In real life, of course, you never know what will happen, because the crooks may have additional considerations (such as the prospect of sleeping with the fishes if they rat out the wrong guy). Consequently the Nash equilibrium calculation does not always predict how people will really behave. Sometimes people temper their choices with considerations of fairness, and sometimes they act out of spite. In Prisoner's Dilemma situations, some people actually do choose to cooperate. But that doesn't detract from the importance of the Nash equilibrium, as economists Charles Holt and Alvin Roth point out. "The Nash equilibrium is useful not just when it is itself an accurate predictor of how people will behave in a game but also when it is not," they write, "because then it identifies a situation in which there is a tension between individual incentives and other motivations." So if people cooperate (at least at first) in a Prisoner's Dilemma situation, Nash's math tells us that such cooperation, "because it is not an equilibrium, is going to be unstable in ways that can make cooperation difficult to maintain."21
Though it is a simplified representation of real life, the Prisoner's Dilemma game does capture the essence of many social interactions. But obviously you cannot easily assess any social situation by calculating the Nash equilibrium. Real-life games often involve many players and complicated payoff rules. While Nash showed that there is always at least one equilibrium point, it's another matter to figure out what that point is. (And often there is more than one Nash equilibrium point, which makes things really messy.) Remember, each player's "strategy" will typically be a mixed strategy, drawn from maybe dozens or hundreds or thousands (or more) of pure "specific" strategies. In most games with many players, calculating all the probabilities for all the combinations of all those choices exceeds the computational capacity of Intel, Microsoft, IBM, and Apple put together.
THE PUBLIC GOOD
It's not hopeless, though. Consider another favorite game to illustrate "defection"—the public goods game. The idea is that some members in a community reap the benefits of membership without paying their dues. It's like watching public television but never calling in to make a pledge during the fund drives. At first glance, the defector wins this game—getting the benefit of enjoying Morse and Poirot without paying a price. But wait a minute. If everybody defected, there would be no benefit for anybody. The free riders would become hapless hitchhikers.
Similarly, suppose your neighborhood association decided to collect donations to create a park. You'd enjoy the park, but if you reason that enough others in the neighborhood will contribute enough money to build it, you might decline to contribute. If everybody reasons the same way, though, there will be no park. But suppose that defecting (declining to pay) and cooperating (contributing your fair share) are not the only possible strategies. You can imagine a third strategy, called reciprocating. If you are a reciprocator, you pay only if you know that a certain number of the other players have decided to pay. Computer simulations of this kind of game suggest that a mix of these strategies among the players can reach a Nash equilibrium.
Experiments with real people show the same thing. One study, reported in 2005, tested college students on a contrived version of the public goods game. Four players were each given tokens (representing money) and told they could contribute as many as they liked into a "public pot," keeping the rest in their personal account. The experimenter then doubled the number of tokens in the pot. One player at a time was told how much had been contributed to the pot and then given a chance to change his or her contribution. When the game ended (after a random number of rounds), all the tokens were then evenly divided up among all the players.
How would you play? Since, in the end, all four players split the pot equally, the people who put in the least to begin with end up with the most tokens—their share of the pot plus the money they held back in their personal account. Of course, if nobody put any in to begin with, nobody reaped the benefit of the experimenter's largesse, kind of like a local government forgoing federal matching funds for a highway project. So it would seem to be a good strategy to donate something to the pot. But if you want to get a better payoff than anyone else, you should put in less than the others. Maybe one token. On the other hand, everybody in the group will get more if you put more in the pot to begin with. (That way, you might not get more than everybody else, but you'll get more than you otherwise would.)
When groups of four played this game repeatedly, a pattern of behavior emerged. Players fell into three readily identifiable groups: cooperators, defectors (or "free riders"), and reciprocators. Since all the players learned at some point how much had been contributed, they could adjust their behavior accordingly. Some players remained stingy (defectors), some continued to contribute generously (cooperators), and others contributed more if others in the group had donated significantly (reciprocators).
Over time, the members of each group earned equal amounts of money, suggesting that something like a Nash equilibrium had been achieved—they all won as much as they could, given the strategy of the others. In other words, in this kind of game, the human race plays a mixed strategy—about 13 percent cooperators, 20 percent defectors (free riders), and 63 percent reciprocators in this particular experiment. "Our results support the view that our human subject population is in a stable … equilibrium of types," wrote the researchers, Robert Kurzban and Daniel Houser.22 Knowing about the Nash equilibrium helps make sense of results like these.
GAME THEORY TODAY
Together with his paper on the bargaining problem (which treats cooperative game situations), Nash's work on equilibria in many-player games greatly expanded game theory's scope beyond von Neumann and Morgenstern's book, providing the foundation for much of the work in game theory going on today. There's more to game theory than the Nash equilibrium, of course, but it is still at the heart of current endeavors to apply game theory to society broadly.
Over the years, game theorists have developed math for games where coalitions do form, where information is incomplete, where players are less than perfectly rational. Models of all of these situations, plus many others, can be built using game theory's complex mathematical tools. It would take a whole book (actually, several books) to describe all of those subsequent developments (and many such books have been written). It's not necessary to know all those details of game theory history, but it is important to know that game theory does have a rich and complex history. It is a deep and complicated subject, full of many highly technical and nuanced contributions of substantial mathematical sophistication.
Even today game theory remains very much a work in progress. Many deep questions about it do not seem to have been given compelling answers. In fact, if you peruse the various accounts of game theory, you are likely to come away confused. Its practitioners do not all agree on how to interpret some aspects of game theory, and they certainly disagree about how to advertise it.
Some presentations seem to suggest that game theory should predict human behavior—what choices people will make in games (or in economics or other realms of life). Others insist that game theory does not predict, but prescribes—it tells you what you ought to do (if you want to win the game), not what any player would actually do in a game. Or some experts will say that game theory predicts what a "rational" person will do, acknowledging that there's no accounting for how irrational some people (even those playing high-stakes games) can be. Of course, if you ask such experts to define "rational," they're likely to say that it means behaving in the way that game theory predicts.
To me, it seems obvious that basic game theory does not always successfully predict what people will do, since most people are about as rational as pi. Neither is it obvious that game theory offers a foolproof way to determine what is the rational thing to do. There may always be additional considerations in making a "rational" choice that have not been included in game theory's mathematical framework.
Game theory does predict outcomes for different strategies in different situations, thou
gh. In principle you could use game theory to analyze lots of ordinary games, like checkers, as well as many problems in the real world where the concept of game is much broader. It can range from trying to beat another car into a parking place to global thermonuclear war. The idea is that when faced with deciding what to do in some strategic interaction, the math can tell you which move is most likely to be successful. So if you know what you want to achieve, game theory can help you—if your circumstances lend themselves to game theory representation.
The question is, are there ever any such circumstances? Early euphoria about game theory's potential to illuminate social issues soon dissipated, as a famous game theory text noted in 1957. "Initially there was a naive band-wagon feeling that game theory solved innumerable problems of sociology and economics, or that, at the least, it made their solution a practical matter of a few years' work. This has not turned out to be the case."23
Such an early pessimistic assessment isn't so surprising. There's always a lack of patience in the scientific world; many people want new ideas to pay off quickly, even when more rational observers realize that decades of difficult work may be needed for a theory to reach maturity. But even six decades after the von Neumann– Morgenstern book appeared, you could find some rather negative assessments of game theory's relevance to real life.
In an afterword to the 60-year-anniversary edition of Theory of Games, Ariel Rubenstein acknowledged that game theory had successfully entrenched itself in economic science. "Game theory has moved from the fringe of economics into its mainstream," he wrote. "The distinction between economic theorist and game theorist has virtually disappeared."24 But he was not impressed with claims that game theory was really good for much else, not even games. "Game theory is not a box of magic tricks that can help us play games more successfully. There are very few insights from game theory that would improve one's game of chess or poker," Rubenstein wrote.25
He scoffed at theorists who believed game theory could actually predict behavior, or even improve performance in real-life strategic interactions. "I have never been persuaded that there is a solid foundation for this belief," he wrote. "The fact that the academics have a vested interest in it makes it even less credible." Game theory in Rubenstein's view is much like logic—form without substance, a guide for comparing contingencies but not a handbook for action. "Game theory does not tell us which action is preferable or predict what other people will do…. The challenges facing the world today are far too complex to be captured by any matrix game."26
OK—maybe this book should end here. But no. I think Rubenstein has a point, but also that he is taking a very narrow view. In fact, I think his attitude neglects an important fact about the nature of science.
Scientists make models. Models capture the essence of some aspect of something, hopefully the aspect of interest for some particular use or another. Game theory is all about making models of human interactions. Of course game theory does not capture all the nuances of human behavior—no model does. No map of Los Angeles shows every building, every crack in every sidewalk, or every pothole—if it showed all that, it wouldn't be a map of Los Angeles, it would be Los Angeles. Nevertheless, a map that leaves out all those things can still help you get where you want to go (although in L.A. you might get there slowly).
Naturally, game theory introduces simplifications—it is, after all, a model of real-life situations, not real life itself. In that respect it is just like all other science, providing simplified models of reality that are accurate enough to draw useful conclusions about that reality. You don't have to worry about the chemical composition of the moon and sun when predicting eclipses, only their masses and motions. It's like predicting the weather. The atmosphere is a physical system, but Isaac Newton was no meteorologist. Eighteenth-century scholars did not throw away Newton's Principia because it couldn't predict thunderstorms. But after a few centuries, physics did get to the point where it could offer reasonably decent weather forecasts. Just because game theory cannot predict human behavior infallibly today doesn't mean that its insights are worthless.
In his book Behavioral Game Theory, Colin Camerer addresses these issues with exceptional insight and eloquence. It is true, he notes, that many experiments produce results that seem—at first— to disconfirm game theory's predictions. But it's clearly a mistake to think that therefore there is something wrong with game theory's math. "If people don't play the way theory says, their behavior does not prove the mathematics wrong, any more than finding that cashiers sometimes give the wrong change disproves arithmetic," Camerer points out.27 Besides, game theory (in its original form) is based on players' behaving rationally and selfishly. If actual real-life behavior departs from game theory's forecast, perhaps there's just something wrong with the concepts of rationality and selfishness. In that case, incorporating better knowledge of human psychology (especially in social situations) into game theory's equations can dramatically improve predictions of human behavior and help explain why that behavior is sometimes surprising. That is exactly the sort of thing that Camerer's specialty, behavioral game theory, is intended to do. "The goal is not to ‘disprove' game theory … but to improve it," Camerer writes.28
As it turns out, game theory is widely used today in scientific efforts to understand all sorts of things. While Nash's 1994 Nobel Prize recognized the math establishing game theory's foundations, the 2005 economics Nobel trumpeted the achievements of two important pioneers of game theory's many important applications. Economist Thomas Schelling, of the University of Maryland, understood in the 1950s that game theory offered a mathematical language suitable for unifying the social sciences, a vision he articulated in his 1960 book The Strategy of Conflict. "Schelling's work prompted new developments in game theory and accelerated its use and application throughout the social sciences," the Royal Swedish Academy of Sciences remarked on awarding the prize.29
Schelling paid particular attention to game-theoretic analysis of international relations, specifically (not surprising for the time) focusing on the risks of armed conflict. In gamelike conflict situations with more than one Nash equilibrium, Schelling showed how to determine which of the equilibrium possibilities was most plausible. And he identified various counterintuitive conclusions about conflict strategy that game theory revealed. An advancing general burning bridges behind him would seem to be limiting his army's options, for example. But the signal sent to the enemy—that the oncoming army had no way to retreat—would likely diminish the opposition's willingness to fight. Similar reasoning transferred to the economic realm, where a company might decide to build a big, expensive production plant, even if it meant a higher cost of making its product, if by flaunting such a major commitment it scared competitors out of the market.
Schelling's insights also extended to games where all the players desire a common (coordinated) outcome more than any particular outcome—in other words, when it is better for everybody to be on the same page, regardless of what the page is. A simple example would be a team of people desiring to eat dinner at the same restaurant. It doesn't matter what restaurant (as long as the food is not too spicy); the goal is for everyone to be together. When everybody can communicate with each other, coordination is rarely a problem (or at least it shouldn't be), but in many such situations communication is restricted. Schelling shed considerable light on the game-theoretic issues involved in reaching coordinated solutions to such social problems. Some of Schelling's later work applied game theory to the rapid change in some neighborhoods from a mixture of races to being largely segregated, and to limits on individual control over behavior—why people do so many things they really don't want to do, like smoke or drink too much, while not doing things they really want to, like exercising.
2005's other economics Nobel winner, Robert Aumann, has long been a leading force in expanding the scope of game theory to many disciplines, from biology to mathematics. A German-born Israeli at the Hebrew University of Jerusalem, Aumann took special
interest in long-term cooperative behavior, a topic of special relevance to the social sciences (after all, long-term cooperation is the defining feature of civilization itself). In particular, Aumann analyzed the Prisoner's Dilemma game from the perspective of infinitely repeated play, rather than the one-shot deal in which both players' best move is to rat the other out. Over the long run, Aumann showed, cooperative behavior can be sustained even by players who still have their own self-interest at heart.
Aumann's "repeated games" approach had wide application, both in cases where it led to cooperation and where it didn't. By showing how game theory's rules could facilitate cooperation, he also identified the circumstances where cooperation was less likely—when many players are involved, for instance, or when communication is limited or time is short. Game theory helps to show why certain common forms of collective behavior materialize under such circumstances. "The repeated-games approach clarifies the raison d'être of many institutions, ranging from merchant guilds and organized crime to wage negotiations and international trade agreements," the Swedish academy pointed out.
While Nobel Prizes shine the media spotlight on specific achievements of game theory, they tell only a small portion of the whole story. Game theory's uses have expanded to multiple arenas in recent years. Economics is full of applications, from guiding negotiations between labor unions and management to auctioning licenses for exploiting the electromagnetic spectrum. Game theory is helpful in matching medical residents to hospitals, in understanding the spread of disease, and in determining how to best vaccinate against various diseases—even to explain the incentives (or lack thereof) for hospitals to invest in fighting bacterial resistance to antibiotics. Game theory is valuable for understanding terrorist organizations and forecasting terrorist strategies. For analyzing voting behaviors, for understanding consciousness and artificial intelligence, for solving problems in ecology, for comprehending cancer. You can call on game theory to explain why the numbers of male and female births are roughly equal, why people get stingier as they get older, and why people like to gossip about other people.
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