesting. On the following episode, Shea Ann made another major miscalculation and was kicked out of the tribe, being eliminated tenth out of 16 contestants. Ted Rogers, a more inconspicuous, but somewhat more experienced player, held out in the game until the last five.

The table shows both the actual and correct moves of both tribes at each stage of the game. (The entry «No move» means that any move is a losing one, provided that the opponent makes the right move.) It can be seen from this table that in almost all cases both tribes made the wrong choice. The only exception was the move of the Chuai Gan tribe, when they ended up with 14 flags, but even that was most likely random, since on the next move the tribe removed 2 out of 11 flags from the field, while they should have taken 3 flags.

Do not judge these tribes too harshly: it takes time and experience to learn how to play even the simplest games. In our classes, we played this game in pairs or in groups of students and came to the conclusion that even first-year students of Ivy League universities (the Ivy League is an association of eight private American universities in seven states in the northeastern United States; the ivy shoots that grow around the old buildings of these universities — Ed.) need to play three or even four rounds before they fully understand the logic of the game and start playing correctly from the first turn. (By the way, how many flags did you choose when we asked you to do this, and how did you justify this choice?) It should be noted that people learn faster when they watch the game from the side than when they play themselves. Perhaps the position of an observer allows you to see the overall picture of the game and build your reasoning more calmly than in the role of a participant.

In order to help you better understand the logic of reasoning in this game, we offer the first task for training thinking (such problems will help you develop your strategic thinking skills). You can find the answers at the end of the book, in the «Solutions» section.

THINKING TRAINING TASK #1

Let us reverse the rules of the game: now you win if you force the other team to take the last flag. Now it’s your turn and you have 21 flags. How many flags will you take?

Now that you have strengthened your thinking by solving this problem, let’s look at other elements of the strategy that are used in games of this class.

Why Backward Reasoning Makes Games Solvable

The game of flags had one property that made it completely solvable, and that was the absence of uncertainty of any kind, whether due to the natural element of chance, the motives and abilities of other players, or their actual actions. At first glance, this is a fairly simple idea, and yet it requires clarification and clarification.

Firstly, at any stage of the game, when the tribe had to make its move, it knew all the conditions of the game, in particular, how many flags were left. In many other games there is an element of pure chance, which is created by nature itself or by the gods of chance. For example, in many card games, the situation is as follows: when a player makes a decision, he does not know for sure what cards the other players have, although he can make some assumptions based on their previous actions. The following chapters deal with games that have a natural element of chance.

Secondly, the tribe that had to make the choice knew the goal of the other tribe — to win. Charlie Brown should also know that Lucy enjoys watching him fall on his back. Players know exactly the goals of another player or players in many simple games and sports, but this is not always the case in the games people play in business, politics, and social interaction. In such games, the motives of the players are a complex combination of selfishness and altruism, the desire for fairness and justice, short-term and long-term considerations, and so on. In order to understand what choice other players will make in the next stages of the game, it is necessary to know their goals, and if there are several such goals, what their priority is. You may never find out about it, so you’ll have to make educated guesses. You should not assume that other people have the same preferences as you or a hypothetical «rational person», but you should carefully analyze their situation. Putting yourself in another person’s shoes is a difficult task, often made more difficult by your emotional attachment to your goals and aspirations. We will discuss this type of uncertainty in more detail later in this chapter, as well as in other chapters of the book. For now, just pointing out that the uncertainty of the motives of other players is exactly the kind of question that it is advisable to address to an independent third party — a strategy consultant.

Finally, players in many games face uncertainty in the choice of other players, sometimes referred to as «strategic uncertainty» to distinguish it from uncertainty due to natural chance (as in the case of dealing cards or bouncing the ball due to uneven fields). There was no strategic uncertainty in the game of flags, as each tribe knew exactly what previous move the other tribe had made. However, in many other cases, the players act simultaneously or in such rapid succession that one does not have time to see the move of the other and make an adequate response move. In football, the goalkeeper who is about to save a penalty must decide which way to move — to the right or to the left, although he does not know in advance where the player who will take the penalty will aim. A good football player tries to deceive the goalkeeper until the last microsecond so that he does not have time to take the necessary actions. The same can be said for serves and dribbles in tennis and many other games. Each participant in a closed auction must make decisions without knowing what choice others are making. In other words, in many games, players make moves at the same time, rather than in a predetermined sequence. The process of reasoning by which the participants in such games have to make their choice is in some respects more complex than the backward reasoning method in games with sequential moves (like the game of flags): each player must understand that others are making an informed choice and, in their own turn, analyze how he thinks, and so on. The following chapters will discuss how to build reasoning and what decision tools to use for games with parallel moves. In this chapter, we will focus exclusively on games with sequential moves, which include the game of flags, as well as the more complex game of chess.

Is the method of backward reasoning always effective in real life?

Building backward reasoning on the game tree is the correct approach to the analysis and solution of games in which players take turns taking turns. Anyone who does not do this (whether consciously or not) prevents the achievement of their own goals; such people should read our book or hire a strategy consultant. However, this is the recommended or standard way to apply the reverse reasoning method. Does this method have some explicit or positive value, like most other scientific methods? In other words, does the application of this method give correct results in games played in real life? Researchers who are tackling the new, exciting fields of behavioral economics and behavioral game theory have conducted experiments that lead to conflicting conclusions.

The most serious critical arguments arise in connection with the game of ultimatum. This is the simplest negotiating game in which only one ultimatum offer is made. The game is played by two players: the one making the offer (player A) and the one answering the offer (player B) and some amount of money, say $100. Player A starts the game by asking Player B to split $100 between the two of them. Player B decides whether or not to accept Player A’s offer. If B accepts, the money is split according to Player A’s offer, each player receives the appropriate amount, and the game ends. If Player B doesn’t accept the offer, both players get nothing and the game ends too.

Pause and think. If you were playing this game as Player A, how would you propose to split the money?

Now consider how this game would be played by two people who act rationally from the point of view of classical economic theory — in other words, if everyone cared only about their own interests and could correctly calculate the optimal strategy for satisfying them. The bidder (A) would think like this: “However I propose to split the money, B has one alternative: get the amount offered or be left with nothing. (The game is played only once, so it makes no sense for B to create a reputation for being intractable or to return player A’s actions with the same coin.) This means that B will accept any offer I make. It is best for me to offer B as little as possible, for example, one cent, if the rules of the game allow it. Therefore, player A would offer B the minimum amount, and player B would accept this offer (This argument is another example of tree logic that does not require the construction of the tree itself).

Take another break and think. If you were playing this game as Player B, would you accept one cent?

The ultimate game has been the subject of numerous experiments.

There have been so many such experiments that it is impossible to list them all. An excellent overview of research on this topic can be found in Colin Camerer, Behavioral Game Theory: Experiments in Strategic Interaction (Princeton, NJ: Princeton University Press, 2003), 48–83, 467. In this book, Colin Camerer analyzes experiments conducted in links to other games, most notably the trust game, such as the game between Charlie and Fredo discussed in . Once again, we draw your attention to the fact that the real behavior of people differs from what can be predicted using the method of inverse reasoning, which takes into account only purely selfish preferences; in real life, people often trust each other.

As a rule, during such experiments, about two dozen people are gathered in one place and randomly divided into pairs. In each pair, the roles of the player making the offer and the player responding to the offer are distributed. New pairs are also formed at random, and the game starts again. In most cases, players do not know with whom they will play in the next round. Thanks to this approach, the experimenter receives several sets of data, working with one group of participants in the experiment during one session, but there is no possibility of forming stable relationships that can affect the behavior of the participants in the experiment. Within this general scheme, different variants of conditions are created in order to study their impact on the final result.

A SIMPLE MIND TRAINING TASK: A DIFFERENT OPTION OF THE ULTIMATUM GAME

In this variant of the ultimate game, player A makes an offer to player B on how to split $100. If B accepts A’s offer, they split the money and the game ends. But if B says no, A must decide whether to make another offer. Each next offer of player A should be more generous. The game ends when player B accepts the offer or when player A stops making offers. What do you think the outcome of this game will be?

In this case, we can assume that player A will make offers until he offers player B $99, keeping $1 for himself. Thus, according to the tree-like logic of reverse reasoning, player B will receive almost all the money. If you were player B, would you stick around until A offers you a 99:1 split? We advise you not to do this.

In all likelihood, your own analysis of how you would act as the first and second players gives you reason to assume that in real life the results of this game should differ from the theoretical predictions presented. And they do differ, and in many cases significantly. Players who make an offer split the money in different ways, but one cent, one dollar, or any other amount below 10 percent of the amount at stake is extremely rare. The average amount (half of the players offer more than this amount, half — less) is in the range of 40 to 50 percent; in many experiments, the most common division is 50:50. Players who respond to offers half the time reject offers that provide them with less than 20 percent of the total.

The irrationality and rationality of caring for others

Why do players offer a large enough share of the total to those who accept offers? There are three possible reasons for this. Firstly, it is possible that the player making the proposal does not know the method of backward reasoning. Second, the proposing players may have other motives other than a purely selfish desire to get the most: for example, altruistic motives or a sense of justice. Third, they may fear that the players they propose to will refuse the amount offered if it is too small.

The first reason is unlikely, since the logic of backward reasoning in this game is not difficult at all. In more complex situations, players are not always able to correctly perform all the necessary calculations, especially if they are new to this game (as we have already seen in the game with flags). But the ultimate game is quite simple even for beginners. Therefore, this phenomenon can be explained by a second cause, a third cause, or a combination of both.

The first results of experiments with the ultimatum game testified in favor of the third reason. Harvard University professor Al Roth and his colleagues came to the conclusion that in the scheme of formation of the threshold of refusals, which prevailed in their group of subjects, offering players chose their offers in such a way as to achieve the optimal ratio of the prospect of obtaining a larger share for themselves and the risk of rejection by responding player. This indicates the presence of outstanding traditional rationality among the players making offers.

However, more recent research on the differences between the second and third possibilities has led to different conclusions. In order to distinguish between altruism and strategy, experiments were carried out using one of the variants of the ultimatum game — the game of dictator. In this game, the first player dictates how the available amount of money should be divided; the other must accept the offer, whatever it may be. In the dictator game, the proponents on average give back substantially less than in the normal ultimate game, but they still give away much more than zero. Therefore, there is some truth in both explanations: the behavior of the offerer in the ultimate game is due to both generosity and strategy.

However, what exactly caused this generosity — altruism or considerations of justice? Both explanations are different aspects of what might be called human concern for others. Such a study allows us to understand the difference between them. In the experiment, which is carried out according to the usual scheme, after the formation of pairs, the roles of the proposing and responding players are distributed by one of the methods of random selection, for example, by tossing a coin. This can give players a sense of equality or fairness. In order to eliminate this element, roles are allocated through a preliminary test, such as a general knowledge test. The winner becomes the bidder, taking it for granted, resulting in an average 10 percent reduction in bids. However, proposals are still much larger than zero, which suggests that there is an element of altruism in the reasoning of the proponents. Keep in mind that they don’t know the responding players personally, so this is about altruism in general, not concern for the well-being of a particular person.

A third variation of individual preference is also possible: the player who makes offers, determining the amount offered, may be guided by a sense of shame. Jason Dana of the University of Pennsylvania, Daylien Kane of the Yale School of Management, and Robin Dawes of Carnegie Mellon University experimented with the following variation of the dictator game. The dictator is offered to allocate $10 to another player. After this amount is allocated, but before it goes to another player, the dictator is offered the following offer: he can receive $9, the other player will receive nothing, but he will not even know that he took part in the experiment. Most dictators accept this offer. Therefore, they are willing to pay one dollar so that the other player does not find out about their greed. (The altruist would rather keep $9 for himself and give the other player $1, rather than keep $9 for himself unless the other person finds out.) Even if the dictator were offered $3, he would still take the money, only the other player would remain in the dark. This is reminiscent of a situation where a person is ready to cross to the other side of the street, just not to give alms to a beggar.

Note the following two aspects of these experiments. First, they are conducted in full accordance with standard scientific methodology: the truth of hypotheses is tested by creating acceptable control options in these experiments. We present here some of the most interesting ones. (Colin Camerer’s book, referenced in the notes to chapter 2, discusses other options.) Second, the social sciences allow for the coexistence of many different causes that explain separate aspects of the same phenomenon. Hypotheses need not be absolutely right or completely wrong; accepting one hypothesis does not mean rejecting all others.

Now analyze the behavior of the players responding to the offer. Why do they refuse, even if they know that the only alternative is to get even less? The reason cannot be that they are looking to build a reputation for being tough negotiators that could bear fruit in future rounds of this game or other division games. The same pair does not play again, and future partners do not have access to data on the past actions of a particular player. Therefore, even if such a motive as reputation is implicitly present, it takes on a deeper form and acts as a general principle of action that the responding player adheres to without any explicit reasoning and calculation in each particular case. He acts instinctively or responds to a suggestion under the influence of emotion. And indeed it is. In a new area of ​​experimental research called «neuroeconomics», the brain activity of the subjects during the adoption of various economic decisions is recorded using functional magnetic resonance imaging (functional magnetic resonance imaging — fMRI) and positron emission tomography (positron emission tomography — PET). Responding players show higher anterior insular cortex activity as the first player’s offers become more unfair. Because the anterior insular cortex is more responsive to emotions (such as anger or dissatisfaction), this result helps explain why second-turners turn down unfair offers. On the contrary, the left side of the prefrontal cortex is activated when the player accepts an unfair offer, which indicates that at such a moment he is consciously controlling his actions, trying to find the optimal balance between the desire to express his displeasure and the desire to get more money.

Many people (especially economists) argue that while bidders do turn down a small fraction of the small amounts offered to them in laboratory experiments, in the real world, where the stakes are often much higher, the odds of rejection are extremely low. In order to test this claim, the researchers conducted experiments with ultimatum games in poor countries, where the amount at stake was equal to the income of the participants in the experiments for several months. The number of refusals did decrease somewhat, but the offers did not become significantly less generous. The consequences of rejection have become more severe for both the players making offers and the players responding to them, so offerers who fear rejection have begun to behave more cautiously.

Human behavior can be partly explained by inherent instincts, hormones or emotions, but some aspects of behavior also depend on cultural affiliation. In the course of experiments that were carried out in different countries, an interesting fact was discovered. Concepts of what constitutes a reasonable offer differed by nearly 10 percent across cultures, although perceptions of qualities such as aggressiveness and toughness varied to a lesser extent. The results of the experiments differed significantly from the rest in only one group: in the Machigenga tribe living in the Peruvian Amazon region, offers were much lower (on average 26 percent) and only one was rejected. Anthropologists explain this by the fact that the Machigenga tribe consists of small families, it is socially isolated and there are no established norms for the distribution of material wealth. By contrast, two cultures had over 50 percent supply; these cultures have developed a tradition of making generous donations when someone is lucky (such a tradition obliges the recipient of a gift to repay with an even more generous donation in the future). This norm or custom carries over to the experiment, although the players do not even know to whom they give money or from whom they receive it.

The evolution of altruism and justice

What conclusion follows from the results of these experiments with ultimatum games, as well as other similar experiments? In many cases, these results are quite different from what one would expect based on the backward reasoning method and the assumption that each player cares only about his own benefit. What exactly can be called a false assumption — the ability of the players to correctly perform the reverse reasoning, or their selfishness, or a combination of both? And what are the possible consequences?

First, let’s analyze the method of backward reasoning. We have seen contestants on the reality show Survivor fail to apply this method correctly in the flag game. They were playing the game for the first time, but even so, there were glimmers of correct reasoning in the discussion of the game. In our experience, students fully master the strategy of the game after playing it or watching it only three or four times. Many experimenters invariably and almost deliberately work with beginners who, in most cases, are just learning to play a particular game. In the real world of business, politics and professional sports, where people are quite experienced in the games they have to play, we must be prepared for the fact that players have much more knowledge and that they most often use the right strategies, chosen either after carefully weighing the options, or with the help of intuition based on knowledge. In more complex games, strategic players can use computers or consultants to perform the necessary calculations; now this practice is quite rare, but will soon become more widespread. That is why we are convinced that the method of backward reasoning should remain our starting point for analyzing such games and predicting their results. If necessary, the first attempts at game analysis can be adapted to a specific context, since beginners can make mistakes, and some games are too complex to be solved without outside help.

It is our deep conviction that a more important lesson can be learned from these experimental studies: people make their choices based not only on their own benefit, but also on other considerations and preferences. This takes us beyond classical economic theory. Game theorists must take into account the importance given to fairness and altruism by the participants in a particular game. «Behavioral game theory pushes the boundaries of rationality rather than discarding it.»

All this is for the best: understanding the motives that guide people deepens our understanding of the process of economic decision-making, as well as the process of strategic interaction. And this is already happening: cutting-edge research in game theory is increasingly taking into account player goals such as justice, altruism, and other aspirations of this kind (and even the “secondary” desire to reward or punish other players in the game whose behavior reflects or violates these principles).

However, we should not stop there; we must take the next step and reflect on why people tend to strive for altruism and justice, and why they often feel such strong anger and resentment when someone violates these principles. This takes us into the realm of conjecture, but one plausible explanation can be found in evolutionary psychology. Groups whose members adhere to such norms of behavior as justice and altruism have fewer internal conflicts than groups that consist of selfish individuals. Consequently, such groups will be able to achieve greater success in collective action, including the creation of wealth that benefits the entire group, and the rational use of common resources. In addition, they spend less effort and resources on internal conflicts. As a result, these groups achieve more both in absolute terms and in competition with groups that lack such norms of behavior. In other words, a certain measure of justice and altruism may have evolutionary survival value.

Some of the biological reasons for refusing unfair offers came to light in an experiment conducted by Terry Burnham.

Terry Burnham is co-author of Mean Genes (Cambridge, MA: Perseus, 2000) and author of Mean Markets and the Lizard’s Brain. How to make money using knowledge about the causes of manias, panics and crashes in financial markets” (M.: Alpina Publisher, 2012). This experiment is the focus of his High-Testosterone Men Reject Low Ultimatum Game Offers, Proceedings of the Royal Society B 274 (2007): 2327–2330.

In his version of the ultimate game, $40 was at stake, and the subjects were graduate students from Harvard University. The player who had to split this amount had only two options: offer another player $25 and keep $15 for himself, or offer $5 and keep $35 for himself. Of those who were offered only $5, twenty students accepted the offer and six rejected it, leaving them and the players who shared the amount to receive nothing. And now the most important thing. It turned out that the six students who refused to accept the offer had testosterone levels 50 percent higher than those who accepted it. Since testosterone levels determine how a person perceives their status and how aggressive they can be, this may be the genetic basis for the evolutionary advantage of what the evolutionary biologist Robert Trivers called «moral aggression.»

In addition to possible genetic factors, in any society there are non-genetic ways of transmitting norms of behavior, in particular, the process of education and socialization of children in families and schools. Parents and teachers teach receptive children the importance of caring for others, sharing with others, and behaving well. Of course, some of these beliefs remain forever in the minds of children and influence their behavior throughout their lives.

In conclusion, we should note that justice and altruism have their limits. In the long term, the progress and success of society is unthinkable without innovation and change, and they, in turn, are impossible without individualism and a willingness to break social norms and discard established stereotypes; egoism often accompanies these qualities. Therefore, a balance is needed between caring for oneself and caring for others.

Very complex trees

As you gain some experience with the reverse reasoning method, you will realize that it can be used in many strategic situations in everyday life or at work, even without building trees. Many other games of medium difficulty are solved by special computer programs that are becoming more and more widely available. However, in complex games (such as chess) it is not possible to find a complete solution using backward reasoning.

In theory, chess is a game that is ideal for backward reasoning because it consists of a series of successive moves.

A more competent analysis of the game of chess from a game theory perspective can be found here: Herbert A. Simon and Jonathan Schaeffer, “The Game of Chess,” in The Handbook of Game Theory, Vol. 1, ed. Robert J. Aumann and Sergiu Hart (Amsterdam: North-Holland, 1992). Chess computers have improved significantly since this article was written, but the overall analysis remains valid. In 1978, Herbert Simon received the Nobel Prize in Economics for his fundamental research on the decision-making process in economic organizations.

Players take turns making their moves; all moves can be observed and cannot be undone; there is no uncertainty about the positions or motives of the players. The rule that a draw is declared if the same position is repeated guarantees the end of the game in a finite number of moves. We can start at the leaves of the tree (or endpoints) and analyze the game backwards. However, theory and practice are two different things. It has been calculated that the total number of vertices in the chess tree is about 10120, that is, 1 followed by 120 zeros. A super-powerful computer, which is 1000 times faster than a conventional personal computer, would take 10103 years to analyze all possible moves through such a tree. It is pointless to wait so long, and the predicted development of computer technology in the foreseeable future is unlikely to significantly improve the situation. So what was the way out for chess players and programmers?

Chess experts have managed to describe the optimal strategies for the last stages of the game. When a small number of pieces remain on the chessboard, experts can look at the end of the game and use backward reasoning to determine whether one side is guaranteed to win or whether the other side can achieve a draw. But in the middle of a chess game, when there are still quite a few pieces left on the board, it is much more difficult to determine the possible development of events. Good chess players can calculate in advance about five pairs of moves, but this will not allow us to simplify the situation to a level that would allow us to play the correct endgame strategy already at this stage.

A practical solution to this problem comes down to the use of two approaches: the analysis of moves by the method of reverse reasoning and the subjective assessment of a chess position. The first is science, game theory: the ability to look ahead and reason backwards; the second approach is the art of the practitioner: the ability to determine the value of a chess position from the number of remaining pieces and the interaction between them without determining a clearly winning strategy of play from that point on. Chess players often refer to this phenomenon as «knowledge,» but you can call it experience, intuition, or art. The best chess players are distinguished, as a rule, by the depth and subtlety of knowledge.

Such knowledge can be accumulated by watching many chess games and many players, and then compiling a set of rules for the game. A lot of work has already been done in this regard with respect to the opening, or the first ten or fifteen moves. There are hundreds of books that analyze different openings and discuss their advantages and disadvantages.

What is the role of computers in all this? There was a time when the project of writing programs that would allow a computer to play chess was considered an integral part of the new field of science — artificial intelligence, whose goal was to create computers that could think like a person. But over the years, this has not been achieved, so scientists began to pay more and more attention to what computers do best — mathematical calculations. Computers calculate more moves ahead and do it faster than humans.

However, experienced chess players are able to discard moves that are likely to be unsuccessful without calculating the consequences four or five moves ahead. This allows them to save time and energy for moves that may be more effective.

Relying solely on mathematical calculations, in the late 1990s, special chess computers such as Fritz and Deep Blue were able to compete with the best chess players.

The rating of chess players is determined by the results of the games; The rating of the best chess computers is comparable to the rating of 2800, which has the world’s strongest chess player Garry Kasparov. In November 2008, Kasparov played a four-game match against the latest version of the Fritz computer, the X3D. As a result, in two games each of the parties won, and two games ended in a draw. In July 2005, the Hydra chess computer inflicted a complete defeat on Michael Adams, who was ranked 13th in the ranking of the best chess players in the world: the computer won five games and one ended in a draw. Perhaps the time is not far off when computer programs will take first place in the rankings and start playing each other in the world chess championships.

What conclusions follow from this story about chess? It shows how the thinking should be in any high difficulty games you may encounter. You need to combine the principle of “look forward and think backward” with experience that will help you evaluate the intermediate positions reached by the end of the pre-calculation period. You will be successful only through this synthesis of the science called «game theory» and the art of playing a particular game, and not by using each of these elements separately.

Think for two

The chess strategy illustrates another important aspect of the backward reasoning method: you must play the game from the point of view of both players. It is very difficult to calculate the best move in a complex game, but it is even more difficult to predict what the other player will do.

If you really had the ability to calculate all possible moves and countermoves, and the other player could do the same, then you could agree in advance on how the game will end. But since your analysis is limited to a few branches of the game tree, the other player may see what you don’t see, or miss what is obvious to you. Be that as it may, your opponent may make a move that you did not foresee.

Effective application of the “look forward and backward” rule requires anticipating what the other player will actually do, not what you would do if you were in his position. The problem is that when you’re trying to put yourself in the shoes of another player, it’s very difficult, if not impossible, to forget the situation you’re in. You know too much about what you plan to do next in the game, and it’s hard not to be influenced by that knowledge when you’re analyzing the game from another player’s point of view. This explains why in chess (or poker) no one plays with himself: it is impossible to bluff or make a surprise move against yourself.

This problem does not have a perfect solution. When trying to put yourself in the other player’s shoes, you must know what he knows and be unaware of what he doesn’t know. The other player’s goals should become your goals, not what you expect them to be. In practice, companies that are trying to calculate the moves and counter moves of competitors in a particular business development scenario hire third-party specialists to play the role of the other side. This way they can be sure that their game partners don’t know too much. In many cases, it is most useful to isolate the opponent’s moves that you did not expect and analyze what led to such a result, so that later you can either try to avoid such a development of events or contribute to it.

To conclude this chapter, let’s return to Charlie Brown and his attempts to decide whether or not to hit the ball. This question has become a real problem for football coach Tom Osborne in the last minutes of his team’s struggle for the title. We believe that he, too, made the wrong decision. The method of backward reasoning will help us understand where he went wrong.

Case Study: The Story of Tom Osborne and the 1984 Orange Bowl

In 1984, the Orange Bowl college football cup match took place between the Nebraska Cornhuskers (Nebraska Corners), which did not suffer a single defeat, and the Miami Hurricanes (Miami Hurricanes), which lost only one match. Since the Nebraska team had better results on the eve of the final game, it was enough for them to draw to end the season in first place.

At the beginning of the fourth period, the Nebraska team was losing with a score of 31:17. The Cornhuskers then began to close the gap. They scored a touchdown, after which the score became 31:23. Nebraska coach Tom Osborne had a major strategic decision to make.

In college football, the team that scores a touchdown receives the right to continue play from a line 2,5 yards from the end zone. The team has two options: either get the ball into the end zone (by running into it or passing the ball to a player who is already in it) and earn 2 more points, or use a less risky strategy by scoring a goal — this gives one extra point.

Coach Osborne chose a safer strategy and the Nebraska team scored a goal for one point. Now the score was 31:24. The Cornhuskers continued to close the gap. In the last minutes of the match, the team earned the last touchdown, narrowing the gap even more: the score was 31:30. It was enough for the Nebraska team to score one more point to win the match and receive the championship title. But such a victory would not bring true satisfaction to the team. Osborne understood that in order to achieve a spectacular victory, the team must win this match.

The Cornhuskers entered the fight for the win, trying to get a two-point lead. Irvin Friar received the ball, but could not deliver it to the opponent’s end zone. The Miami and Nebraska teams ended that year tied. But since the Miami team defeated Nebraska in the final match, it was this team that took first place in the standings.

Put yourself in the shoes of Coach Osborne. Could you do more?

Example Analysis

Osborne was accused by many fans of trying to win instead of being content with a draw. But we are not interested in this question. Osbourne was willing to take risks to win, but he did it wrong. He would have done better if his team had first tried to score two points by getting the ball into the end zone. If it were possible to do this, the team could try to score a goal and get another point; in case of failure, another attempt should have been made to score two points after a touchdown.

Let’s analyze this situation in more depth. When Osbourne’s team was down 14 points, he already knew he needed two touchdowns and three extra points. He decided to earn first one point, and then two more. If both attempts were successful, the order in which they were attempted would not matter. If the one-point goal had not been scored, and the two-point ball delivery to the end zone had been successful, then the order would not have mattered either, since the match would have ended in a draw and the Nebraska team would have won the championship. The order of the game would only matter if the Nebraska team made a failed attempt to score two points. According to Osborne’s plan, this led the Nebraska team to defeat not only in the final game, but in the entire championship. If instead the team tried to earn two points first, then even a failed attempt would not necessarily lead to defeat. The score would remain 31:23. Having earned the next touchdown, the team would have reduced the gap and the score would have become 31:29. A successful attempt to score two points after a touchdown would have given the Nebraska team a tie and they would have taken first place!

Moreover, it would have been a draw after a failed attempt to win, so no one would criticize Osborne for playing for a draw.

We also heard a counter-argument: if Osborne decided to score two points first, but this attempt was unsuccessful, this would mean that his team is playing for a draw. In this regard, the team risked losing morale, and then it would hardly have earned a second touchdown. On the other hand, if the team played two decisive points at the very end of the match, the players would be aware that everything is at stake, and would act accordingly. This argument is flawed for several reasons. First, if the Nebraska team tried to score two points only after the second touchdown and this attempt was unsuccessful, they would lose. But if the team hadn’t managed to score two points after the first touchdown, they still had a chance to tie. And although this chance could be even less, “something” is more than nothing. The argument about the dynamics of the game is also untenable. While the Nebraska team could indeed step up the offensive in the final game of the championship, you can expect the Miami team to increase their defense in the process. This match was important for both teams. As for the offensive momentum, if Osbourne’s team scored two points after the first touchdown, it would increase their chances of earning another touchdown. In addition, this would allow the team to end the match in a draw, scoring three goals against the opponent.

One of the general takeaways from this story is that if you have to take a risk, in most cases it’s best to take it as soon as possible. This is obvious to those who play tennis: everyone knows that you need to take risks on the first serve, and the second should be taken more carefully. In this case, if the first attempt fails, it does not mean that all is lost. You still have time for other activities that will help catch up or even move forward. This approach (taking risks as early as possible) applies to many areas of life, whether it be career choices, investments, or romantic dates.

For a better understanding of the principle of «look ahead and think backwards,» review the following case studies in Chapter 14: How to Pick the Best Location; «Red — I win, black — you lose»; ““Shark Repeller” with the opposite effect”; «Hard man, soft proposal»; «Three-sided duel»; «Win without knowing how.»

Chapter 3

Many contexts — one concept

What do the following situations have in common?

• Two gas stations or two supermarkets located in close proximity to each other, from time to time, start tough price wars between themselves.
• During the election campaign, both the Democratic and Republican parties in the United States often adhere to centrist politics, trying to win over undecided voters; in doing so, they ignore the main supporters of the far left and the far right.
• “The diversity and productivity of New England fisheries has reached an unprecedented level. However, the trend of overfishing over the past century has led to the extinction of one species of fish after another. Atlantic halibut, sea bass, cod, yellowtail flounder… have joined the ranks of those species that are now considered extinct from the point of view of commercial fishing.”
• At the end of Joseph Heller’s famous novel Catch-22, World War II is almost over. Yossarian does not want to be among those who die last: this will not affect the outcome of the war in any way. He explains this to Major Denby, the senior officer. Denby asks him, «But, Yossarian, imagine what happens if every American thinks like that?» Yossarian answers him: “Only a complete fool thinks differently. Am I wrong?».

Answer: These are all examples of the prisoner’s dilemma.

There are no prizes for the correct answer, because the prisoners’ dilemma is the topic of this chapter. However, we decided to take this opportunity to draw your attention (as we did in ) to the fact that the general conceptual model of game theory will help to understand a number of diverse and seemingly unrelated phenomena. We should also note that neighborhood stores don’t constantly fight price wars, and political parties don’t always gravitate towards the centrist position. Analysis and examples of how participants in such games can avoid or solve such a dilemma is an important part of the current chapter.

As with the interrogation of Dick Hickok and Perry Smith (characters of the novel In Cold Blood discussed in chapter 1), each participant in the game has his own reasons for doing something that will entail adverse consequences for both, since each of them only looks after its own interests. If one confesses to a crime, it is better for the other to confess too, in order to avoid a harsh sentence; if one decides to refrain from confessing, the other can greatly alleviate his lot if he confesses. In fact, prisoners are under such intense pressure in such a situation that they are tempted to plead guilty, whether they are guilty (as in Cold Blooded Killer) or innocent, but the police fabricated a case against them (as in Los Angeles Confidential). -Angeles).

The same thing happens with price wars. If the Nexon gas station charges a low price, Lunaco would be better off lowering their prices as well so as not to lose customers; if Nexon charges a high price for its gasoline, Lunaco can win over many buyers by lowering the price. But if both filling stations sell gasoline at a low price, neither of them will earn anything (although such a price is only at hand for customers).

If the Democrats adopt an election platform geared towards centrist politics, the Republicans risk losing all those voters, and thus the election, if they work only with their core economic and social rights supporters; if Democrats only patronize their core minority and labor union supporters, then Republicans can win moderate voters over to their side, and thus win the election by a large margin of votes, sticking to a more centrist position.

If all anglers fish in moderation, a large catch by one angler will not deplete fish resources; if everyone else begins to actively increase the fishery, then any individual fisherman would act foolishly, trying to protect fish resources alone. As a result, overfishing occurs and some species become extinct.

In Catch-22, it is Yossarian’s logic that makes it so difficult to continue fighting in an already lost war.

A bit of history

How did game theorists invent and name this game that covers so many aspects of economic, political, and social interaction? This happened at the initial stage of the history of the development of the discipline. Harold Kuhn, who himself was one of the pioneers of game theory, spoke about this at a symposium held as part of the 1994 Nobel Prize ceremonies.

In the spring of 1950, Albert Tucker, while on vacation, came to Stanford, and since there were not enough rooms, he was placed in the office of the psychology department. One day, one of the psychologists knocked on his door and asked what he was doing. Tucker replied, «I’m working on game theory.» The psychologist asked if he would agree to conduct a seminar on this topic. For this workshop, Tucker came up with the prisoner’s dilemma as an example to illustrate game theory, the Nash equilibrium, and the paradoxes that accompany socially undesirable equilibria. Since this was a truly fundamental example, it became the subject of dozens of scientific papers and a number of serious books29.

Other scholars tell a slightly different story. In their opinion, the mathematical structure of the game was described before Tucker by two mathematicians — Merrill Flood and Melvin Drescher of the Rand Corporation (a research center that was once a stronghold of the Cold War). Tucker’s genius was that he came up with a story that illustrates mathematical calculations. And it was really brilliant, because the presentation of an idea can decide its fate: a memorable presentation contributes to the rapid spread of an idea among thinking people, while a boring and dry one can lead to the fact that the idea will not receive due attention or be forgotten altogether.

Visual representation

We will illustrate the method of solving this game with a business example. Two competing mail order companies, Rainbow’s End and BB Lean, specialize in apparel. Every autumn they print and send out winter catalogs. Both companies must adhere to the prices indicated in their catalogs throughout the winter season. The catalog preparation period is much longer than the mailing window, so both companies must make pricing decisions at the same time, without any knowledge of the competitor’s decisions. Both companies know that their catalogs are aimed at a general audience of potential buyers who know how to shop smart and look for low prices.

As a rule, almost identical assortment of goods is published in both catalogs. Let’s say one of these items is a cu in high-quality chambray. Such a cu costs each company $20.

This price includes not only the cost of buying US dollars and the Chinese manufacturer, but also the cost of transportation to the United States, customs duties, and storage of goods until the order is completed. In other words, this price includes all costs associated with this product.

Both companies estimate that if they each bid $80 for this item and sell 1200 units, that would generate a profit of (80–20) × 1200 = $72. It also turned out to be the best price for both companies: if they can agree to charge the same price, $000 is the price that will give both companies maximum profit.

Each company calculated that if one of them reduced the price by $1 and the other left it unchanged, then the company that lowered the price would attract 100 buyers: 80 buyers who switched from another company and 20 new ones (for example, those who decided to buy a CU that would not be bought at a higher price, or buyers who wanted to order a catalog item instead of buying it at a local mall). Thus, every company is tempted to charge a lower price in order to attract more buyers. The purpose of this whole story is to understand what such a decision can turn into.

Let’s start with the assumption that both companies have to choose between two prices: $80 and $70.

The clarification that companies have only two price options is only necessary to describe the analytical method for solving such games in the simplest possible way. In the next chapter, we will give companies more leeway with regard to pricing.

If one company lowers the price to $70 and the other leaves the price at $80, the first company will attract 1000 buyers to its side, while the second will lose 800. Therefore, the company that lowered the price will sell 2200 NIS, and the other company will have the volume sales will drop to 400 units; the profit is (70–20) × 2200 = $110 for the company that cut the price and (000–80) × 20 = $400 for the other company.

What happens if both companies cut the price to $70 at the same time? If the price drops by $1, companies will have existing buyers and 20 new ones will appear. Therefore, if both companies reduce their price by $10, they will each sell 10 × 20 = 200 units more than the previous 1200 units. Thus, each company will sell 1400 units and make a profit of (70-10) × 1400 = $70.

Imagine the possible profit of both competing companies in a visual form. We can’t use a game tree like the ones in Chapter 2 for this. In this example, two players are playing at the same time. Neither player can make another move based on information about what the other player has done or what move he can make in return. Instead, each player must analyze what the other player is thinking at the same time. The starting point for such “reasoning about reasoning” is to visualize all the consequences of each combination of possible choices that both companies can make simultaneously. Since each of them has only one alternative: $80 or $70, this means that there are four possible combinations. The easiest way to display them is in the form of a table consisting of columns and rows, which we will call the game table, or payoff table. The Rainbow’s End (abbreviated as RE) selection will appear in the rows of this table, while the BB Lean (BB) selection will appear in the columns. Each of the four cells in the table, corresponding to each choice of RE in a row and BB in a column, contains two digits representing each company’s profit from selling US dollars (in thousands of dollars). The number located in the lower left corner of the cell corresponds to the player for whom the rows are highlighted; the number in the upper right corner of the cell — to the player for whom the columns are selected.

Thomas Schelling invented this way of presenting the payoff of both players in one table, which makes it possible to clearly distinguish which payoff corresponds to each player. He modestly wrote the following about this: “If I am ever asked if I have made any contribution to game theory, I will answer … yes, this is a method of staggering all the payoffs in one matrix.” In fact, Schelling developed many other important game theory concepts: focal points, certainty, commitment, threats and promises, fracture, and more. In the following chapters we will often refer to Thomas Schelling and his work.

In the language of game theory, these numbers are called payoffs.

Generally, the larger the win value, the better for the player. However, in some cases (as in the case of remand prisoners) the gain is measured in terms of the number of years they have to spend in prison, so each prisoner wants to get as few years as possible. The same happens if the win is a place in the ranking, where the first place is the best. Analyzing the table of the game, it is necessary first of all to determine how the payoff in this game is interpreted.

In order to make it completely clear which wins correspond to each of the players, in the presented table, the corresponding fragments of the cells are highlighted in different shades of gray.

Before we start looking for a solution to this game, we would like to draw your attention to one aspect of it. Compare pairs of wins in four cells. The best result for RE does not always mean the worst result for BB, and vice versa. In particular, for both companies, the situation in the upper left cell is better than in the lower right one. At the end of this game, there doesn’t have to be a winner and a loser: it’s not a zero-sum game. We discussed in Chapter 2 that Charlie Brown’s investment game is also not a zero-sum game like most of the games we encounter in real life. In many games, such as the Prisoner’s Dilemma, the main question is how to avoid losing or win both sides.

Dilemma

Let us analyze the course of reasoning of the manager of the company RE. “If BB chooses $80, I can get $110 instead of $72 by dropping the price to $70. If BB chooses $70, my winnings are $70 if I also set that price, and only $24 if I leave $80. For me, the better option (in fact, the best one, since I have only one alternative) remains unchanged, no matter what the BB decides. Therefore, I don’t have to think about what they think at all; I just need to bid $70 first.”

If a game with parallel moves has such a property (namely, the optimal choice of a player does not depend on the choice of other players), this greatly simplifies the reasoning of the players, as well as the analysis that game theorists do in such cases. Therefore, the presence of such a property greatly simplifies the solution of the game. Game theorists call it the dominant strategy. A player is said to have a dominant strategy if that strategy is better than all other strategies, regardless of which strategy or combination of strategies the other player or players choose. There is a simple rule for participating in games with parallel moves:

RULE #2: If you have a dominant strategy, apply it.

In we brought to your attention the general principle of developing optimal strategies for games with sequential moves. That was our rule #1: look ahead and think backwards. In games with parallel moves, things are not so simple. Nevertheless, the «reasoning about reasoning» that is needed in games with parallel moves can be formulated as three simple rules. These rules, in turn, are based on two simple concepts: dominant strategies and equilibrium. Rule #2 is presented in this chapter; Rules Nos. 3 and 4 will be formulated in later chapters.

The prisoner’s dilemma is an even more specific game: in it, not one, but both players (or all players) have a dominant strategy. The manager of BB thinks exactly the same way as the manager of RE; in order to grasp this idea well, you must analyze the BB manager’s reasoning for yourself. Once you do this, you will see that $70 is the dominant strategy for BB as well.

The result of applying this strategy is shown in the lower right cell of the game table: both companies charge $70 and make a profit of $70 each. It is necessary to pay attention to the following aspect of the prisoners’ dilemma, which makes it such an important game. When each of the players applies their dominant strategy, both get worse results than they would if they trusted each other and agreed that each would choose a different, dominated strategy. In our example, this would mean that each company would price its product at $80 in order to achieve the outcome shown in the top left cell of the game matrix, which is a profit of $72.

In fact, $80 is the amount that provides both companies with the maximum profit. They could charge that price if they teamed up and formed a cartel in their industry. A rigorous proof of this statement requires mathematical calculations, so just take our word for it. Readers who wish to review these calculations can access them on the book’s website.

For this, it would not be enough for only one company to charge $80 for its product: this would entail very bad consequences for this company. Either way, both companies must charge a high price, which is very difficult to achieve in practice, given the temptation for each to charge a lower price than its competitor. If both companies pursue their own selfish interests, they will not be able to get the best result for both of them. Such a conclusion contradicts what classical economic theories teach us, starting with the theory of Adam Smith.

Of course, such a price reduction is beneficial to consumers who are not active participants in this game. Therefore, it is in the interest of society as a whole to ensure that the two companies cannot solve the pricing dilemma. In the United States and elsewhere, this function is performed by antitrust policy.

This raises a number of questions, some of which are related to more general aspects of game theory. What happens if only one participant in the game has a dominant strategy? What if no player has a dominant strategy? If each player’s optimal choice depends on what the other player chooses at the same time, can they guess each other’s choice and find a solution to this game? We will analyze the answers to these questions in the next chapter, which deals with a more general approach to solving games with parallel moves, namely the Nash equilibrium. In this chapter, we will focus on solving the prisoner’s dilemma.

In a generalized description of the prisoner’s dilemma, the two strategies available to each player are denoted as «cooperate» and «betray» (or in some cases, «deceive»); we will adhere to these terms. Betrayal is the dominant strategy for every player; if both players choose this strategy, their payoff will be less than in the case of choosing the cooperation strategy.

Preliminary Considerations for Resolving the Prisoner’s Dilemma

There are good reasons for players facing a prisoner’s dilemma to agree on joint action that would allow them to avoid solving it. For example, in New England, anglers may agree to limit their catch in order to preserve fish stocks for the future. The only problem is how to ensure the implementation of such agreements in conditions where each side is tempted to deceive the other (for example, to catch more fish than the quota allows). What does game theory say about this? And what happens in such cases in real life?

More than fifty years have passed since the prisoner’s dilemma was first formulated. During this time, the theoretical foundations of this dilemma have been improved, and a large amount of data has been accumulated, both in the process of observing what happens in real life and in the course of laboratory experiments. Let’s analyze this material and see what lessons we can learn from it.

The strategy of cooperation has a downside: the desire to avoid betrayal. The player can be motivated to choose a cooperative strategy instead of the dominant betrayal strategy by promising him sufficient rewards. In addition, he can be deterred from using the betrayal strategy with adequate punishment.

The reward method is problematic for several reasons. The reward can be internal in nature: one player pays another for choosing a cooperation strategy. In other cases, the reward may be external: a third party interested in cooperation between two players pays them for choosing this strategy. Be that as it may, the reward cannot be given to the player until he makes his choice, otherwise he will simply put it in his pocket, after which he will refuse to fulfill the agreement. On the other hand, if the reward is simply promised, the player may not believe this promise: when he chooses a cooperation strategy, it is possible that the one who made this promise will break it.

However, despite all these difficulties, the reward is effective and useful. With maximum creativity and imagination, players could simultaneously make promises to each other and make them trustworthy by placing the promised reward in an escrow account controlled by a third party.

Based on this idea, James Andreoni and Hal Varian developed an experimental game called Zenda. See Preplay Communication in the Prisoners’ Dilemma, Proceedings of the National Academy of Sciences 96, no. 19 (September 14, 1999): 10933–10938. We played this game in class and found that it promotes cooperation between players. However, in real conditions, this is much more difficult to achieve.

In real life, the situation is more often different: since the players interact in several directions, cooperation in one of them is rewarded with a return service in something else. So, female chimpanzees share food or look after other people’s cubs in exchange for help in grooming. In some cases, a third party may be interested in providing cooperation in the game. For example, in order to put an end to conflicts around the world, the United States of America and the European Union sometimes promise economic assistance to the participants in the confrontation as a reward for the peaceful resolution of the conflict. This is how the United States rewarded Israel and Egypt for their cooperation in signing the Camp David Peace Accords in 1978.

A more common method of solving the prisoner’s dilemma is punishment. It has direct effect. In L.A. Confidential, Sergeant Ed Exley promises Leroy Fontaine, one of the suspects, that if he agrees to become a government witness, he will receive a lighter sentence than the other two suspects, Ray Coates and Tyrone Jones. But Leroy knows that when he gets out of prison, the friends of these two can be waiting for him at large!

Punishment, which seems more natural in this context, is made possible by the fact that most of these games are part of a continuous interaction. Cheating may provide one player with a short-term advantage, but it will damage his relationship with another player and cost him much more in the long run. If the price is high enough, this is what can keep the player from cheating (In 2005, Robert Aumann received the Nobel Prize in Economics for his outstanding contribution to the development of a general theory of cooperation in repetitive games).

Baseball provides a clear example of this situation. In the American Baseball League, pitchers hit batters 11 percent to 17 percent more often than in the National Baseball League. According to University of the South Swaney professors Dag Drainen and John-Charles Bradbury, this is due to the designated hitter rule32. In the American Baseball League, pitchers are not hitters. Therefore, an American League pitcher who hits a batter may not fear retaliation from an opposing pitcher. The chance of hitting a pitcher is very small, but it’s four times more likely if he hit someone in the previous half of an inning. In this case, the fear of getting a retaliatory strike is obvious. Ace pitcher Kurt Schilling explained it this way: «Are you really ready to throw a ball at someone if you play against Randy Johnson?».

Randall David Randy Johnson is an American professional baseball player. The speed of his feeds often exceeded 160 km / h. — Approx. ed. At the time, Schilling was a pitcher for the Arizona Diamondbacks of the National Baseball League, and Cy Young Trophy winner Randy Johnson was his teammate. Source: Ken Rosenthal, “Mets Get Shot with Mighty Clemens at the Bat,” Sporting News, June 13, 2002.

In a situation where one player punishes another for cheating, the strategy is “an eye for an eye, a tooth for a tooth”. This strategy turned out to be a real discovery made during the most famous experiment with the prisoners’ dilemma. In the next section, you will learn about this experiment and its lessons.

Equal Response Strategy

In the early 1980s, University of Michigan political scientist Robert Axelrod challenged game theorists around the world to develop strategies for solving the prisoner’s dilemma in the form of computer programs. They were divided into pairs, each of which acted out the prisoner’s dilemma 150 times. Based on the points scored, a rating of the programs that took part in the tournament was made.

The winner was Anatoly Rapoport, professor of mathematics at the University of Toronto. His winning strategy turned out to be one of the simplest: «an eye for an eye, a tooth for a tooth.» For Robert Axelrod, this result was a big surprise, so he decided to hold another tournament, increasing the number of participants. Rapoport again submitted a program based on the same strategy — and won again.

The strategy of equal response is one of the variants of the rule of conduct «do to others as they do to you.»

The book Exodus (21:22-25) says: “When people fight, and they hit a pregnant woman, and she throws it away, but there is no [other] harm, then take from the [guilty] fine, which the husband of that woman will impose on him. women, and he must pay it through intermediaries; and if there is harm, then give life for soul, eye for eye, tooth for tooth, hand for hand, foot for foot, burning for burning, wound for wound, bruise for bruise. The New Testament preaches cooperative behavior. Matthew 5:38-39 says, “You have heard that it was said, an eye for an eye, and a tooth for a tooth. But I say to you: do not resist evil. But whoever strikes you on your right cheek, turn to him the other also.” Thus, we have moved from the rule “Do to others as they do to you” to the golden rule: “And as you want people to do to you, so you do to them” (Gospel of Luke 6:31). If people always adhered to the golden rule, the prisoner’s dilemma would simply not exist. Thinking more broadly, the conclusion becomes clear: while cooperation may reduce your winnings in a particular game, the possible rewards in life after death may make this strategy worthwhile even for an egoist. Do you think there is no afterlife? Blaise Pascal’s wager is that if you act on this assumption, the consequences can be disastrous, so it’s better to choose another option.

More precisely, this strategy involves cooperation in the first stage, after which the actions that the opponent took in the previous stage are repeated.

According to Robert Axelrod, a strategy of equal response relies on four principles that must be present in any effective strategy for the recurring prisoner’s dilemma: clarity, benevolence, retribution, and forgiveness. The strategy of equal response actions is very simple and clear: the opponent does not need to think long and hard about your next move or calculate it. At the heart of such a strategy is benevolence: it never initiates deception. There is an element of retaliation in this strategy: it doesn’t let cheating go unpunished. In addition, this strategy encourages forgiveness: the participants in the game do not hold grudges against each other for too long and are ready to resume cooperation.

One of the most impressive characteristics of the Equal Response Strategy is that it performed best in the entire tournament, although it did not (and could not) beat any of the competing strategies in a direct confrontation with them. At best, this strategy can only equalize the score with the opponent. Therefore, if Axelrod were to rate each game on a winner-take-all basis, the equal-response strategy would only have losses and draws, and thus would not win the entire tournament.

Since there is one winner for every loser, this inevitably results in one of the contestants ending up with more wins than losses, and others with more losses than wins. (The only exception is when each match ends in a draw).

However, Axelrod did not evaluate paired games between computer programs on a winner-take-all basis: his tournaments took into account such a factor as willingness to cooperate. The great advantage of this strategy is that it brings rivals closer together. In the worst case, this strategy can fail due to one betrayal, but then only a draw.

The equitable response strategy won this competition precisely because it stimulated cooperation without allowing exploitation. Other strategies were either too trust-based and open to exploitation, or too aggressive and encouraged players to knock each other out of the game.

And yet we believe that the strategy of equal response is erroneous. The slightest mistake or misinterpretation of the results leads to a complete failure of the strategy. This shortcoming was not so obvious in the artificial environment of competition between computer programs, since errors and misinterpretations were simply ruled out there. However, if this strategy is applied in the real world, errors and misconceptions are inevitable, and the result can be disastrous.

The problem with the strategy of equal response is that both sides of the confrontation repeat each other’s mistakes and delusions. One side punishes the other for betrayal, and this sets off a chain reaction. The opponent responds to the punishment with a return blow, which entails another punishment. In such a confrontation, there may not come a moment when one of the parties would accept punishment without a retaliatory strike.

Suppose Flood and Drescher are playing a strategy of equal response. At first, none of them goes to betrayal, so for a while everything goes well. Then, say, in the 11th round of the game, Flood mistakenly chooses the «betray» strategy or stops at the «cooperate» strategy, but Drescher mistakenly believes that Flood chose betrayal. Either way, Drescher will choose «betray» in round 12, but Flood will choose «cooperate» because Drescher chose cooperation in round 11. In the 13th round, they will switch roles. The situation where one of the players chooses cooperation and the other chooses betrayal will be repeated over and over again until the next mistake or delusion restores cooperation between rivals or forces each of them to choose betrayal.

Such cycles or retaliatory strikes often occur during actual conflicts between Israelis and Arabs in the Middle East, or between Catholics and Protestants in Northern Ireland, or between Hindus and Muslims in India. On the border between the states of West Virginia and Kentucky, there was a memorable feud between the Hatfields and the McCoys. Fiction, too, provides vivid examples of how such actions can lead to a never-ending cycle of retaliation, as in the case of the feud between the Grangerfords and the Shepherdsons in Mark Twain’s novel (M. Twain, The Adventures of Tom Sawyer and Huckleberry Finn / Translated by Nina Daruses. – M.: NIGMA, 2013).

“What was the quarrel about, Buck?” Because of the earth?

— I don’t know. May be.

So, who fired first? Grangerford or Shepherdson?

“God, how do I know!” After all, it was so long ago.

“And no one knows?”

— No, dad, I think he knows, and some of the old people know; they just don’t know what caused the quarrel in the very first time.

A strategy of equal response does not offer the possibility of stopping this vicious cycle. It is too retaliatory and not sufficiently stimulating for forgiveness. In the next versions of the competition, which arranged by Robert Axelrod, the possibility of errors and misconceptions was foreseen; as a result, other, more disinterested, strategies have shown their superiority over the strategy of equivalent responses.

In 2004, Graeme Kendall of the University of Nottingham organized a competition to celebrate the twentieth anniversary of the first tournament hosted by Robert Axelrod. The winner was a group of researchers from the University of Southampton. The Southampton group proposed a strategy consisting of 60 elements: 59 «warriors» and one «queen». All of these programs started with the same combination of characters so that the programs could recognize each other. The strategy was designed in such a way that the warrior programs sacrificed themselves, giving the queen an opportunity to succeed. In addition, warrior programs refused to cooperate with rival programs in order to reduce their score. Having an army of warriors willing to sacrifice themselves is indeed one way to increase your payoff, but it doesn’t tell us anything about how to solve the prisoner’s dilemma.

Here we can learn something even from monkeys. In one experiment with crested tamarins, one of the monkeys was given the opportunity to pull a lever so that the other could get food. However, in order to pull the lever, force had to be applied. Theoretically, it would be more beneficial for each monkey to do nothing while the partner pulls the lever. But tamarins have learned to cooperate in order to avoid retribution. Their cooperation continued until one of the monkeys committed treason twice in a row. This strategy is a variation of the tit-for-tat strategy, namely, two tit-for-tat.

Later experiments

The prisoner’s dilemma has been the subject of thousands of experiments involving different numbers of players, with repetitions and with different interpretations of the conditions of the game. Here are some important findings from these experiments.

The first and most important conclusion is that the players choose the cooperation strategy quite often, even if two players get paired only once. On average, almost half of the players prefer cooperation. The most impressive proof of this fact came from the game show Friend or Foe on the Game Show Network. Teams of two people were asked fairly simple questions. The money received by the participants for the correct answers went to the «trust fund»; over 105 episodes, such a fund accumulated between $200 and $16. In order to share this money, the two participants had to solve the following dilemma.

Each player had to write the word «friend» or «foe» on a piece of paper. If both wrote «friend», the money was divided equally. If one player wrote «enemy» and the other wrote «friend», the player who wrote «enemy» got all the winnings. But if both players wrote «enemy», neither of them got anything. Whatever the other side did, each player could get at least as much as his opponent (or even more) if he wrote «enemy» and not «friend.» Nevertheless, almost half of the participants in the show wrote the word «friend.» Even as the prize pool increased, the likelihood of players choosing to collaborate remained the same. People were equally willing to cooperate when three and five thousand dollars were at stake. Felix Oberholzer-Gee, Joel Waldfogel, Matthew White and John List came to the same conclusions in the course of research.

If you’re wondering if a TV show can be considered scientific research, consider this fact: The contestants were paid out over $700. This prisoner’s dilemma experiment had the best funding in the history of this kind of experiment. In addition, many important conclusions were drawn from the results of the quiz. It turned out that women are more willing to cooperate than men: 53,7 percent (in the first season — 47,5 percent). In the first season, the participants of the show did not have the opportunity to see the results of other competitions before making a decision. But in the second season, the results of the first 40 episodes were announced, which made it possible to see a pattern. The participants of the show learned from the experience of their predecessors. If the team consisted of two women, the coefficient of cooperation increased to 55 percent, and when the team included one woman and one man, this coefficient fell to 34,2 percent. For men, in this case, the coefficient of cooperation also decreased to 42,3 percent. In general, the willingness of the participants in the show to cooperate decreased by ten points.

When a group of participants in an experiment is paired several times, forming new pairs each time, the number of people who choose to collaborate decreases over time. However, this number does not reduce to zero; instead, a small group of participants in the experiment is formed, invariably preferring cooperation.

If the same couple plays the base Prisoner’s Dilemma game many times in a row, in most cases a very significant sequence of mutual cooperation is formed; this continues until one of the players, at the very end of the series of games, chooses the strategy of betrayal. This is exactly what happened in the first prisoner’s dilemma experiment. As soon as Merrill Flood and Melvin Drescher came up with this game, they challenged two of their colleagues to play it 100 times. In 60 rounds of the game, both participants chose a cooperation strategy. A long period of mutual cooperation lasted from round 83 to round 98, until in round 99 one of the players chose the strategy of betrayal.

If we follow the strict logic of game theory, then in reality this should not have happened. If the game repeats exactly 100 times, it is a series of games with simultaneous moves, which means that we can apply the logic of backward reasoning to it. Determine what will happen in the hundredth round. This is the last round of the game, so betrayal cannot be punished in subsequent rounds. In this case, according to the principle of dominating strategy, both players must choose the betrayal strategy in the last round. But as soon as such an assumption is accepted, the last one becomes, in fact, the 99th round. Although the players have one more round to go, the choice of betrayal strategy in round 99 cannot be penalized in round 100, as the choice made in this round is predetermined. Therefore, the logic of the dominant strategy applies to the 99th round as well. This reasoning can be continued up to the first round. However, in a real game, whether in a lab or in the real world, players tend to ignore this logic and try to capitalize on mutual cooperation. Behavior that at first glance may seem irrational (rejection of the dominant strategy) turns out to be the right choice, provided that other players behave equally irrationally.

Game theorists offer the following explanation for this phenomenon. There are people in this world who always do to others as they are treated; such people are ready to cooperate as long as others do the same. Suppose you are not one of those nice enough people. If you behaved as your personality type requires in a finite rep game, you would start by cheating. This would reveal your character to another player. In order to hide the truth (at least for a while), you will have to behave with dignity. Why would you do it? Let’s say you start the game doing the right thing. If the other player is not one of those who always pays in the same coin, he will think that you may be one of those few people who can be called decent. Temporary cooperation can bring certain benefits, so the other player, wanting to get this benefit, will try to respond to your decency in the same way. This will benefit you too. Of course, when you do this, you (as well as the other player) plan to switch to a betrayal strategy by the end of the game. However, at the initial stage of the game, both of you can maintain mutually beneficial cooperation. Although each player waits for the moment when they can take advantage of the other’s decency, this mutual deception benefits both of them.

In some experiments, instead of dividing the subjects into pairs and conducting a series of games with the prisoner’s dilemma, a large game is organized with the participation of the entire group. We want to give here a particularly interesting and instructive example. Professor Raymond Battalio of Texas A&M University organized the following game with 27 students38. All the students, supposedly the owners of the hypothetical companies, had to decide (simultaneously and independently of each other, writing their decision on a piece of paper) how much output their companies would produce: 1, which would help keep aggregate supply low and prices at high, or 2, which will allow you to receive additional income at the expense of others. Depending on the number of students who choose output 1, the money will be paid to them according to the following scheme:

This diagram is shown in a graphical form.

The game is designed so that students who choose 2 («betray») always get 50 cents more than students who choose 1 («cooperate»), but the more students choose 2, the smaller their total winnings. Suppose all 27 students start with choice 1; in this case, each of them will receive 1,08 dollars. Now imagine that one of them switches to option 2. There are 26 students left in the game who chose 1; they will each receive $1,04 (4 cents less than the original plan), but the student who changes strategy will receive $1,54 (46 cents more). This payoff distribution does not depend on the initial number of students intending to choose 1 over 2. In this case, option 2 is the dominant strategy. Each student who switches from strategy 1 to strategy 2 increases his payoff by 46 cents, but at the same time reduces the payoff of each of the remaining 26 participants in the game by 4 cents. When all the participants in the game start to act selfishly, trying to get the maximum winnings, each of them will receive 50 cents. If they could successfully combine their efforts and choose a course of action that would minimize their total gain, each of them would receive $1,08. How would you play this game?

When this game was played in practice (once without group discussion, another time with discussion in order to work out a concerted action), the number of students who were willing to cooperate and chose option 1 ranged from 3 to 14. In the last game, in in which the students joined forces, there were 4 of them. The total winnings amounted to $15,82, which was $13,34 less than in the round of the game in which the students managed to agree. “I will never trust anyone again in my life!” muttered the student who was most in favor of concerted action. But what was his choice? “Well, I chose 2,” he said. Yossarian would understand him.

Modern experiments with multiplayer prisoner’s dilemma games use a variant called the «common fund contribution game». Each player is given some initial amount, say $10. After that, he decides how much of this amount he will keep for himself and how much he will give to the general fund. Then the experimenter doubles the amount accumulated in the general fund and divides it equally among all participants in the game (both those who contributed to the general fund and those who kept the entire amount for themselves).

Suppose there are four players in the group: A, B, C, and D. Regardless of the actions of the other players, if A decides to contribute $1 to the pool, after doubling, the pool will increase by $2. But $1,5 will go to Players B, C, and D; Player A himself will receive only 50 cents. Therefore, player A will lose even more money if he increases his contribution to the general fund; on the contrary, he will benefit if he reduces the amount of this contribution. And this situation develops regardless of what contribution to the general fund other players make (and whether they make it at all). In other words, player A’s dominant strategy is to contribute nothing to the pool. The same is true for players B, C, and D. According to this logic, each participant in the game should expect to be able to become a «free rider» — to benefit from the actions of other players without making any contribution to the general fund. If all four players stick to their dominant strategy, the pool will be empty and each player will simply keep their original $10. If everyone tries to ride a «hare», the bus will not budge. On the other hand, if each player contributed their entire amount of $10 to the pool, after doubling, the pool would be $80, and each player’s share would be $20. But each of them has their own motives in such a game. This is their dilemma.

The game with contributions to the general fund is not only the object of laboratory experiments or theoretical research; it plays out in the real world in those cases of social interaction where a common good can only be created through the voluntary contribution of group members, but access to it cannot be denied to those members of the group who have not contributed to the common cause. Such a situation arises, for example, in cases of flood control or rational use of natural resources: dikes or dams cannot be built so that the flood waters flood the fields only of those villagers who did not take part in the construction of flood control structures. With regard to the rational use of gas and fish resources, in practice it is simply impossible in the future to exclude those who consumed them excessively in the past from these resources. This is what creates the dilemma in a multi-player game: each player is tempted to shy away from work or don’t contribute to the common fund, hoping to benefit from the contributions of other group members. When all players think this way, the result of joint actions is very small or non-existent, which negatively affects the entire group. This situation is so common that in all areas of social theory and social policy there is a need for a deep understanding of the methods of solving this dilemma.

Perhaps the most interesting variant of this game is when players are given the opportunity to punish those who violate the default social agreement of cooperation. However, the associated costs must be borne by all participants. After the game with contributions to the general fund is completed, the information about the contribution of each player is brought to the attention of all the others. Then the second stage of the game is played, during which each player can take actions aimed at reducing the payoff of other players, but this will cost him some amount (usually about 33 cents) for every dollar of the reduction he chose. In other words, if Player A decides to cut Player B’s payoff by three dollars, Player A’s payoff will be cut by one dollar. The money released as a result of such a reduction is not transferred to anyone else, but is returned to the experimenter’s fund.

The results of this experiment suggest that its participants tend to punish violators of social agreements (the so-called social deceivers) by exacting a significant amount of money from them. The prospect of punishment significantly increases the amount of contributions to the general fund in the first stage of the game. In all likelihood, punishment is an effective way to achieve cooperation that benefits the entire group. However, the fact that people actually resort to this method seems surprising only at first glance. Punishing others at your own expense is already a contribution to the common good. This is a dominated strategy, but if it encourages the deceiver to behave better in the future, it benefits the whole group, and the punisher receives only a small share of this benefit. Therefore, the punishment must be the result of something more than a purely selfish calculation. And indeed it is. During a series of experiments, positron emission tomography of the brains of players was carried out (This is discussed in the article by Thomas Hayden, “Why We Need Nosy Parkers,” US News and World Report, June 13, 2005. More information can be found here: DJ de Quervain , U. Fischbacher, V. Treyer, M. Schellhammer, U. Schnyder, and E. Fehr, “The Neural Basis of Altruistic Punishment,” Science 305, no. 5688 (August 27, 2004): 1254–1258.). It turned out that the use of punishment activates the dorsal striatum — the area of ​​​​the brain that is responsible for pleasure and satisfaction. In other words, people really get psychological benefit or pleasure by punishing the violators of collective agreements. In all likelihood, this instinct has deep biological roots and has been selected because it provides an evolutionary advantage.

In his book Passions Within Reason (New York: WW Norton, 1988), Cornell University economist Robert Frank argues that emotions, in particular guilt and love, have also evolved, and social values ​​such as trust and honesty were shaped and reinforced as a counterweight to a person’s transient tendency to deceive their fellow human beings, and to provide the long-term benefits of cooperation. And Robert Wright, in The Non-Zero-Sum Game (Nonzero. New York: Pantheon, 2000), develops the idea that the mechanisms that provide mutually beneficial outcomes in a non-zero-sum game explain much of human cultural and social evolution.

How to achieve cooperation

Based on all these examples and experiments, a number of prerequisites and strategies for successful cooperation can be identified. The following is a systematic description of these concepts, as well as examples of their application in real life.

A successful punishment system must satisfy a number of requirements.

Deception definition. You cannot punish someone for cheating without establishing the fact of cheating. If the deception is detected quickly and accurately, the punishment can be immediate and adequate. This reduces the gains from deception and increases the costs associated with it, and therefore increases the likelihood of successful cooperation. So, airlines constantly monitor the prices of competitors; if American Airlines wanted to cut their fares from New York to Chicago, United Airlines could do the same in no more than five minutes. However, companies wishing to lower their prices can do so through secret deals with customers, or hide price cuts in complex deals with many different conditions such as delivery times, product quality, warranties, and so on. In the most extreme case, each company can track only its own sales and profits, which also depend on a number of random factors, including the actions of other companies. For example, a company’s sales may be driven by fluctuations in demand, not just covert price cuts by a competitor. In this case, the process of detecting deception and punishing for it is not only slow, but also not quite correct, which increases the temptation to deceive again.

In conclusion, it should be noted that when two or three companies operate in the same market segment at the same time, they must establish not only the fact of fraud, but also who committed this fraud. Otherwise, the punishment is not targeted, but vague and can unleash a price war that will hurt everyone without exception.

The nature of punishment. Next, you need to decide what exactly the punishment should be. Sometimes players have the opportunity to punish other participants in the game literally immediately after the discovery of deception, even in the process of a single interaction. As we noted when discussing the dilemma that arose before the heroes of the film «LA Confidential», the friends of Ray Coates and Tyrone Jones will wait until Leroy Fontaine is released from prison and punish him for becoming a state witness in order to get a lighter sentence. In an experiment involving students at the University of Texas, when they could identify who violated the Option 1 agreement, they were able to apply social sanctions, such as ostracism, to the deceivers. In this case, few students took that risk for 50 cents.

Other types of punishment arise from the very structure of the game. This typically occurs in repetitive games, where gaining from cheating in one round of the game results in a loss in future rounds. Whether this is enough to keep the player who allows this possibility from cheating depends on the size of the gain and loss, as well as the importance of future events in relation to the present. We will analyze this aspect next.

Clarity. A potential deceiver must understand the limits of acceptable behavior as well as the consequences of deception. If these aspects of the game are too complex, the player may cheat by mistake or because he failed to calculate his moves and played intuitively. Let’s take this example as an illustration. Suppose Rainbow’s End and BB Lean are constantly playing the price-setting game and RE concludes that if RE’s average discounted earnings over the past 17 months are 10 percent below the real average rate of return on industrial capital over the same period, this will mean that the BB went on a deception. BB knows nothing about this rule directly; the specialists of this company will have to draw indirect conclusions about which rule is applied in RE by observing the actions of this company. However, the rule stated here may be too complicated for the BB, so it’s not such a good way to keep the BB from cheating. But the strategy of equivalent response actions is completely understandable: if BB goes on a deception, RE will immediately respond to this by lowering prices.

Inevitability. Players must be confident that betrayal will be punished and cooperation rewarded. Lack of such confidence is a major problem with some international agreements, such as the World Trade Organization (WTO) Trade Restriction Agreements. When one country complains that another has violated the agreement, the WTO initiates an administrative process that drags on for months or even years, and the punishment depends not on the actual circumstances of the case, but on the requirements of international politics and diplomacy. It is unlikely that such measures to ensure the implementation of agreements can be called effective.

The size. How severe should the punishment be? Apparently, there should be no restrictions here. If the punishment is severe enough to keep players from cheating, there simply won’t be a need to apply it. Therefore, it is possible to introduce a sufficiently serious punishment, which will really become a deterrent. For example, the WTO would include in its charter a clause on the use of nuclear weapons against a country, which would thwart the organization’s attempts to keep protectionist tariffs at the low levels agreed upon by WTO members. You probably shuddered in horror — partly because you allow the possibility of a mistake that will entail such consequences. If mistakes are indeed possible (as is the case in real life), punishment should be set at the minimum level that ensures successful deterrence under all circumstances. In the most extreme cases, it may even be advisable to forgive individual cases of violation of agreements. A company that is struggling to survive can be allowed to cut prices slightly without fear of retaliation from competitors.

Repeatability. Let’s go back to the price game between Rainbow’s End and BB Lean. Suppose they manage to keep prices at the optimal level for both companies year after year — $ 80. But RE managers are considering lowering the price to $70. According to their calculations, this would bring RE an additional profit of $110 — $000 = $72. However, this can destroy trust between companies. RE should understand that in the future, BB will also decide to lower the price to $000, and both companies will only be able to earn $38 a year. Had RE stuck to the original deal, both companies would have earned $000 each. Therefore, if RE lowers the price, it will cost her $70 — $70 = $000 each future year. Is a one-time win of $72 worth losing $000 a year for years to come?

The interest rate is one of the main factors on which the balance between the present and the future depends. Let’s say the interest rate is 10 percent per year. RE can deposit its $38 into an account and earn $3800 a year. This fully covers the losses of the company in the amount of two thousand dollars for each successive year. Therefore, it is profitable for RE to deceive a competitor. But if the interest rate is only five percent per year, then the $38 will earn the company only $1900 each subsequent year, which is less than the $2 loss that RE incurs because of the breach of agreement, so the company decides not to cut the price. The interest rate at which equilibrium will be reached is 38∕0,0526 = 5,26, or XNUMX percent per year.

The basic idea behind all of this reasoning goes like this: if the interest rate is low enough, the future has a relatively higher value. For example, if the interest rate is 100 percent, the future has a low value relative to the present: a year later, one dollar is only 50 cents now, because in a year you can turn that 50 cents into one dollar, earning another 50 cents in interest for this year. But if the interest rate is zero, then a year later one dollar will be worth as much as it is now.

If you read the financial press, you’ve probably come across this phrase many times: «Interest rates and bond prices move in opposite directions.» The lower the interest rate, the higher the bond price. Bonds are essentially a promise of income in the future, so they represent the importance of the future. This is another way to keep the role of interest rates in mind.

In our example, the more realistic interest rate is five percent, so the temptation for each company to cut the price $10 below their mutually optimal price of $80 is well balanced, and concurrence in a repetitive game is not always possible. In Chapter 4, we will see how much the price can fall if the shadow of the future does not hang over the participants in the game, and the temptation to deceive the opponent becomes irresistible.

Another important factor to consider in games of this nature is the likelihood of further interaction. If the CU is a temporary fashion item that may not sell for the next year, the prospect of future losses will not quell the temptation to cheat a competitor this year.

However, Rainbow’s End and BB Lean sell many other merchandise besides this CU. Won’t an attempt to lower the price per dollar lead to a competitor’s response to all other goods? And is the prospect of such a serious retaliatory move not enough to refrain from a strategy of betrayal? Alas, it is not so easy to achieve sustainable cooperation between companies, based on the practical value of their interaction across the entire range of products. The prospect of retaliation across all commodities is inextricably linked to the prospect of immediate gains from a series of deceptions across all of these, not just one. If the payoff tables for all products were identical, profits and losses would increase exactly as many times as the number of products each company has, and this would not affect the grand total. Therefore, successful punishment in the dilemma with many goods is not so obvious and depends rather on differences between the goods themselves.

The third important point related to this topic is the projected change in business volume over an extended period. Such a change can have two aspects: a steady rise or fall and fluctuations. If the business continues to grow, a company that is currently considering betrayal needs to consider that it risks more serious losses in the future due to loss of cooperation. On the contrary, if the volume of business is reduced, companies are more tempted to betray and get the most out of it now, knowing that their future is at stake. In terms of hesitation, companies are more likely to cheat competitors when there is a temporary upswing: in this case, cheating will provide them with more profit now, while the negative impact of the loss of cooperation will hit them only in the future, when the volume of business will be at an average level. Therefore, it can be assumed that price wars should occur during a period of high demand. But this is not always the case. If a period of low demand occurs due to a general economic downturn, consumers have reduced incomes and they shop more cautiously: their loyalty to a particular company may change, and their reaction to price differences may accelerate. In this scenario, the price-cutting company has the right to expect that it will be able to attract more customers to its side by poaching them from a competitor, and therefore, to receive more profit due to such deception in the near future.

And finally, the composition of the group of players plays a big role in achieving cooperation. If it is stable and is expected to remain so in the future, this helps to maintain cooperation. New players who are not interested in cooperation or who do not have a history of participating in this cooperation agreement are less likely to adhere to it. If an existing group of players expects new players to emerge in the near future who will violate the default cooperation agreement, this increases their willingness to deceive competitors and take advantage of at least some additional benefits now.

Kant’s categorical imperative and the prisoners’ dilemma

Sometimes one hears claims that participants in the prisoner’s dilemma choose to cooperate because they are making a decision not only for their own sake, but also for the sake of their opponent. In fact, this is a delusion, but people act in this way, as if this is really the case.

Each participant in the game wants the other player to choose cooperation, and decides for himself that his opponent adheres to the same decision-making logic as he does. According to such a participant in the game, the other player must draw the same logical conclusions that he himself did. Therefore, if a player chooses to cooperate, he assumes that the other player will do the same; if he chooses the strategy of betrayal, he concludes that this will cause the other player to betray as well. This is reminiscent of Kant’s categorical imperative: «Act only according to that maxim which you would like to see as a universal law.»

Of course, this is very far from the truth. The actions of one player have no effect on the other player. And yet people feel that their actions, even if they are subtle, can influence the choice of other participants in the game.

The power of this way of thinking was shown in an experiment conducted by Eldar Shafir and Amos Tversky with students at Princeton University. The researchers asked the students to play a prisoner’s dilemma game. However, unlike the usual pattern of playing such a game, in some cases they told one player what the other had done. When students were told that the other player had chosen a betrayal strategy, only three percent responded with cooperation. When players were told that their opponent had chosen to cooperate, this increased the proportion of players who chose to cooperate to 16 percent. Consequently, the vast majority of students still preferred to act on the basis of their own interests. Nevertheless, many of them were ready to return cooperation for cooperation, even if for the sake of this they had to sacrifice something.

What do you think happened when the students weren’t told about the choice of other players? Is the share of participants in the game willing to cooperate still between 3 and 16 percent? No, it has increased to 37 percent. At first glance, this may seem to make no sense. If you didn’t choose to cooperate, neither when you knew the other side chose the strategy of betrayal, nor when you knew the other side chose the strategy of cooperation, why on earth would you choose to cooperate if you don’t know what the other side did?

Eldar Shafir and Amos Tversky call this «quasi-magical thinking» — the belief that by taking action you can influence the actions of the other side. When people are told what the other side has done, they realize they cannot change it. But if they don’t know it, they feel like their actions might have some impact, or that the other side will somehow apply the same line of reasoning and come to the same conclusions as they do. Since the option «cooperate, cooperate» is more preferable than the option «betray, betray», they choose cooperation.

It should be noted that such logic is completely illogical. Your thoughts and actions have no influence on the thoughts and actions of other people. They have to make decisions without knowing what you think and what actions you take. Nevertheless, the fact remains that if the members of a society had such quasi-magical thinking, they would not fall prey to the prisoner’s dilemma and would benefit more from interacting with each other. Can human society deliberately instill such a mindset in its members for this great purpose?

Examples of Prisoner’s Dilemma in Business

Armed in the previous sections with a set of tools such as experimental data and theoretical concepts, let’s go out of the laboratory into the real world, analyze some examples of the prisoner’s dilemma and try to find a solution to it.

Let’s start with the dilemma faced by competing companies in an industry. Their common interests are best served by monopolizing or cartelizing this industry and keeping prices high. However, each of these companies will be able to achieve more for themselves if they violate the agreement and go to lower prices in order to take business away from competitors. What can these companies do in such a situation? Some of the factors contributing to successful collusion between companies (such as increased demand or the lack of a new player to disrupt the situation) will remain — at least in part — out of their control. However, companies can set the stage for successful deception detection and develop an effective punishment strategy.

Such collusion is easier to achieve if companies regularly meet with their representatives and exchange information. In this case, they will try to reach a compromise during the negotiation process on what methods of work are considered acceptable and what constitutes fraud. The negotiation process and its outcome contribute to bringing clarity to the current situation. If an event occurs that can be regarded as a fraud (in the absence of evidence to the contrary), one more meeting will make it clear that this is in fact an innocent accidental mistake or a deliberate fraud. Therefore, this approach can prevent unnecessary punishments. In addition, the meeting helps to develop adequate actions in case punishment is still needed.

The problem is that successfully resolving such a dilemma within a group of industry participants is detrimental to the public interest. Consumers are forced to pay inflated prices, and companies do not bring part of their inventory to the market in order to keep prices high. As Adam Smith said, “people of the same profession rarely get together even for fun, but their meetings end in a conspiracy against society or a plan to increase prices” (Smith A. Research on the nature and causes of the wealth of nations. M .: Eksmo, 2007.). Governments that seek to protect the interests of society intervene in such situations and enact antitrust laws that prohibit companies from colluding.

Not all governments are equally concerned about the public interest. Some of them serve the interests of manufacturers and turn a blind eye to the formation of cartels or even facilitate it. We don’t name these governments here, lest they ban our book in their countries!

In the United States, the Sherman Antitrust Act prohibits any conspiracy to restrict trade and monopolize an industry. Price collusion or collusion to secure market share are the most common examples of such collusion. The U.S. Supreme Court has not only banned formal agreements of this kind, but also ruled that any explicit or implicit agreement between companies that leads to price fixing is a violation of the Sherman Act, regardless of the original intentions. Violation of antitrust laws can lead not only to corporate fines, but also to imprisonment of top company officials.

All this does not mean that companies do not try to use illegal methods and get away with it. In 1996, Archer Daniels Midland (ADM), a leading US agricultural processing company, and its Japanese partner, Ajinomoto, were caught in such a conspiracy. They entered into market sharing and pricing agreements for various products such as lysine (which is made from corn and is used to fortify chicken and pig feed). The purpose of the agreements was to keep prices high at the expense of consumers. These companies were guided in their actions by this principle: «Competitors are our friends, and consumers are our enemies.» Company misconduct was discovered when one of the ADM negotiators became an FBI informant and made many audio recordings and even several video recordings of meetings between company representatives (Kurt Eichenwald describes this case brilliantly and with humor in The Informant. See: Eikhenwald K. Informator, St. Petersburg: Azbuka-klassika, 2009).

The most famous case of antitrust violation, used as a case study in business schools, occurred in the market for large turbines for power generation. In the 1950s, there were three companies operating in the American turbine market: GE (General Electric (GE) — an American diversified corporation, a manufacturer of many types of equipment. Note ed.) was the largest of them — it accounted for 60 percent of the market; Westinghouse is about 30 percent of the market and Allied-Chalmers is about 10 percent. They retained these market shares and maintained high prices through an elaborate scheme of coordination. Here’s how she worked. Electricity companies announced a tender for the turbines they were going to buy. If the invitation to tender came from the 1st to the 17th day of the lunar month, Westinghouse and Allied-Chalmers would have to bid very high prices for their turbines so that their bids would surely lose the tender, and GE would win by mutual agreement. tender by offering the lowest price (which was still a monopoly price providing high profits). Similarly, Westinghouse became a known winner if the invitation to tender was received from the 18th to the 25th lunar day, and Allied-Chalmers from the 26th to the 28th lunar day. Because the electricity companies sent invitations to tender outside of the lunar calendar, over time, each of the producers gained a share of the market. Any attempt to break the agreement would immediately be noticed by competitors. However, since it never even occurred to the Justice Department officials to tie the winners of the tender to the lunar cycle, this collusion was protected from the threat of detection. In the end, the authorities finally figured out the situation, some of the leaders of these three companies were sentenced to prison terms, and such a lucrative conspiracy failed. Subsequently, other attempts were made to apply various collusive schemes.

In 1996-1997, a variant of the turbine scheme was used in the bidding process for a mobile frequency license auction. A company that wanted to obtain a license to use a frequency in a given city communicated to others its decision to compete for that license by listing that city’s telephone code as the last three digits of the bid. Other companies gave her the opportunity to win the auction. Provided that the same group of companies participates in a large number of such auctions over a long period, and that the antimonopoly authorities do not discover the scheme, it can operate for a long time.

More often, however, companies in an industry attempt to reach an implicit or default agreement without direct contact. This eliminates the risk of criminal prosecution for violating antimonopoly laws, although there are other measures at the disposal of the antimonopoly authorities aimed at destroying even tacit collusion. The complexity of this situation lies in the fact that such an agreement is not entirely clear to its participants, and attempts to violate it are difficult to detect. However, companies are able to find a way to overcome both.

Instead of setting fixed prices, they may agree to divide the market along territorial lines, product categories, or any other such characteristic. In this case, the fraud is much easier to spot: your salespeople will immediately know that a competing company has taken a part of the market from you.

The process of detecting price cuts (especially in retail) can be simplified and retaliation made quick and automatic through schemes such as promises to match or even outperform competitors and create the best customer experience. Many companies selling electronic appliances and other household products publicly state that they will sell their product at a price below that of any competitor. Some of them even promise that if you find a lower price for the same product within a month of buying it, they will refund the difference or even double the amount. At first glance, it might seem that such strategies stimulate competition by guaranteeing low prices. However, even a superficial analysis of this situation from the point of view of game theory shows that in reality these strategies can have exactly the opposite effect. Let’s say Rainbow’s End and BB Lean have adopted this strategy and agreed to charge $80 for their US dollars. Now each of them knows that if they lower the price to $70, the competitor will immediately know about it. In fact, the biggest trick of this strategy is that it shifts the task of uncovering the deception to consumers, who are most interested in finding low prices. And the potential deal breaker also knows that his competitor can immediately retaliate by lowering their prices without even having to wait for a catalog to be released next year.

Promises to match or even outperform competitors in terms of price or product quality can be quite sophisticated and evasive. In a competition between Pratt & Whitney (P&W) and Rolls-Royce (RR) to supply jet aircraft engines for the Boeing 757 and Boeing 767 aircraft, P&W promised all potential buyers that its engines would be eight percent more fuel efficient. fuel than RR engines, otherwise P&W will refund the difference in fuel costs.

The principle of most favored nation states that the seller will offer the best price to everyone, not to selected consumers. If this promise is taken at face value, it may seem that manufacturers guarantee low prices. But let’s take a deeper look at the situation. This principle means that the manufacturer cannot compete by offering selective discounts in order to lure customers away from a competitor, while at the same time selling products at the same high prices to regular customers. Such a manufacturer is forced to go for a general price reduction, which costs him more, because it leads to a reduction in the profit margin on all sales. As you can see, this creates a clear advantage for the cartel: there is less gain from deception, which makes it more likely that the cartel will retain its influence.

The Federal Trade Commission, a branch of the US antitrust enforcement system, has been investigating DuPont, Ethyl, and other gasoline antiknock manufacturers. Based on the results of the investigation, the commission came to the conclusion that such practices are anti-competitive and prohibited companies from including such clauses in contracts with clients.

This decision was not made without controversy. Commission Chairman James Miller expressed a dissenting opinion. He wrote that «perhaps such items reduce the cost to buyers of finding the best price and help them find the best combination of price and value.» More information on this can be found here: In the matter of Ethyl Corporation et al. // FTC Docket 9128, FTC Decisions 101. — January — June 1983. — P. 425-686.

Tragedy of the Commons

At the beginning of this chapter, we mentioned the overexploitation of fish resources. Such problems are due to the fact that each person seeks to consume as many resources as possible for personal gain, shifting the consequences of their actions to all other people or to future generations. University of California professor Garrett Hardin called this phenomenon the tragedy of the commons and used as one example the overgrazing of common grazing lands in England in the 47th and XNUMXth centuriesXNUMX. The problem of overuse of common resources has become widely known under the very name that Hardin used — «tragedy of the commons.» A more significant example of this problem has now emerged — global warming. No one benefits personally from reducing carbon emissions, but if everyone pursues only their own interests, everyone will suffer.

This is a prisoner’s dilemma with many participants, similar to the one faced by Yossarian in the novel Catch-22, trying not to risk his life during the war. Undoubtedly, society is aware of the consequences of not resolving such dilemmas and is making attempts to correct the situation. But how do you determine how successful they are?

Indiana State University political scientist Elinor Ostrom, along with her colleagues and students, has done extensive research on trying to solve the dilemma of the tragedy of the commons—in other words, the problem of rational consumption and conservation of public resources, as well as preventing overexploitation and rapid depletion of natural resources. The researchers studied both successful and unsuccessful attempts and identified the prerequisites for effective coordination of efforts in this area.

First, there must be clear rules regarding the members of the group of players in a given game — those who have the right to use the relevant resources. Typically, this criterion is geographic location or place of permanent residence, as well as ethnicity, skills or membership, which can be sold at auction or provided for an entrance fee.

The introduction of property rights is what actually happened in England, where there were two waves of «fencing» when public lands were given to private owners: first by local aristocrats during the Tudor period, and then by laws passed by Parliament in the XNUMXth and XNUMXth centuries. When the land is privately owned, an invisible hand closes the gate only as far as necessary. The landowner imposes grazing fees in order to increase rental income, and this leads to a reduction in pasture use. This approach improves overall economic efficiency, but changes the distribution of income: payment for grazing makes the owner of the pasture even richer, and the pastoralists poorer. But even if we do not take into account the problem of income distribution, this plan is still not always feasible. In the absence of international governance, it is very difficult to define and enforce ownership of international waters or of sulfur and carbon dioxide emissions: fish and pollutants move from one ocean to another; the wind carries sulfur dioxide across borders, and carbon dioxide rises from any country into one atmosphere. That is why issues such as whale fishing, acid rain and global warming must be addressed through direct surveillance methods, but it is not an easy task to achieve the conclusion of appropriate international agreements.

Second, there should be clear rules defining what is allowed and what is not allowed. Such rules include restrictions on the period of use (open or closed season for hunting or fishing), location (fixed area or rotation of areas for fishing), technology (size of fishing nets), and finally on the amount or proportion of the resource (for example , the amount of firewood that one person is allowed to collect and take out of the forest).

Thirdly, a transparent and understandable system of penalties for violation of the listed rules should be introduced for all parties. This system need not be a detailed set of rules; general norms of behavior in a stable community can be no less transparent and effective. Sanctions against rule breakers can range from verbal reprimand or social ostracism to fines, loss of future rights, and in some extreme cases, imprisonment. The severity of each type of punishment can also be different, but it is important to adhere to the principle of gradually tightening the punishment. In the case of the first possible violation of the rules, the most commonly used method is to establish direct contact with the violator and demand that he solve the problem that has arisen. For the first or second infringement, fines are fairly low, and only increase if the infringements continue or become more serious.

Fourth, an effective violation detection system must be introduced. The best method is for violations to be detected automatically in the daily activities of the player. For example, a fishing company that has both good and bad lots can rotate the rights to the good lots. An angler given a good fishing spot will immediately notice if the intruder is using that spot; he has a personal interest in reporting the violation to the rest of the group and in obtaining adequate sanctions against the violator. Another example is the introduction of requirements that harvesting in the forest or other common areas should be done only by groups: this ensures mutual control and eliminates the need to hire guards.

In some cases, the rules that define permitted actions need to be developed considering whether there are realistic ways to detect violations. For example, it is sometimes difficult to control the size of the catch, even if the angler has the best intentions. Therefore, fishing rules based on the number of fish caught are rarely used. Quotas for the quantity of a given resource are more effective when the quantity is easier to track and can be accurately measured, such as water supplied from reservoirs and harvested forest products.

Fifth, when such rules and systems for their enforcement are created, it is very important that information about this is available to potential users of resources. Although each user may be tempted to break the rules once the rules are in place, they all have an equal interest in creating an effective system of rules. When creating it, users can apply their knowledge of the relevant resource and the technology of its use, knowledge of the practical feasibility of measures to detect violations, as well as an understanding of the degree of confidence of their group members in various types of sanctions for violation of the rules. As practice shows, in the case of centralized or hierarchical management, many of these aspects are interpreted incorrectly, so such a system for regulating common resources is ineffective.

In general, Elinor Ostrom and her colleagues are optimistic about the possibility of effectively solving many problems of collective activity through the use of information about local conditions and systems of rules, but at the same time she warns against excessive perfectionism: “This dilemma will never disappear, even in those systems that work best. <...> No monitoring or application of sanctions will be able to reduce the temptation to zero. Instead of thinking about overcoming the tragedy of the commons, it is necessary to create systems of self-organization that will cope with this problem better than others.

Harsh laws of nature

As one would expect, the prisoner’s dilemma arises not only in humans, but also in other biological species. In situations such as building a nest, foraging for food, and escaping predators, an animal may act either selfishly (for its own benefit and that of close relatives) or for the benefit of the entire group. What circumstances contribute to effective collective action? Evolutionary biologists have done research on this subject and have come across some amazing examples and insights. Here is one such example.

When the British biologist John Haldane was asked if he would risk his life for another person, he replied: “For two of his brothers or for eight cousins, yes.” You share half of your genes with your sibling (other than identical twins) and one-eighth with your cousin; therefore, such actions increase the expected number of copies of your genes that will be passed on to the next generation. Such behavior is perfectly justified from a biological point of view: the process of evolution gives preference to it. Such a purely biological basis for cooperative behavior among close relatives explains the surprising and complex interaction that is observed in ant colonies and swarms of bees.

In the absence of similar genetic links, altruism among animals is rare. However, mutual altruism arises and persists between members of a group of animals with a lower genetic identity if the interaction between these animals is stable and long-term. Here is a slightly scary but interesting example: vampire bats living in Costa Rica live in colonies of about ten individuals, but hunt separately. Every day can be good for some vampires and bad for others. Successfully hunted vampires return to the hollow tree in which their group dwells, and can share the prey by vomiting the blood they brought from the hunt. A bat that does not receive its portion of blood for three days is threatened with death. Vampire mouse colonies employ effective methods of mutual insurance against such a threat through prey sharing.

University of Maryland biologist Gerald Wilkinson investigated the underlying causes of this behavior by collecting bats from different locations and grouping them into one group. He systematically drew blood from some of them and observed whether other mice shared blood with them. As a result, Wilkinson discovered that mice only share blood if one of the group members is on the verge of death, but not before. In all likelihood, bats are able to distinguish real need from temporary bad luck. Even more interesting is that only those mice who knew each other from previous groups shared blood among themselves, and that they were more willing to share with those members of the group who had already come to their aid. In other words, bats are able to identify other mice and remember their behavior in the past, which leads to the formation of an effective system of mutual altruism.

Case Study: The Early Bird Dilemma

The Galapagos Islands are the habitat of Darwin’s finches. These volcanic islands have very difficult conditions for life, so evolutionary selection is quite strong there. Changing the size of a finch’s beak even by a millimeter can be a decisive factor in the fight for survival.

Each island has its own food sources, and the finch’s beak reflects these differences. On the large island of Daphne, the main food source is the cactus. On this island, birds with the aptly named «cactus finches» have evolved so that their beaks are ideally suited to collecting pollen and nectar from cactus flowers.

These birds do not play a conscious game against each other. However, each adaptation of their beak can be considered their strategy. Strategies that give finches a foraging advantage ensure survival, selection of mating partners, and more offspring. The finch’s beak is the result of this combination of natural and sexual selection.

However, even if at first glance everything is as good as possible, genetics sometimes throws up surprises. An old saying goes: who gets up early, God gives him. On Daphne Island, it was the finch that became the early bird that gets the nectar. Instead of waiting until nine o’clock in the morning for the cactus flowers to open on their own, some finches tried something new. They themselves opened the flowers of the cactus in order to profit before anyone else.

At first glance, this may seem to give the finches an advantage over rivals arriving a little later. The only problem is that when the birds open the flower, they often break off the stigma. Jonathan Weiner explains it this way:

The stigma is the tip of a thin-walled tuye that protrudes like a long, straight straw at the center of each flower. When the stigma is broken, the flower becomes sterile. The male gametes contained in the pollen cannot combine with the female gametes of the flower. As a result, the cactus flower withers without setting fruit51.

If a cactus flower fades, the main food source of cactus finches disappears. One can guess what the result of such a strategy would be: no nectar, no pollen, no seeds, no fruits — and as a result, no cactus finches. Does this mean that in the course of evolution, finches faced a prisoner’s dilemma, due to which this species may become extinct?

Example Analysis

This is not entirely true for two reasons. Finches occupy a certain territory, which means that these birds (and their descendants), if there are no cacti left in their habitat, may die. In that sense, destroying a food source for the birds that will inhabit the area next year is not worth the extra pollen. Consequently, birds with such a deviation from the norm will not have an advantage over others. However, the conclusion will be different if this strategy is widely adopted. The finches will expand their foraging, and even if there are birds left to wait for the flowers to open naturally, it still won’t save the stigmas of their cacti. After that, famine will set in, and then those birds that have had strong advantages from the very beginning will have the best chance of surviving. In this case, an additional sip of nectar will play a decisive role.

Here we see adaptation according to the principle of tumor cells. If the population remains small, it may become extinct. But if the population grows too large, this strategy becomes the best. Since such a strategy becomes profitable even on a relative scale, the only way to stop this process is to destroy the entire population and start all over again. If there are no finches left on Daphne Island, there will be no one else to break the stigmas of flowers and the cacti will begin to bloom again. When a couple of lucky finches arrive on the island, they will have the opportunity to repeat the whole process from the very beginning.

The game we are talking about here is very reminiscent of the prisoners’ dilemma, this is one of the variants of the game «hunting for a deer», which was analyzed at one time by the philosopher Jean-Jacques Rousseau (There is another interpretation of the game «hunting for a deer», which was described by Jean-Jacques Jacques Rousseau We will return to it in the next chapter). If during such a hunt all participants work together to catch a deer, they succeed. The problem arises only when a hare runs past one of the hunters. If there are too many hunters chasing the hare, there are not enough left to catch the deer. In this case, it is better for everyone to start chasing hares. In deer hunting, the optimal strategy is to pursue a deer if and only if you are sure that all other hunters will do the same. You have no reason not to chase a deer, unless you don’t trust other hunters.

As a result, we get a game of trust. There are two ways to play this game: everyone works together — and life is beautiful, or everyone pursues only their own interests — and life becomes terrible, cruel and short. This is not the classic prisoner’s dilemma, in which each person has an incentive to cheat the other players, no matter what actions they take. In this case, there is no reason to break the rules if you are sure that others do the same. But do you trust them? Even if you do, can you rely on them to believe you? Or can you trust that they will trust that you trust them? As Franklin Roosevelt said (in a different context), we have nothing to fear but fear itself.

To put your knowledge of the prisoner’s dilemma into practice, review the following case studies from Chapter 14: How much is one dollar worth? and «The Problem of King Lear».

Chapter 4

Role of coordination

Fred and Barney are rabbit hunters living in the Stone Age. One evening, when they were drinking together, a conversation began between them about business. After exchanging views, they realized that by joining their efforts, they could hunt a much larger animal, such as deer or bison. One who hunts alone cannot expect to be able to overwhelm such a large animal as a deer or a bison. But if the hunters united, each day of hunting deer or bison would bring six times more meat than a day of hunting rabbits alone. Such cooperation brings great advantages: each hunter will get three times more meat from hunting a large game than from hunting rabbits.

Fred and Barney agreed to hunt big game the next day and returned to their caves. Unfortunately, they had drunk too much the day before and both of them forgot which animal they should hunt — deer or bison. The hunting areas for these animals are in opposite directions. There were no mobile phones in those days, and this was all before Fred and Barney were neighbors, so they couldn’t get to each other’s cave quickly to figure out where to go. The next morning, each had to decide for himself.

In order to decide where to go, the two hunters will have to play a game with simultaneous moves. If we denote the amount of meat that each hunter receives per day of hunting rabbits as one unit, then the share of each of them in the case of successful coordination of efforts in hunting deer or buffalo will be three units. Therefore, the paytable in this game looks like this:

This game is quite different from the prisoner’s dilemma discussed in the previous chapter. Let’s analyze the most important difference. Fred’s optimal choice depends on what Barney does, and vice versa. There is no optimal strategy for either player, regardless of the actions of the other; unlike the prisoner’s dilemma, there are no dominant strategies in this game. Therefore, each player must analyze the possible choice of the other player and, taking this into account, seek his own optimal strategy.

Fred thinks like this: “If Barney goes to where the deer graze, then I will get a large share of the prey if I go there, and I will not get anything if I go to the land of the buffalo. If Barney goes to the land of the buffalo, it should be the other way around. Rather than take the risk of going to one of these areas and finding that Barney has gone the other way, shouldn’t I hunt rabbits myself, as I have always done, even though it will bring me less meat? In other words, shouldn’t I take one for sure instead of risking three or nothing? It depends on what I think Barney will do, so I need to put myself in his shoes and think about what he thinks. But he also wonders what I will do, and tries to put himself in my place! Is there an end to these repetitive reflections on reflections?”

Trying to find the square of a circle

John Nash’s Perfect Equilibrium was developed as a theoretical tool for finding the «squaring the circle» of thinking about thinking about other players’ choices in strategy games.

For those readers who have not seen Russell Crowe’s A Beautiful Mind as Nash, or who have not read the best-selling biography of John Nash by the same title, Sylvia Nazar, we want to clarify the following. John Nash developed the fundamental concept of equilibrium in games in 1950, after which he wrote many more works of great importance for mathematics. After several decades of severe mental illness, Nash recovered; in 1994 he received the Nobel Prize in Economics. This was the first Nobel Prize awarded for research in the field of game theory.

The idea is to find a solution in which each participant in the game chooses the strategy that best suits his interests in response to the strategy of the other player. If such a situation develops in the game, neither player has any reason to change his choice unilaterally. Hence, this is the potentially stable outcome of a game in which the players make individual and simultaneous choices of their strategies. Let’s start by illustrating this idea with a few practical examples, then discuss the extent to which the Nash equilibrium can predict the outcomes of various games; at the same time, we will justify the reasons for cautious optimism and for using the Nash equilibrium as a starting point for the analysis of almost all games.

Let’s analyze this concept using the price game between Rainbow’s End and BB Lean as an example. In Chapter 3, they only had two options for the USD price: $70 and $80. Each company was strongly tempted to lower that price. Now let’s increase the number of choices by giving them the opportunity to change the price by one dollar in a lower price range, from $42 to $38.

The $1 price increment and limited price range are chosen here only to make it easier to get started with this game. The following describes an example in which each company can choose a price from a continuous range of values.

The previous example says that if both companies bid $80, they would each sell $1200. If one of the companies cuts the price by one dollar and the other leaves it unchanged, then the company that lowered the price will attract 100 buyers: 80 buyers who switched from some other company and 20 new ones — these could be buyers who decide to buy from .e.shku, which would not be bought at a higher price. If both companies cut the price by one dollar, existing customers will not change their habits, but each company will have 20 new customers. Therefore, if both companies charge $42 instead of $80, they will each receive 38 × 20 = 760 buyers over and above the original 1200. In this case, each company will sell for $1960 and make a profit of (42–20) × 1960 = $43. Performing similar calculations for other price combinations, we get the following payoff table for this game:

THINKING TRAINING TASK #2

Try building this spreadsheet in Excel.

This spreadsheet may seem complicated, but it’s actually very easy to build using Microsoft Excel or any other spreadsheet program.