2. Game Theory: A Beautiful Mind
1I encountered the word Nash. I was a student at Hebrew University in Jerusalem and I came across Nash in an introductory Game Theory course. For me, Nash was then just a short and catchy adjective attached to two abstract concepts that are central to game theory: Nash Equilibrium and the Nash Bargaining Solution. If the concept of equilibrium were named Cournot (who had already thought of this concept in a narrow context in 1838) or ”Alpha Equilibrium” or even ”Smiley,” it would really have made no difference to me. I must have realized that the word Nash was connected to a person. And if I had asked myself who Nash was, I probably would have guessed he was an English intellectual who died at the beginning of the twentieth century.
2I arrived at Princeton for the first time. I heard from other students that a crazy genius would roam around the campus, sitting for hours in the cafeteria with a pile of computer printouts in front of him, reading newspapers he collected from abandoned tables. This crazy person could be seen riding back and forth on the ”Dinky,” a short rail line connecting Princeton Junction to the campus. It was rumored that he had an account on the university’s computer and was busy with mysterious calculations. They said there was one student who had dared to approach him and speak with him, and that he had readily helped the student. It was hard for me to identify him; a lot of weird people roam the lawns at Princeton.
3One afternoon, I was about to give a lecture in the Economics Department. Before the lecture, I asked my host to take me to the cafeteria and show me this ethereal character. My host looked toward one of the tables, lowered his eyes and whispered to me: ”That’s John Nash.” The man whose name rolled off my tongue more often than any other name in my professional life and whose work was the basis for the bargaining model I was about to present that day, was hunched over the table, wrapped in a long and shabby coat, wearing worn-out sneakers, and not looking at anyone. He was a solitary figure in the cafeteria, which looked like a dining hall in an Oxford college. I did not approach him, of course. I was not merely shy; I was afraid to speak with someone who was crazy.
4Whoever invented the name ”game theory” was also a genius in public relations. Who would be interested in this theory if it were called ”A Collection of Models of Rational Decision-Making in Interactive Situations”?
5The word ”game” has a mischievous, youthful and accessible ring to it. We all play board games, social games, political games and other games. Add to this the fact that the basic terminology of game theory includes words like ”strategy” and ”solution,” and it becomes a real celebration. After all, we are all frustrated experts in war strategy and everyone is searching for a solution to the conflict in the Middle East. So here, maybe, just maybe, we have found the lost box of tricks with the hidden secrets that will improve our game skills. But it is not quite so simple. I will try to explain, but first I will reiterate that some of the things I am going to say here are not uncontroversial.
6I think that the body of knowledge called game theory is a collection of formal models that enables us to analyze strategic and rational patterns of human thought. It sounds terrible, so I will elaborate a bit.
7In game theory, as in most economic theories, the operating units are rational decision makers. The decision maker, who is called a player in game theory, seeks to attain a well-defined objective. Whenever he is required to carry out an activity, he behaves (or at least can be regarded as behaving) in the following way: he asks himself what is desirable and what is possible, and does what is best in his (subjective) view, given what is possible (objectively speaking).
8Not every decision problem is a game theory problem. Let’s say that I am about to leave my house and I am contemplating whether or not to take an umbrella. I have four possible scenarios in mind: ”I have an umbrella and it’s raining,” ”I have an umbrella and it isn’t raining,” ”I don’t have an umbrella and it’s raining,” and ”I don’t have an umbrella and it isn’t raining.” I have some sense of how much pleasure or discomfort I would experience in each of the four situations. Whatever I decide the outcome is uncertain, which in our professional jargon we call a lottery. If I take an umbrella with me, there is a chance I will find myself opening it and will stay dry, and there is also a possibility that I will drag it with me for no reason. If I do not take an umbrella, there is a chance that I will get wet, and there is also a chance that I will be able to enjoy strolling around without having to schlep an umbrella around. I assess the chance of rain, something I have no control over that is determined by nature, compare the two possibilities, and decide. I do not have to predict anyone’s action or imagine anyone else’s calculations. It is just me, alone. I face a decision problem that is not a game.
9On the other hand, let’s say that I believe there is a rain god up above who holds the key to the floodgates of heaven and that this rain god has interests of his own. Maybe he cares about me, but maybe he really doesn’t like me, and has it in for me. My forecast of the rain god’s behavior will be based not only on the meteorological service, but also on an analysis of the god’s considerations. Maybe he will want me to get caught in a heavy storm if I make light of his powers and ignore threatening skies. And maybe if I believe in his being merciful and leave the house without an umbrella he will reward my faith in him by scattering the clouds.
10Or, let’s say that I plan to go for a romantic walk in the rain under one umbrella with a friend, who will be coming from her home, and that she too must decide whether to take an umbrella when she leaves her house. My rational decision will be affected by my expectations of my friend’s behavior and my expectations will be based, among other things, on the fact that both of us dream of walking in the rain under one umbrella.
11In such situations, I must exercise strategic thought: I ask myself what the rain god (or my friend) will do. My expectations develop as I ask myself how they analyze the situation and what they think about me. Game theory deals with situations in which each of the players is a rational decision maker and puts himself in the shoes of the other before making a decision.
12I brought my son Yuval to the first meeting of the beginners’ group of a chess class. The veteran Tel Aviv teacher who directed the activity began with a direct appeal to the children. She encouraged the children, who came from various neighborhoods, to become friends and added, with feeling: ”Children, I think chess is very important because it teaches you to look at other children from their point of view.” I said to myself: what a nice approach she has, teaching the children to think about others through the game of chess. I recalled the words of the game theorist, John McMillan, in the summary of a chapter on negotiating:
What advice for negotiators does Game Theory generate? The most important ideas we have learned… are the value of putting yourself in the other person’s shoes and looking several moves ahead.
13I was excited – I recognized game theory in the words of a children’s teacher. But after the initial excitement, I wondered: game theory and the game of chess perhaps encourage a person to think about a situation from the perspective of the other, but only in order to do the best thing for himself. The chess teacher confused strategic thinking with developing empathy. Strategic thinking does not encourage a person to think about the wellbeing of the other. Game theory speaks of a player who steps into the shoes of the other in order to assess what the other will do, and he does this for his own benefit, in accordance with his own preferences. But these preferences may reflect affection, sympathy and benevolence, or hatred, bitterness and revenge. An educational task such as developing empathy remains in the realm of the kindergarten teacher, not the chess teacher and not the expert in game theory.
14Let’s play (you can experiment with the situations discussed here and in the rest of the chapter: http://www.openbookpublishers.com/exsites/136). Here is a well-known situation called the Traveler’s Dilemma or Basu’s Game. Originally, it involved a story about two travelers returning from the same holiday resort, both bringing the same souvenir in their suitcases. The suitcases are lost and the lost-baggage clerk at the airport must compensate them for the real value of the loss. The clerk has only a vague idea of the value of the lost object: between $180 and $300. Therefore, he plans to award compensation according to the declarations of the two travelers, but he knows they are liable to exaggerate the value of the loss and that they do not feel obliged to tell the truth. He takes the two travelers into separate rooms and asks each of them to declare the value of the souvenir – an amount between $180 and $300. He promises to compensate both of them according to the lower of the two values they declare. In addition, if they give different prices, he will impose a ”fine” of $5 on the traveler who states the higher price and will ”award a prize” of $5 to the traveler whose demand is more modest.
15In order to consider this situation in terms of game theory, we must answer several questions:
- Who are the players?
- For each player, what choices does he face? (We sometimes call these choices: ”strategies.”)
- For each combination of choices by the players, what will be the result of the game? (Unlike nongame decision problems, the consequence of a player’s decision does not depend solely on the action he chooses. It depends also on the actions the other players choose.)
- For each player, what are his preferences? (That is, how does he rank the possible results of the game?)
16Only after answering all of these questions can we say that we have defined the story as a game in the accepted sense of game theory.
17In the case of the Traveler’s Dilemma, the answers to the four questions are:
- The players are the two travelers.
- For each player, the possible choices are amounts between $180 and $300.
- Each player receives a number of dollars equal to the smaller of the two numbers chosen by the players, plus $5 if the number he chooses is lower than the number the other player chooses, or minus $5 if the number he chooses is higher than the number the other player chooses.
- Each player is interested in receiving as much money as possible.
18The answer we gave to the fourth question does not derive from the initial description of the situation. The need to answer all of the questions made us describe the players’ preferences. In answering the fourth question, we assumed that each player is only interested in the amount of money he will receive at the end of the game, and that he is not interested in what the other player receives or the ratio between the two sums. This assumption is not obvious and is probably not realistic. Game theory certainly allows for an analysis of this situation even when we attribute additional considerations to the players, such as ”I do not want to be seen as petty in trying to earn a few dollars at the other guy’s expense,” ”It is important to me that both of us together receive as much as possible,” or ”I do not want to be the sucker of that smart-aleck.” But in game theory, an unfortunate convention has taken root: when we state that game theory says the outcome of the game will be this or that, we assume that the only thing a player cares about is the amount of money he will receive at the end of the game.
19Clearly, each player faces a strategic decision here. The best action from one player’s perspective depends on what he expects the other player to do. The rationality of a player is defined in game theory as choosing the best action in light of his beliefs about the behavior of the other players. This definition does not impose constraints on these beliefs and, in particular, does not demand that the beliefs be reasonable in any sense.
20In the current case of the Traveler’s Dilemma, it seems that the choice of $300 is not rational for me as a player in the game. Regardless of what I believe the other player will do, there is another declaration of price that would result in my receiving more than $300. If I believe that the other player will choose $300, my choice of $300 would bring me $300, while if I chose $299 I would receive $304. And if I believe that he will choose a number N, which is less than 300, then the choice of $300 would compensate me with only $(N-5), while if I choose the number N, for example, I would receive $N.
21Even if I am not sure about the other player’s price declaration, the action $300 cannot be rational.
Let’s assume that I am certain that he will choose a number no greater than M, and that I attribute a positive probability to both of the other possibilities: that he will choose exactly M or a number less than M. We will show now that choosing the number M-1 is preferable to choosing the number 300. To be precise, we will show that there are circumstances in which declaring M-1 would yield a higher sum than declaring 300, and that there are no circumstances in which declaring 300 would yield higher revenue than declaring M-1. If the other player declares M and I declare M-1, I will receive $M+4. And if I declare 300, I will receive at most $M. If the other player chooses M-1 and I do the same, I will receive $M-1 and that is more than the $M-6 I would receive if I choose 300. Finally, if the other player chooses a number less than M-1, the payment I receive will be the same regardless of whether I choose M-1 or 300.
22So we have rejected the rationality of choosing $300. Could it be rational from my perspective to choose the number $299? Yes. For example, if I am sure the other player will choose the number 300, it will be best for me to choose 299. But if I put myself in the shoes of the other player and assume that he also is rational, I will reach the conclusion that he will surely refrain from choosing the number 300 too. Thus, the highest number he might choose is 299. Consequently, following the principle of the previous argument, the choice of 299 is also not a rational choice for me. And if I again put myself in the shoes of the other player and assume that he puts himself in my shoes, I will reach the conclusion that he believes I will not choose 300 and, therefore, he will also refrain from choosing the number 299. Accordingly, the choice of 298 is also not rational from my perspective. And now, in order to decide what is good for me, I must step into the shoes of the other and think how he puts himself in my shoes and imagines my stepping into his shoes. It is difficult for me to keep track of this reasoning. This circular thinking simply drives me crazy. It is like the sentence: ”I think that she does not think that I think that she will smile at me if I go up to her.” It is a proper sentence and has meaning, but it is annoying.
23Nash equilibrium: In the introductory chapter, we said that a solution concept is an array of rules by which an economic tale is allowed to develop from its beginning to its end. In the context of game theory, the beginning of the tale is a description of the game, and its end is the actions the players choose. Nash equilibrium is a solution concept that extricates us from the whirlpool of ”I think that he thinks that I think…” The concept is built upon two assumptions:
First, the action attributed to a player is the best for him in light of what he believes the other players will do.
Second, the player’s belief regarding the actions of the other players is correct.
24In other words, according to Nash, the answer to the question ”What will happen?” is consistent with the assumption that each individual is able to step into the other’s shoes, correctly predict his moves, return to his own shoes and choose the best action from there.
25There are also other interpretations of Nash equilibrium. For example, sometimes we think of Nash equilibrium as a stable norm of behavior. A norm of behavior is a rule that tells individuals what to do in the various situations they are liable to encounter. A norm of behavior in a game situation is stable if the action prescribed by the norm for each player is the best one for him when he expects the other players to act according to the same norm. In other words, a stable norm of behavior is a rule of behavior that advises the players in the game to act in accordance with Nash equilibrium. For example, in the context of the Traveler’s Dilemma, the norm of asking for as much as you can is not stable because it is worthwhile for each individual to deviate from the norm and to demand a bit less. (The slightly more modest demand is rewarded with an extra payment.) On the other hand, the norm of being as modest as possible in your demands is stable in this game.
26Nash was not the first one to use the concept of equilibrium, but he was the first to formulate it in an abstract way and not in the context of a specific game. There is no guarantee that a Nash equilibrium exists in a game; there are games that do not have an equilibrium. Nash showed that if a game meets certain conditions, a Nash equilibrium does indeed exist. This is the original mathematical part of his work.
27Back to Basu and the travelers: what is the Nash equilibrium in Basu’s Game? One player chooses $250 and the second chooses $240 – this is not an equilibrium. When the second player declares $240, it is not optimal for the first player to declare $250. The first player can receive higher compensation if he chooses, for example, $239. Similarly, there is no equilibrium in which the two players choose any two different numbers.
28If both of the players choose $240, this is also not an equilibrium. Each player can enlarge his payoff by subtracting a dollar from his demand. Similarly, there is no equilibrium in which both players choose another identical number. There is only one exception and it is when both players choose $180. One cannot go lower. Choosing $180 is the best for a player who believes that the other player will choose $180.
29Conclusion: The Traveler’s Dilemma has a unique equilibrium in which both players choose $180. We asked: ”What will happen?” And here, in this game, the Nash equilibrium gives us an unequivocal answer. (Incidentally, this is quite a rare situation.) Hallelu-Nash.
30How is the answer the Nash equilibrium provides in this game related to human behavior in reality? We do not know how people would behave in such a situation in real life. At best, it is possible to bring people to a laboratory and watch them play a similar game. This would be a costly experiment. Alternatively, it is possible to ask people, like the readers of this book, to imagine themselves in this type of situation. People are quite imaginative and like to play such games. The participants can be encouraged to take the game seriously by awarding a symbolic payment.
31In my view, as I already noted in the previous chapter, such a reward is completely unnecessary and may even be detrimental. I have data on the responses of more than 13,000 people who were asked to answer (via the website http://gametheory.tau.ac.il) the question of what they would do if they were players in the game. One third of the respondents were students in game theory courses in about thirty countries, and half were people invited to eleven public lectures on ”Game Theory and John Nash” that I delivered in recent years in seven countries. Before each lecture, I asked the audience to answer a number of questions, including those in Basu’s Game. Of course, the sample is not representative of the world’s population. But our objective is not to describe, even approximately, how the world’s population plays the game. Our interest is to peek into the considerations of the people participating in this game, and we would be happy to get even a rough impression of how common these considerations are world wide.
32Among the respondents, 45 % chose the price $300, a choice we have just termed irrational. Only 20 % of the respondents chose $180, the choice recommended by the concept of Nash equilibrium. Another 19 % chose between $295 and $299, while the choices of the remaining 16 % were in the broad range of $181 to $294.
33The players who chose a number in the range of 295 to 299 apparently used strategic thinking. That is, they said to themselves something like: The other player will choose 298, so I will choose 297, or to be sure I’ll even go down to 296. On what basis do I determine that the respondents in this range were more strategic than the respondents who chose a number from 181 to 294? First, I do this based on introspection: I can find a reasonable explanation for the choiceof297, but cannot find one for choosing a number such as 236. And I also have additional support for the assumption that the choice of a number in the range of 295 to 299 is the outcome of more serious thinking: I also recorded the response time of the participants in the experiment – that is, the time from the moment the computer sent them the question until it received their answer. The response time of those whose choice was in the range of 295 to 299 was significantly greater (a median of 107 seconds) than the response time of those quoting in the range of 181–294 and of those who chose the number 300 (a median of 77 seconds).
34I suspect that most of the respondents who chose the number 180 are members of the Victims of Game Theory organization, who implemented the concept of Nash equilibrium without exercising common sense. If you had to play against a randomly selected participant in the experiment, I would not recommend that you choose an equilibrium action. According to the rules of the game, you will be able to win a maximum of $185. On the other hand, a simple calculation shows that if you choose 298 or 299, and you play against one of the subjects in my database you can expect to receive an average payment of $262.
35And here is another interesting fact: The distributions of the answers in various countries (among audiences at my public lectures) were similar.
36I have no substantiated explanation for this fact. Perhaps people are divided into different types and their type determines their choice in this game. For example, perhaps there are four types: the instinctive (who chooses the number 300 in this game); the sophisticated (who chooses a number in the range of 295–299); the disciplined (who took a course in game theory and, obeying the equilibrium, chooses 180); and the capricious (who chooses a number randomly). And perhaps every society has a similar distribution of these four types. Of course, this is a far-reaching hypothesis that I cannot substantiate. If this hypothesis were firmly based, we would expect that we could assess the distribution of types and estimate the distribution of behaviors in various games.
37Despite the fact that the distributions of answers in the seven countries are similar, there are also significant differences, particularly in the percentage of those choosing the game theory solution, 180. This fact is perhaps related to the percentage of respondents who had taken a course in game theory. I find support for this assumption in the results from 9,300 students in game theory courses who were presented with the same question. The percentage of those choosing the equilibrium action was 23 %, compared with 14 % in the more diverse population of those attending public lectures on game theory. This increase in the share was derived primarily from a decline in the percentage of those choosing 300. This outcome strengthens the suspicion that a small group of students had internalized the ideas presented in the game theory course, even to the point of choosing the equilibrium point when this action was not really the most intelligent thing to do.
38Does Basu’s Game show that the concept of Nash equilibrium does not explain the way people play? Not necessarily. Nash equilibrium is a solution concept applied to a description of a game that includes not only the procedure of the game, but also the preferences of the players. We saw that the popular choice of the maximum number, 300, is not rational if the players care only about the sum of money they will receive at the end of the game. But the choice of $300 can be optimal for a player whose preferences include considerations such as unpleasantness about profiting at the expense of others or embarrassment for appearing to be petty. Therefore, the norm of behavior of seeking to collaborate with the other player, expressed in this game by choosing $300, is not a Nash equilibrium if the players care only about the cash in their pocket, but this norm is stable in the sense of Nash equilibrium if the players’ preferences are such that the unpleasantness caused to a player who made a few dollars at the expense of the other player is the same as losing more than $4.
39Here is another game. Two individuals, who are capable of reaching an agreement, participate in the Ultimatum Game. There are several possible agreements that they would prefer to accept rather than remain in disagreement. Some of the agreements are better for one player and others are better for the other player. One of the players in the game is assigned the role of the proposer and the second player is the responder. The proposer must offer a proposal for an agreement and the responder must accept or reject the proposal. If the responder accepts the proposal, the agreement is executed. If the responder rejects the proposal, the two players part ways without an agreement.
40Ultimatum games are common in everyday life: in the labor market, unskilled workers can only say yes or no when offered a job. They usually say yes even when offered meager wages. In every visit to the supermarket, you are actually participating in an ultimatum game. You want a carton of milk. The supermarket places an ultimatum before you – either buy it at the set price or don’t buy it. There is no point in arguing with the cashier at the supermarket about the price of milk. Sometimes the situation occurs not in real life, but rather in the imagination of someone who believes that he can make the other side an offer he cannot refuse. How disappointed he is when it turns out that he was mistaken in his in assessment of the considerations of the other side.
41To keep it simple, we will focus on an ultimatum game involving $100. The agreement is any allocation of the sum between the two individuals. If there is no agreement, the two individuals will receive nothing. In order to complete the description of the situation, we must describe the preferences of the players. And again, as usual, we will assume (something that is not self-evident) that each player is only concerned with the sum of money he gets at the end of the game, and that he is interested in getting as much as possible.
42As always, in order to analyze the game, we must apply a solution concept. For this type of game, the conventional practice in economics is to apply a solution concept called perfect equilibrium. This is a version of Nash equilibrium, adapted for games involving a series of moves. We will now apply this concept, which was discussed in the opening chapter of the book, to the Ultimatum Game. A perfect equilibrium describes a pair of plans of action, one for the proposer and one for the responder. The proposer’s plan of action is the proposal he intends to offer to the responder. The responder’s plan of action is a policy that describes which proposals he will accept and which he will reject. Perfect equilibrium demands that the pair of plans of action meet two requirements. The first requirement pertains to the proposer’s plan of action: the proposal he intends to offer to the responder must be the best one for the proposer, taking into account the responder’s policy of acceptance. The second requirement pertains to the responder’s plan of action: if the responder intends to accept a certain proposal, he would not benefit by rejecting it; and if the responder intends to reject a certain proposal, it would not be preferable for him to accept it. In other words, after each proposal that the proposer might offer (and not only the proposal he is supposed to offer according to his plan of action), the responder’s plan of response is best for him.
43Here is one perfect equilibrium of the game. The proposer’s plan: offer $1 to the responder. The responder’s plan: accept any offer except $0. Now let’s check whether the plan of each player is optimal in every possible scenario in which he is likely to implement it.
44The proposer starts the game. His decision problem is to choose one of 101 possible proposals. According to the responder’s strategy, each proposal of a positive number of dollars, X, will be accepted and is ”worth” $100-X to the proposer. The proposal $0 will be rejected and is therefore worth $0 to the proposer. Thus, the best action for the proposer is indeed to offer $1 to the responder.
45The responder will have to act after receiving the proposal. If the proposal is a positive sum, it is best for him to accept it, because rejecting it would mean receiving nothing. If the proposal is $0, then he would be left with the same $0 regardless of whether he accepts or rejects the proposal. Therefore, rejecting the proposal, as planned, is an optimal action from his perspective.
46It is easy to confirm that the game also has a second perfect equilibrium in which the proposer offers $0 and the responder plans to say yes to any proposal offered to him, including the insulting offer of $0.
47We will now see that in every perfect equilibrium the proposer ends the game with at least $99. In other words, there is no perfect equilibrium in which the proposer receives less than $99. Accepting any proposal that gives the responder at least $1 is better for him than the alternative of rejecting the proposal and receiving nothing. Therefore, in every perfect equilibrium, the responder plans to accept every offer that gives him at least $1 and, consequently, the proposer can receive at least $99. In a perfect equilibrium, the proposer chooses an optimal plan and it awards him at least $99.
48Let’s summarize: game theory ”predicts” what many people think. The player who is able to issue an ultimatum has a great advantage over the player who can only accept or reject the proposal. The first player will receive all (or almost all) of the pie that is to be allocated.
49The Ultimatum Game and its analysis by means of the concept of perfect equilibrium serve as a basis for many other and more complex economic models. I know very few games that have been given so much attention. Is the outcome of the model consistent with behavior in real life? This question has been examined in various experiments in many cultures – with money, with lots of money and without money – and with a large number of participants. On the book’s website, the Ultimatum Game is expressed in this way:
Imagine you and someone you do not know can share $100. It is up to you to propose how to divide the $100 between the two of you, and the other player will need to accept or reject your proposal. If he rejects the proposal, neither of you will receive anything.
What sum will you offer the other player?
50I have data on the choices of about 12,300 people, most of them students, who were asked this question. Nearly half of the participants (49 %) offered the other player the fair offer of $50. About 9 % took advantage of their position to some extent and offered the other player a sum in the range of $40 to $49 – that is, just a little less than $50. Some 12 % offered the other player only $1 (and only 1 % made the even more absurd offer of $0). And 13 % offered the responder a sum in the broad range of $2 to $39.
51One group, not mentioned above, stood out: the 18 % of the participants who offer the other player more than $50. Some of them apparently got confused between receiving and giving – for example, writing the number 60 and expecting to receive $60 if this offer was accepted. But the fact that 7 % of the proposers offered a proposal in the range of 51 to 59 (compared to only 3 % who made an offer in the parallel range of 41 to 49) suggests that there is a significant group of ”annoyingly” generous people who feel better if they receive less than what they are entitled to under the conventional rules of fairness.
52The figure presents distributions of the proposals of nearly 5,000 people invited to public lectures on game theory in seven countries. The distributions are similar, but there are also differences. I might be tempted to say that the Dutch are tougher than the Israelis because the Dutch, on average, offer $4 less than the Israelis. I might also be impressed by the fact that the highest percentage of those offering the fair allocation is in the U.S. (59 %). But I do not attribute significance to these findings: at most, they are interesting observations. One result that is perhaps more significant pertains to the differences between the genders: in nearly all of the universities, the average proposal by women was higher by $3 to $5 than that of men. Some 56 % of the women chose the fair allocation, compared to only 46 % of the men. No more than 8 % of the women offered a proposal consistent with the game theory solution, compared to 14 % of the men. In other words, the women are more generous and less attracted to the game theory analysis.
53What would people do as responders? The following question also appears on the book’s website:
You and a person you do not know can share $100. He offers a proposal and you can only accept or reject it. If you reject it, both of you will receive nothing.
He offers you $10 (and if you accept the offer, he will receive $90).
Would you accept the offer?
54I have the responses of 8,100 participants. Some 63 % of them accepted the proposal. The percentage of those accepting the proposal ranged from 52 % in Bangkok to 72 % in Tel Aviv. If I took the results of the survey seriously (and I do not), I would predict, based on these data, that a large number of Israelis would acquiesce to an international ultimatum to return more or less to the 1967 borders. (As a matter of fact, I think this is true – not because of these data, but based on my familiarity with Israeli society.) And another finding related to gender: about 60 % of the women accepted the offer of $10, compared with 65 % of the men. Are women tougher? Are they less materialistic? Or are they endowed with a more developed sense of fairness? In any case, in my view they are more pleasant.
55As is customary in economic models, at least until recent times, the classic analysis of the game assumes that each player is interested only in the sum of money he himself will receive at the end of the game. Is this description of the players’ preferences in the game correct? It is very doubtful.
56First, many people also show interest in a fair allocation and not only in the sum of money they receive. There are even those who are happier if the sum of money is split equally between the two players than if they receive the entire sum. A situation the literature calls the Dictator Game clearly reveals the existence of these considerations. In the Dictator Game, one player – called the dictator – is asked to divide a sum of money between himself and another anonymous player. The other party plays no active role in the game, so it is actually a decision problem. Thousands of students in game theory courses were asked to imagine a situation in which they play the game as the dictator. About 36 % took the entire sum, about 35 % divided the sum equally between them and the anonymous player, and nearly all of the others allocated something to the other player, but less than the sum they allocated to themselves. On average, the dictator gave the other player about a quarter of the sum to be allocated. These facts indicate that in such situations most people are not so selfish and also consider the fairness of the allocation.
57People care not only about the sums of money they will receive at the end of a game, but also about the way they obtain the money. Let’s say that $100 is about to fall from the sky, with $10 falling into my hands and $90 into the hands of my neighbor, and I have the ability to prevent this shower of dollars. The choice I face is between the allocation of $10 to me and $90 to the neighbor, or a situation in which both of us receive nothing. I would be considered malicious if I preferred that we both receive nothing rather than my neighbor receiving much more than I do. On the other hand, let’s say that in the Ultimatum Game I receive an offer of $10, which leaves $90 in the hands of my brazen neighbor, the proposer. In this case, I would undoubtedly surrender the $10 as long as it would put the proposer in his place. I thought that nearly everyone feels like me, but to my surprise only about a half of the respondents in the survey declared that they would reject the proposal to receive $10 out of the $100.
58It goes without saying that I would not use classic game theory analysis as a basis for advising a person who is about to deliver an ultimatum demanding ”all” or ”nearly all” of the pie. There is reason to believe that at least 75 % of the participants would reject the offer of $1 and that (almost) everyone would accept the 50–50 offer. Thus, a proposer who acts according to the recommendation of game theory would receive $0 in 75 % of the cases and the average (expected) number of dollars in his pocket would be less than $25. If he acts in a less sophisticated way and proposes the fair allocation, he would almost certainly receive an affirmative response and could expect to have nearly $50 in his pocket. The participants in the experiment who make the embarrassing offer of just $1 because they learned this in a game theory course are again the distinguished members of the Victims of Game Theory organization. And if they played the game in real life, their achievements would be inferior to those who had not become wise by studying game theory.
59This does not prevent some strategic experts from treating the game theory solution of the Ultimatum Game as a sacred rule. Some say that a certain prime minister of Israel issued an ultimatum to the head of the Palestinian Authority that granted the Palestinians 90 % of the territory of the West Bank and Gaza Strip. The rais refused and the rest is written in another bloody page in the chronicles of the Middle East. When the other side believes that it deserves 100 % or that the pie is part of another pie and 90 % of the small pie is less than 50 % of the large pie, it is likely to reject even an offer of 90 % of the small pie.
60I was a young, fresh lecturer in the Department of Economics in Jerusalem. One day, a letter from the Nobel Prize Committee landed in my mailbox. I was surprised. I later understood that the selection committee for the prize was seeking to identify important fields and worthy candidates by approaching researchers who were in the early stages of their careers. I began my response to the committee’s query with banal remarks about the centrality of game theory in economic theory. I listed many fields in which game theory is used. When I came to candidates, I cited four names. And then, I added a paragraph on a fifth candidate, John Nash. I wrote that Nash, the outstanding person in the group, lives in Princeton and had stopped working due to personal problems, but that the three articles he wrote during 1950 to 1953 were the most important and most inspiring in game theory and in all of economic theory since the book Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern. Nash was clearly worthy of the Nobel Prize. My remarks had at most a marginal influence on his winning the prize nine years later. And I was left to wonder: was I motivated only by academic judgment or did I also want to rectify the injustice suffered by someone who had been abandoned and had not received the recognition he would have gained if he had not been mentally ill? Is it possible and is it desirable to separate the goal of correcting human injustice from pure academic assessment?
61Let’s return to the games. The next game, in a slightly different form, was discussed in the introductory chapter.
Imagine that you are the manager of a chain of coffee shops competing against two other chains. A new beachfront residential neighborhood is being built, with seven huge apartment blocks equal in size and equidistant from one another. The towers are numbered from left to right: 1, 2, 3, 4, 5, 6, 7.
Each of the three chains plans to open a branch in one of the blocks. The three coffee shops will be very similar. The manager of each of the chains must decide in which block to open a branch, with the goal of attracting as many customers as possible. He must make a decision immediately, before knowing where his two competitors will set up their coffee shops.
The clientele is expected to consist of residents of the seven blocks and each customer is expected to patronize the cafe that is closest to his apartment.
In which block will you set up the coffee shop of the chain you manage?
62Hotelling spoke about a main street and stores. Here, the main street becomes seven blocks on a beachfront and the stores become coffee shops. I admit that this is an expression of my fondness for coffee shops. While I try to boycott the coffee chains that annihilate the intimate neighborhood coffee shops that I love so much, I have compromised here in order to make the story more realistic.
63The assumptions in the story are reasonable, even if they do not perfectly describe reality. In real life, coffee shops are not absolutely identical. Some people prefer to patronize a particular coffee shop even if it is further from where they live. The coffee shops in our game compete with their rivals only in terms of location, while in reality competition is frequently conducted via the price and quality of the coffee. We assume that the decisions are made simultaneously: each player chooses his location without knowing the location of the other two. In real life, the players try to preempt their rivals or, conversely, wait until the picture becomes clearer.
64Let’s assume first that only two (and not three) chains are competing in our example, and proceed to translate the story into a game. The two players in the game are the managers of the chains. Each chain must choose a tower, a number from 1 to 7. The chain’s objective is to maximize the number of its clients, a number that depends on its location and the location of the other chain and is calculated as follows: for each block, we will check which chain has the nearest cafe and we will count the residents of the block as clients of that chain. If the two cafes are located at an equal distance from the tower, the residents of the tower will divide equally between them. For example, if Chain A opens a branch in Tower 4 and Chain B opens a branch in Tower 6, the residents of Towers 1, 2, 3, 4 will be customers of Chain A and the residents of Towers 6 and 7 will be customers of Chain B, while the residents of Tower 5 will be split between the two chains. In this case, Chain A’s market share will consist of the residents of 4.5 towers, and the market share of Chain B will consist of the residents of 2.5 towers.
65With two players in the game, there is a single Nash equilibrium: when both of the players set up their coffee shops in the middle block (No. 4). A unilateral move by one of the players from the center will diminish his clientele. The proof that there is no other equilibrium is similar to the one we saw in the introductory chapter: a situation in which one or more blocks separate the two cafes is not an equilibrium because if either of them moves to one of the blocks located between them, it would increase its market share. If the two cafes are located in adjacent towers, one of them enjoys less than half of the market and if it moves to the block in which the competitor is located, it would increase its market share to 50 %. If the two cafes are located in the same block, and it is not the middle one, then each of them enjoys half of the market and can increase its share to at least 4/7 of the market by moving to the middle block.
66Indeed, if I asked the readers to consider the game with two competitors, I am sure that an overwhelming majority would choose to locate in the middle block, thus confirming the game theory prediction. I have the results of a survey conducted among 8,100 participants. About 68 % of them chose No. 4. This sounds like good news for those who look to game theory to help predict what will happen in real life. As I noted in the opening chapter, this game is important and resembles familiar real life situations. In as early as 1929, Hotelling said the following about the two players choosing to locate themselves in the middle:
So general is this tendency that it appears in the most diverse fields of competitive activity, even quite apart from what is called economic life. In politics it is strikingly exemplified. The competition for votes between the Republican and Democratic parties does not lead to a clear drawing of issues, an adoption of two strongly contrasted positions between which the voter may choose. Instead, each party strives to make its platform as much like the other’s as possible.
67But… our joy is premature. What happens if three competitors, instead of two, operate on the beachfront?
68It is not difficult to see that Hotelling’s Game with three players does not have a Nash equilibrium. I will just explain here why locating the three cafes in the middle tower is not an equilibrium. If a player thinks that his two competitors will choose to locate their shops in No. 4, then his choice to locate his shop there will bring him a third of the total number of residents in the neighborhood. That is, the number of his customers will be equal to the number of residents in 2. blocks. On the other hand, if he opens a shop in No. 3 or 5, his market share will be the residents of 3 blocks. Thus, the choice of No. 4 is not optimal for a player who anticipates that his two competitors will locate in the center. Therefore, setting up all three cafes in the middle block is not a Nash equilibrium.
69Hotelling’s Game with three players is a symmetrical game in the sense that the description of the game does not discriminate between the players. In games of this type, it is customary to look at the concept called symmetric equilibrium, an extension of the concept of Nash equilibrium. One can think of this kind of equilibrium as the distribution of behaviors in a large population of individuals, with each individual programmed to play the game in a specific way. The distribution of behaviors describes the percentage of individuals in a population who would choose Block No. 1 in the game, the percentage who would choose No. 2, and so on. Each individual in the population is programmed to locate himself in a particular apartment block and expects to play the game against two random competitors from within the population. In a symmetric equilibrium, none of the individuals programmed to set up in a certain block would be able to increase his anticipated market share by moving to another one.
Let’s look at the following distribution of behaviors: Half of the population is programmed to choose Block No. 3 and the other half is programmed to choose No. 5.
A player who must decide where to locate his shop and knows that his competitors will be randomly selected from this distribution faces uncertainty regarding the location of his two competitors. He expects a probability of ¼ that both will locate their shops in No. 3, a probability of ¼ that both will locate their shops in No. 5, and a probability of ½ that one competitor will set up in No. 3 and the other in No. 5. Therefore, if he locates his shop in No. 4, there is a probability of ¼ that his clientele will include the residents of blocks 4, 5 ,6, 7; a probability of ¼ that his clientele will include the residents of Blocks 1, 2, 3, 4 ; and a probability of ½ that his clientele will only include the residents of No, 4. Consequently, the average market share of a player who locates his shop in No. 4 is equal to the number of residents in 2.5 blocks. On the other hand, the choice of No. 3 or No. 5 leads to an average clientele of the residents of only 7/3 blocks. (This is evident for reasons of symmetry, even without doing the calculation.)
All of this demonstrates that when a player expects that his two competitors are randomly selected from this population, the average market share he receives if he locates his shop in Block No. 4 will be greater than the average market share he can expect if he chooses what he was programmed to do. Therefore, this distribution is not a symmetrical equilibrium.
70It is possible to show (and this demands a little work) that Hotelling’s Game with three players has only one symmetric equilibrium, when 40 % of the players choose No. 3,20 % choose No. 4 and 40 % choose No. 5.
71The following table presents the equilibrium distribution, alongside the distribution of choices of 7,400 people, most of them students in economics and game theory courses:
72As the game theory solution predicts, the proportion choosing the middle tower, 43 %, is significantly lower than the proportion choosing this tower in Hotelling’s Game with two stores, 70 %. But there is no similarity between the distribution predicted by game theory and the distribution in the survey. The choice of the middle tower remains the most common even in a game with three players. This choice reflects the instinct (which we already noted in the previous chapter) of people when faced with a linear group of alternatives to choose the alternative in the center. An examination of the participants’ response times supports the hypothesis that the choice of the middle tower is the instinctive action. The median response time (54 seconds) of the participants who chose Tower 4 is similar to the median response time of the individuals who chose the towers at the end of the line, which is clearly an irrational choice. On the other hand, the median response time of those who chose Towers 2, 3, 5, 6 was much higher (80 seconds).
73Despite the lack of agreement between Nash equilibrium and the experimental evidence, game theory has become established as a central tool in economics. Nash equilibrium became an accepted solution concept that is used to predict behavior in so-called non-cooperative games – that is, games in which the players operate independently and do not form groups (coalitions) who make coordinated decisions. In the 1950s and 1960s, game theory languished at the margins of economics. Von Neumann and Morgenstern’s Theory of Games and Economic Behavior, which many regard as the beginning of game theory, was published at Princeton during World War II and was immediately recognized as an enormous intellectual achievement. Nonetheless, for half a century the study of game theory barely extended beyond the mathematics and operations research departments. Only in the 1970s did game theory penetrate into the core of economics. If till then a market and competitive equilibrium constituted the major tool of economic analysis, they were now joined by the related duo of a games and Nash equilibrium. Since the 1980s, countless people have delighted in declaring that game theory is useful in all fields: competition between few competitors and company takeovers in economics, strategic voting and negotiation between countries in political science, the relations between flowers and butterflies and the evolution of animals in biology, moral issues in philosophy, developing communication protocols in computer science, and even the biblical stories of the binding of Isaac and the judgment of Solomon – all have been examined with the tools of game theory.
74Game theory won media acclaim in 1994 thanks to ”the mother of all auctions”: some $7 billion in communications frequencies were sold by the Federal Communications Commission (FCC) in a public auction planned in consultation with game theorists. The bidders in the tender also hired game theory experts to advise them. In the media, and not only there, this event was seen as definitive proof of the applicability of game theory. I have my doubts.
75I personally know some of the people who planned this tender and similar tenders. They are undoubtedly bright and intelligent. They are also people with two feet firmly on the ground. However, to the best of my understanding, they based their recommendations on basic intuitions and human simulations, and not on sophisticated models of game theory. I do not find any basis for claiming that it was game theory that helped them in planning the tender. At most, these advisors were intimately familiar with a specific type of strategic considerations that we often study in game theory.
76During the years that game theory flourished, John Nash was diagnosed as a paranoid schizophrenic. He was hospitalized a number of times in mental hospitals, received insulin treatments, heard voices that crowned him ”King of Antarctica,” and communicated with other worlds via the pages of The New York Times.
77I sat in my office at Princeton. It was evening, the door was open. John Nash walked by in the hallway and seemed to be looking for something. He entered my room and asked politely whether I knew the fax number of someone at Stockholm University. The number he had was a six-digit number and since all telephone and fax numbers in the U.S. have seven digits, Nash figured that there was a digit missing in the number he had. Based on what I knew, I explained to him that in Stockholm the telephone and fax numbers have six digits (as they did at the time). He felt relieved. I exploited the moment to do something that I had wanted to do for quite a while and had not dared. I mustered the courage and handed him a copy of a text book about game theory that I had written together with Martin Osborne. Nash took the book. I do not remember him thanking me. He mentioned that he already had two books on game theory on his shelf and now he would have ”two plus one = three books.” And then, leafing through the book, he said in surprise: ”I see that my name appears here.”
78Is game theory useful? The popular literature is full of nonsense about the applications of game theory. Here is an example from a serious newspaper, the Financial Times (17 April 2002):
Kofi Annan is famously active in seeking advice from a variety of sources, and recent propositions from the United Nations’ secretary general suggest he has found a new font of inspiration as he casts about for tips on how to solve the seemingly intractable crisis in the Middle East: game theory. Even Mr Annan’s language has taken a turn for the mathematical. In several of his most recent speeches, the career bureaucrat has called on the US to help Israel and the Palestinians abandon their ‘logic of war’ for a ’logic of peace.’
79And here is another example: February 2006 was a tense month in Thailand. The opposition demanded the prime minister’s resignation. Seven months later, the pressure culminated in a military coup. During the same month, I happened to be in Bangkok and delivered one of the public lectures about game theory that I mentioned earlier. I emphasized my opinion that game theory is not relevant to practical questions. Of course, I did not make any reference to the political situation in Thailand. The closest I came to making a reference to Thailand was when I complimented the audience for being particularly generous in the Ultimatum Game. That was sufficient for a reporter from the Thai newspaper The Nation, who attended the lecture and wrote about it the following day, to lead with a headline: ”Time to go to the polls, game theory says.”
80There is disagreement in the game theory community regarding the applicability of the theory. Some believe that the function of game theory is to provide useful predictions of behavior in strategic situations. The economist Hal Varian wrote in a review of the film A Beautiful Mind: ”Mr. Nash’s contribution was far more important than the somewhat contrived analysis about whether or not to approach the beautiful woman in the bar. What he discovered was a way to predict the outcome of virtually any kind of strategic interaction” (The New York Times, 11 April 2002). We will get to the beautiful woman at the bar later, but I have absolutely no idea how Varian reached the conclusion about the predictive ability of Nash equilibrium. Even when a game has a single equilibrium, there remains a huge disparity between the prediction of game theory and reality. In addition, in many games there are multiple Nash equilibria and this narrows their potential to predict. And this is before noting the fundamental difficulty of predicting the behavior of individuals when they are exposed to a prediction and are likely to respond to it. Incidentally, the article (published in the journal Econometrica in 1951) for which Nash was awarded the Nobel Prize is devoid of any pretension of usefulness in economics. The only ”economic” example you will find there is a simplification of a poker game with three players.
81Economists such as Avinash Dixit and Barry Nalebuff believe in the power of game theory to enhance strategic intelligence. The study of game theory is supposed somehow to foster the ability to play in strategic situations. But even they do not regard game theory as a collection of guidelines on ”how to...”.
82Game theory rhetoric switches between usefulness on the one hand, and awareness that it is dealing with simplified models, on the other. All in all, it seems to me that game theory tends to present a false front of usefulness. The Thai journalist evidently heard in my remarks what he wanted to hear. Nonetheless, I do not believe that he would write such a headline if I were a physicist or mathematician. Something in the language we use in economics and game theory creates an illusion that we understand and leads to the hasty application of ideas.
83My view of game theory is consistent with my approach to economic models in general, as explained in Chapter 0. Game theory does not try to describe reality or be normative. Game theory investigates the logic of strategic thinking. But just as logic does not make people truthful or guide judges to just decisions, game theory does not assist players in playing games. If game theory has a practical aspect, it is derived indirectly. It enables us to conduct an orderly discussion of the concept of rationality in interactive situations. It enriches the discussion of economics and other fields of social sciences by focusing on strategic considerations, some of which we might not have been aware of. It is entertaining. And that is something; but it is not what people generally describe as useful. Incidentally, sometimes I wonder why we need to address the question of the usefulness of game theory at all. Does academic research have to be judged according to the immediate and practical benefit it brings?
84Despite my reservations about the predictive ability of game theory, I do not deny the fact that people’s behavior in game situations follows certain rules or patterns, which can be discovered by observing events in the world or by experimental results. But it is connected only loosely (if at all) to game theory analysis. Here is another example:
85Treasure Hunt was my favorite radio program during my childhood. The program was broadcast once every four weeks at 9 pm. The theme music sounded as if it came straight from the courts of medieval knights. The treasure hunter in the studio would receive the riddle and turn to the audience with questions, and the listeners would call the studio and offer their answers for a price. If the treasure hunter identified the location of the treasure and his emissary reached the treasure before 10:20 pm, he would receive 1,000 lira, minus the payments he made to listeners. I would collect in advance all of my Land of Israel books and would concoct innovative solutions to infiltrate the busy phone lines. My excitement reached a peak during the few occasions when the treasure hunter purchased the answer of ”a schoolboy from Jerusalem” for 5 lira. When I think of examples for the game theory course, my thoughts wander to Treasure Hunt. This radio program was also the inspiration for the following game that appears on the book’s website:
Your competitor will have an opportunity to open only one box. Your goal is for the competitor not to find the treasure.
In which box will you hide the treasure?
86This situation can be thought of as a game with two players – the hider and the seeker. The hider has four strategies, one for each box in which he can hide the treasure. The seeker has four strategies, one for each box in which the treasure may be hidden. A pair of choices by the hider and the seeker leads to one of two possible outcomes: the seeker finds the treasure or the seeker does not find the treasure. The seeker prefers the first outcome and the hider the second.
87There is an absolute conflict of interests between the two players in this game. The hider wants to reduce the probability of the seeker finding the treasure. The seeker seeks to increase the probability of finding the treasure. Anything that is good for one player is necessarily bad for the other player. In professional jargon, it is customary to call this type of game a zero-sum game. Public figures and columnists frequently use this concept, appropriately or inappropriately, in order to embellish their remarks with learned terms.
88In this game, a candidate for Nash equilibrium is a pair of choices: the box in which the hider places the treasure and the box the seeker opens. If the two choices are identical, the seeker finds the treasure, and thus the hider would have done better to place the treasure in a different box. If the two choices are not identical, the treasure will not be found, and thus the seeker would have done better to open a different box. Therefore, this game does not have a Nash equilibrium.
89The situation is different when the description of the game includes the possibility that a player chooses a box randomly. According to this approach, the hider chooses four non-negative numbers that must add up to 1. In the professional jargon, we call this type of choice a mixed strategy. Each number corresponds to the probability of the treasure being placed in one of the boxes. For example, the choice of (0.3, 0.2, 0.2, 0.3) means that there is a probability of 30 % that he will hide the treasure in each of the two outermost boxes and a 20 % probability for each of the two innermost boxes. Or, the choice of (1, 0, 0, 0) means that the hider definitely places the treasure in the left-most box. Similarly, for the seeker, a mixed strategy is a choice of four non-negative numbers that add up to 1, with each number corresponding to the probability that he will open a particular box.
90Each player can be regarded as someone who spins a roulette wheel with four slots. Each slot in the roulette table corresponds to one of the boxes and its relative area corresponds to the probability that the player chooses the respective box. The result of the spin of the roulette wheel determines which box the player will choose. We do not necessarily think of the player as someone who actually spins the wheel. The randomness can be a result of a process that occurs inside the mind of the player when he decides to choose one of the four boxes. And, for someone looking on from the side, it may seem that the player uses a random method to choose the particular box.
91A pair of mixed strategies defines the probability that the treasure will be found. For example, if the hider places the treasure in the four boxes (A, B, A, A) with the probabilities (0.3, 0.2, 0.2, 0.3) and the seeker opens the boxes with the probabilities of (0.1, 0.4, 0.4, 0.1), the seeker will have a 22 % probability of finding the treasure. There is a probability of 30 % that the treasure will be hidden in the left-most box and a probability of 10 % that the seeker will open this box. Consequently, there is a probability of 3 % that treasure will be hidden in the left-most box and that the seeker will also open this box. Similarly, there is an 8 % probability that the treasure will be hidden and found in the box marked ”B”, and so on.
92A candidate for Nash equilibrium is a pair of mixed strategies, one for the hider and one for the seeker. In order for the pair of strategies to be an equilibrium, neither of the players benefits by switching his strategy to another strategy. That is, when the hider is aware of the seeker’s strategy, the hider does not have an alternative strategy that reduces the probability that the treasure is found; and when the seeker is aware of the hider’s strategy, the seeker does not have an alternative strategy that increases the probability of finding the treasure.
93Each of the following two reasons is sufficient to determine that the aforementioned pair of strategies is not an equilibrium: (i) The seeker’s strategy is to open the boxes with probabilities (0.1, 0.4, 0.4, 0.1). The hider can reduce the probability that the treasure will be found from 22 % to 10 % if he places the treasure in one of the outermost boxes with a probability of 1. (ii) The hider places the treasure in boxes with probabilities (0.3, 0.2, 0.2, 0.3). The seeker can increase the probability of finding the treasure from 22 % to 30 % if he searches for the treasure in one of the outermost boxes.
94It is possible to confirm that in the game’s only equilibrium each of the players chooses each box with a probability of ¼ and, therefore, the probability of finding the treasure is ¼.
95Of course, one can also think of the Treasure Hunt Game in ways other than Nash equilibrium. Here is one alternative. Let’s say that you are participating in the game in the role of the hider. You are a pessimistic person and believe that whichever mixed strategy you choose the other player will correctly guess this strategy and then choose the best action for him (and therefore the worst from your perspective). If you choose a mixed strategy in which all of the probabilities are not equal, then there is at least one box in which you could hide the treasure with a probability greater than ¼. The pessimistic approach leads you to think that the seeker will surely open this box and, therefore, the treasure will be found with a probability of more than ¼. On the other hand, if you hide the treasure with a probability of ¼ in each of the four boxes, you will ensure that the seeker finds the treasure with a probability of only ¼. Consequently, in light of the pessimistic expectations regarding your competitor’s moves, you would hide the treasure with equal probabilities in each of the boxes. In game theory, we call this type of strategy a max-min strategy.
96Similarly, we can see that the seeker’s max-min strategy is to search for the treasure at equal probabilities in each of the four boxes. Thus, we found that in the Treasure Hunt Game the equilibrium strategies and the max-min strategies are identical. This is no coincidence. We noted that the game is a zero-sum game: Whatever is good for me is bad for the other player, and vice versa. A central result in game theory, called the max-min theorem, teaches us that in every zero-sum game (and not only in the Treasure Hunt Game), Nash equilibrium strategies are identical to max-min strategies. In this way, it becomes clear that the two ways of looking at the game, which appear to be so disparate at first glance, lead to identical conclusions in games with absolute conflicts of interest. This is game theory at its best.
97Let’s return to the Treasure Hunt Game. Imagine for a moment that you are the seeker. If you believe in the predictions of game theory, you are not particularly anxious, because you know that the treasure is hidden with equal probabilities in each of the boxes. Therefore, it makes no difference to you which box is opened. But in light of the survey results of 5,500 hiders, I hope for your sake that you will indeed devote some attention to your choice. The distribution of the choice of boxes (A, B, A, A) is: (19 %, 25 %, 34 %, 22 %). The middle box marked ”A” is the most popular choice in nearly every group of students participating in the survey. (I have no idea why the box marked ”B” was the most popular choice in one very large group, at Tilburg University in Holland.) If these data predict the behavior of the hider you are competing against, then if you choose the middle box marked A, your chances of finding the treasure will increase to 34 %.
98In another survey, 3,500 students were randomly assigned to be hiders or seekers. Among the hiders, the distribution of choice was strikingly similar to the large sample population (17 %, 25 %, 35 %, 23 %). Among the seekers, the results were even more dramatic (11 %, 27 %, 47 %, 15 %). This result leads me to think that if I had hidden the treasure I would have placed it in the left-most box and had a 89 % chance of keeping it for myself, which is much higher than the game theory prediction of only 75 % probability of success in hiding the treasure from the seeker.
99So here is a ”useful” finding. Let’s say that I was a strategic advisor to the U.S. Army in its efforts to wipe out Saddam Hussein in 2003, and let’s assume that there were four palaces in Baghdad arranged along the Tigris River like the four boxes in the diagram, that one of them (marked with the letter ”B”) was the most prominent palace, and that it was possible to bomb only one palace.
100If I had assumed that they had not read the results of this study in Baghdad, then I would have recommended attacking the central palace marked A.
101If I had thought that Saddam Hussein’s advisors had read the results of this study and had assumed that the U.S. Army was unaware of the study, I would have concluded that his advisors would recommend to him that he hide in one of the outermost palaces and I would have recommended to the Americans that they attack one of them.
102And if it was well known that they had read about the study in both Washington and Baghdad – I would have had no idea what to advise.
103In any case, even if this finding is useful, and can be used for good or bad (depending on which side uses it), it is totally unrelated to an analysis of the game with the tools of game theory.
104For years, I believed that teaching game theory is not helpful and is even harmful because it can potentially encourage selfishness and deviousness .In 1998, a wonderful group of students gathered at Tel Aviv University for a seminar on economic theory. Most of the participants were pursuing academic studies as part of their military service and, because they appeared in uniform, the seminar was nicknamed ”the officers’ seminar.” I proposed to them that they tackle the mission of proving that the teaching of game theory is harmful. We composed a questionnaire that included a series of imaginary decision problems. It seemed to us that certain decisions express a tendency toward egoistic and manipulative behavior. We asked students who were about to start a game theory course, as well as other students who had already completed the course, to respond to the questionnaire. We expected that a comparison of the responses by the two groups would show that studying game theory at the undergraduate level makes the students more selfish and devious. But nothing of the sort happened. We did not find any effect of game theory on anything. But I still believe such an effect exists.
105Dozens of journalists wrote about Nash’s winning of the Nobel Prize in Economics in 1994. Everyone applauded game theory. Sylvia Nasar was the only one who recognized that the announcement of the prize was more than just a triumph for game theory – it was an event with a strong human touch. She published an article that stretched over two full pages in The New York Times, describing Nash in three stages: a young and handsome genius; a sick person roaming the idyllic campus of Princeton like a ghost; and finally, the happy ending: Nash recuperates, returns to activity and wins the Nobel Prize. The article touched people’s hearts and led to the writing of the book in which Nasar delved into the depths of Nash’s mind and delusions. The book’s success led to its being made into a film. From the public perspective, the book and film focused the attention of millions of people on mental illness and the discrimination against those who suffer from it, and instilled new hope in patients and their families.
106There are a number of inaccuracies in the film. For example, after Nash is notified that he has been awarded the Nobel Prize, there is an impressive scene: a ceremony of academic deference in which we see Nash sitting in the coffee room of Fine Hall and one professor after another comes up to him and places a pen on the table (you can watch this scene on this book’s website: http:// http://www.openbookpublishers.com/exsites/136). I happened to be there.
107In the morning, the announcement was made that John Harsanyi, Reinhard Selten and Nash would jointly receive the Nobel Prize ”for their pioneering analysis of equilibria in the theory of non-cooperative games.” Princeton University reacted to the announcement with embarrassment. Nash had no formal affiliation with the university at the time. Only out of kindness had the university provided him with a computer account. Nash was umbilically connected to the Mathematics Department at Princeton and resided in the town of Princeton, which was also the home of most of the people in the world who cared about him. How do you explain to the world that the Nobel Prize recipient is from Princeton, but not from Princeton University? In consultation with the governors of the university, the Mathematics Department decided to arrange a modest celebratory gathering to toast Nash’s award. In the department’s tea room, a number of math professors gathered, along with students who happened to be there and a very small number of economics professors. The senior university officials were there and perhaps a photographer too. The beginning of the ceremony was delayed for reasons that were unclear. Glasses were raised, and a few very brief congratulatory remarks were made. The guest of honor maintained the right to remain silent. When the participants finished their drinks, a general uneasiness spread over the room. Nash stood alone in the center of the room. No one approached him. He walked toward the table of refreshments and said to me and others who stood near the table: ”The cookies today are better than they usually are.” To the best of my knowledge, there was and is no pen ceremony anywhere. No such ceremony even took place metaphorically. Nash received an office from the university and an ID card that enabled him to enter Princeton’s faculty club. But no one addressed Nash as ”master” as in the film.
108What happens when Hollywood tries to explain game theory? In one scene in the movie (which, again, you can view on the book’s website: http://www.openbookpublishers.com/exsites/136. openbookpublishers.com/exsites/136), a group of students, including the young Nash, enter a pickup bar. Which girl to approach? The proud, stunning blonde who will only react to you if no one else approaches her, or one of the many unexceptional brunettes at the bar with whom your chances are much higher? You do not need to have experience in bars to realize that there is a place here for strategic thinking. The rational courtier will choose his objective in line with his expectations of his friends’ behavior.
109We will simplify the story and assume that Nash goes to the bar with only one friend (although in the movie, there are four men in the group). Each contemplates whether to pursue the striking blonde or turn his attention to one of the many brunettes in the bar. In keeping with the film, we will assume that if both men pursue the blonde, she will turn her back on both of them and they will end up alone that evening. If only one of them pursues the blonde, he will win her coveted company while his friend will have to settle for one of the less desirable brunettes. If both of them refrain from approaching the blonde, both of them will be able to spend the evening with a brunette.
110We can present the story in the following table:
and the other man chooses
111One player controls the choice of row and a second player the choice of column. Each square contains a description of the result from the perspective of the player who chooses the row. For example, if the row player chooses to pursue the blonde and the column player turns to the brunette, the result from the perspective of the row player is winning the blonde.
112This description brings us closer to presenting the situation in the conventional language of game theory. In each square, we insert a number that represents the amount of utility the player derives from the result described in the square. If the number assigned to one square is higher than the number assigned to another square, this means that the player prefers the result in the former to the result in the latter.
113The highest number was assigned to the result in which the row player wins the blonde’s company. The lowest number was assigned to remaining alone. In the middle are the two possibilities in which the row player finds consolation in the arms of a brunette. His consolation will be greater if the other player does not find a way to the blonde’s heart.
114In the game theory literature, this game is called Chicken. It has two Nash equilibria. In one equilibrium, Nash pursues the blonde and his friend suffices with the brunette. In the second equilibrium, Nash suffices with the brunette and his friend pursues the blonde.
115Just for fun, I asked the audience at the lectures on ”Game Theory and John Nash” to play the game. About 46 % of some 3,000 men and 48 % of about 1,500 women chose the blonde. The Dutch and the Israelis were the boldest: 56 % of the men and women chose the blonde. Actually, I do not know whether this finding is an expression of strategic boldness in bars or simply differences in the preference of hair color.
116The discussion of the game in the movie is confused and misleading. Nash in the movie says that he discovered a contradiction in a principle that has been accepted in economics since the time of Adam Smith:
Adam Smith said the best outcome for the group comes from everybody trying to do what’s best for himself. He was wrong. The best outcome comes from everybody trying to do what’s best for himself and the group.
117The message attributed to Smith in the film is simplistic. In what sense does each individual’s concern for himself generate the best result? What is best for a group that includes individuals with disparate interests? In any case, the discovery attributed to Nash – who argued that if people act in a self-centered way the result for the group is actually liable to be worse than if people act in a socially responsible way – is not well demonstrated in the bar scene.
118The idea is presented clearly in the game called the Prisoner’s Dilemma, a game that has lost some of its luster after being used so often to illustrate game theory. The Prisoner’s Dilemma involves two prisoners who are apprehended while committing a misdemeanor and are suspected of committing a serious crime. But the Prisoner’s Dilemma Game can also be expressed as follows: there are a number of residents on a street, and the level of cleanliness in the street depends on whether or not the residents of the street keep it clean. Let’s say that each resident believes that, regardless of what the neighbors do, his exertion in maintaining cleanliness is more bothersome to him than the marginal increase in environmental damage that would result if he discontinued this effort. At the same time, let’s also assume that every resident would prefer a situation in which all residents maintained the cleanliness and the street remained clean, rather than having all residents, including himself, neglecting the cleanliness and living amidst the stench of garbage. Of course, each resident only controls his own actions.
119This game does not require a player to enter the other’s shoes. Whatever the player believes regarding the behavior of the other players, it would be best for him to not make an effort to preserve the cleanliness. Consequently, if all of the residents are rational, the sad result is that no one maintains the cleanliness and the street smells. Each player acts only to promote his own interests and the result is worse than if all of the residents maintain the street’s cleanliness. To the best of my knowledge, the Prisoner’s Dilemma Game was formulated by Albert Tucker, and he has no connection to Nash’s contribution to game theory.
120Nash is depicted in the movie as someone who managed to gain control of the delusional voices he heard, as someone who discovers the true meaning of life when he lovingly offers his wife a white kerchief, and as someone who returns to the academic world. If I were to choose an ending to the movie I would choose a different one, a bit more melancholy. In his autobiography, Nash noted that he also lost something during his recuperation process. He characterizes the period of irrationality as a time of ” dream-like delusional hypotheses” and describes his return to ”thinking rationally again in the style that is characteristic of scientists,” as a process that ”is not entirely a matter of joy” because rationality of thought ”imposes a limit on a person’s concept of his relations to the cosmos.”
121The story of Nash is the journey of his transformation from a descriptive term in an abstract mathematical concept into a human being. The title crazy genius is for us an invitation to an encounter with other worlds. We try to imagine the voices that Nash heard and ask where he was during the thirty years when he walked among us but was not with us. People who are different scare us, but we are also attracted to them. Curious and scared, we confront our prejudices and try to accept that someone who is mentally ill, even if he is different from us, deserves to be one of us.
122As for me, I was fortunate to be present at stages of Nash’s journey and the march of game theory from the margin of economics to its core. And what do I find in game theory? A beautiful mind. It is interesting because it touches upon the way we think about the world. It is beautiful because it offers lucid formulations for what appears to be confusing. It has the same interest and beauty that I find in philosophy, mathematics, logic and literature. If in a circuitous way that I am unaware of it can also be useful – then so much the better. But, in my view, usefulness is neither the criterion nor the essence.
List of illustrations
|Title||Traveler's Dilema Distribution of declared values|
|Title||Ultimatum Game Distribution of Proposals|
© Open Book Publishers, 2012