11
Pascal’s Wager
Games, probability, information, and ignorance

Although this may seem a paradox, all exact science is dominated by the idea of approximation.

—Bertrand Russell

As a teenager in 17th-century France, Blaise Pascal seemed destined for mathematical greatness. He wrote a genius-caliber treatise on geometry at age 16 and invented a rudimentary computer to assist the calculations of his tax-collector father. But as an adult, Pascal was seduced by religion, forgoing math to produce a series of philosophical musings assembled (after his death) into a book called Pensées. He died at 39, leaving a legacy, in the words of the mathematician E. T. Bell, as “perhaps the greatest might-have-been in history.”1

Still, Pascal remains a familiar name in today’s mathematics textbooks, thanks to a favor he did for a French aristocrat who desired assistance with his gambling habit. What Pascal offered was not religious counseling on the evils of gambling, but mathematical advice on how to win. In his correspondence on this question with Pierre Fermat, Pascal essentially invented probability theory. What’s more, out of Pascal’s religious ruminations came an idea about probability that was to emerge centuries later as a critical concept in mathematics, with particular implications for game theory.



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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature 11 Pascal’s Wager Games, probability, information, and ignorance Although this may seem a paradox, all exact science is dominated by the idea of approximation. —Bertrand Russell As a teenager in 17th-century France, Blaise Pascal seemed destined for mathematical greatness. He wrote a genius-caliber treatise on geometry at age 16 and invented a rudimentary computer to assist the calculations of his tax-collector father. But as an adult, Pascal was seduced by religion, forgoing math to produce a series of philosophical musings assembled (after his death) into a book called Pensées. He died at 39, leaving a legacy, in the words of the mathematician E. T. Bell, as “perhaps the greatest might-have-been in history.”1 Still, Pascal remains a familiar name in today’s mathematics textbooks, thanks to a favor he did for a French aristocrat who desired assistance with his gambling habit. What Pascal offered was not religious counseling on the evils of gambling, but mathematical advice on how to win. In his correspondence on this question with Pierre Fermat, Pascal essentially invented probability theory. What’s more, out of Pascal’s religious ruminations came an idea about probability that was to emerge centuries later as a critical concept in mathematics, with particular implications for game theory.

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature When it comes to making bets, Pascal observed, it is not enough to know the odds of winning or losing. You need to know what’s at stake. You might want to take unfavorable odds if the payoff for winning would be really huge, for example. Or you might consider playing it safe by betting on a sure thing even if the payoff was small. But it wouldn’t seem wise to bet on a long shot if the payoff was going to be meager. Pascal framed this issue in his religious writings, specifically in the context of making a wager about the existence of God. Choosing to believe in God was like making a bet, he said. If you believe in God, and that belief turns out to be wrong, you haven’t lost much. But if God does exist, believing wins you an eternity of heavenly happiness. Even if God is a low-probability deity, the payoff is so great (basically, infinite) that He’s a good bet anyway. “Let us weigh the gain and the loss in wagering that God is,” Pascal wrote. “Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is.”2 Pascal’s reasoning may have been theologically simplistic, but it certainly was mathematically intriguing.3 It illustrated the kind of reasoning that goes into calculating the “mathematical expectation” of an economic decision—you multiply the probability of an outcome by the value of that outcome. The rational choice is the decision that computes to give the highest expected value. Pascal’s wager is often cited as the earliest example of a math-based approach to decision theory. In real life, of course, people don’t always make their decisions simply by performing such calculations. And when your best decision depends on what other people are deciding, simple decision theory no longer applies—making the best bets becomes a problem in game theory. (Some experts would say decision theory is just a special case of game theory, in which one player plays the game against nature.) Still, probabilities and expected payoffs remain intertwined with game theory in a profound and complicated way. For that matter, all of science is intertwined with probability theory in a profound way—it’s essential for the entire process of

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature observation, experiment, and measurement, and then comparing those numbers with theory. And probability arises not only in making measurements and testing hypotheses, but also in the very description of physical phenomena, particularly in the realm of statistical physics. In the social sciences, of course, probability theory is also indispensable, as Adolphe Quetelet argued almost two centuries ago. So game theory’s intimate relationship with probabilities, I’d wager, is one of the reasons why it finds such widespread applicability in so many different scientific contexts. And no doubt it’s this aspect of game theory that has positioned it so strategically as an agent for merging social and physical statistics into a physics of society—something like Asimov’s psychohistory or a Code of Nature. So far, attempts to devise a sociophysics for describing society have mostly been based not on game theory, but on statistical mechanics (as was Asimov’s fictional psychohistory). But game theory’s mixed strategy/probabilistic formulas exhibit striking similarities to the probability distributions of statistical physics. In fact, the mixed strategies used by game players to achieve a Nash equilibrium are probability distributions, precisely like the distributions of molecules in a gas that statistical physics quantifies. This realization prompts a remarkable conclusion—that, in a certain sense, game theory and statistical mechanics are alter egos. That is to say, they can be expressed using the same mathematical language. To be more precise, you’d have to say that certain versions of game theory share math identical to particular formulations of statistical mechanics, but the deep underlying connection remains. It’s just that few people have noticed it. STATISTICS AND GAMES If you search the research literature thoroughly, though, you will find several papers from the handful of scientists who have begun to exploit the game theory–statistical physics connection. Among them is David Wolpert, a physicist-mathematician at NASA’s Ames Research Center in California.

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature Wolpert is one of those creative thinkers who refuse to be straitjacketed by normal scientific stereotypes. He pursues his own intuitions and interests along the amorphous edges separating (or connecting) physics, math, computer science, and complexity theory. I first encountered him in the early 1990s while he was exploring the frontiers of interdisciplinary science at the Santa Fe Institute, discussing such issues as the limits of computability and the nature of memory. In early 2004, Wolpert’s name caught my eye when I noticed a paper he posted on the World Wide Web’s physics preprint page.4 His paper showed how to build a bridge between game theory and statistical physics using information theory (providing, incidentally, one of the key inspirations for writing this book). In fact, as Wolpert showed in the paper that attracted my attention to this issue in the first place, a particular approach to statistical mechanics turns out to use math that is equivalent to the math for noncooperative games. Wolpert’s paper noted that the particles described by statistical physics are trying to minimize their collective energy, like the way people in a game try to reach the Nash equilibrium that maximizes their utility. The mixed strategies used by players to achieve a Nash equilibrium are probability distributions, just like the distribution of energy among particles described by statistical physics. After reading Wolpert’s paper, I wrote him about it and then a few months later discussed it with him at a complexity conference outside Boston where he was presenting some related work. I asked what had motivated him to forge a link between game theory and statistical physics. His answer: rejection. Wolpert had been working on collective machine learning systems, situations in which individual computers, or robots, or other autonomous devices with their own individual goals could be coordinated to achieve an objective for the entire system. The idea is to find a way to establish relationships between the individual “agents” so that their collective behavior would serve the global goal. He noticed similarities in his work to a paper published in Physical Review Letters about nanosized computers. So Wolpert sent off one of his papers to that journal.

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature “The editor actually came back and said ‘Well, … what you’re doing just plain isn’t physics,’” Wolpert said. “And I was annoyed.” So he started thinking about physics and game theory. After all, a bunch of agents with their own agendas, yet pursuing a common goal, is entirely analogous to players in a game seeking a Nash equilibrium. “And then I said, OK, I’m going to try to take that and completely translate it into a physics system,” he recalled.5 Games deal with players; physics deals with molecules. So Wolpert worked on the math that would represent a player’s strategy like a molecule’s state of motion. The mix of all the players’ strategies would then be like the combined set of motion states of all the molecules, as ordinarily described by statistical physics. The formulas he came up with would allow you to calculate a good approximation to the actual set of any individual player’s strategies in a game, given some limited knowledge about them. You could then do exactly the same sort of calculation for the combined strategies of all the players in a game. Basically, Wolpert showed how the math of statistical physics turns out to be the same as the math for games where players have limited rationality. “Those topics are fundamentally one and the same,” he wrote in his paper. “This identification raises the potential of transferring some of the powerful mathematical techniques that have been developed in the statistical physics community to the analysis of noncooperative game theory.”6 Wolpert’s mathematical machinations were rooted in the idea of “maximum entropy,” a principle relating standard statistical physics to information theory, the math designed to quantify the sending and receiving of messages. The maximum entropy (or “maxent”) idea was promoted by the maverick physicist Edwin Jaynes in a 1957 paper that was embraced by a number of physicists but ignored by many others. Wolpert, for one, calls Jaynes’s work “gloriously beautiful” and thinks that it’s just what scientists need in order “to bring game theory into the 21st century.” Jaynes’s principle is simultaneously intriguing and frustrating. It seems essentially simple but nevertheless poses tricky complications. It is intimately related to the physical concept of entropy, but is still subtly different. In any event, its explanation requires a

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature brief excursion into the nature of probability theory and information theory, the essential threads tying game theory and statistical physics together. PROBABILITY AND INFORMATION For centuries, scientists and mathematicians have argued about the meaning of probability. Even today there exist separate schools of probabilistic thought, generally referred to by the shorthand labels of “objective” and “subjective.” But those labels conceal a tangle of subarguments and technical subtleties that make probability theory one of the most contentious and confusing realms of math and science. In a way, that’s a bit surprising, since probability theory really lies at the very foundation of science, playing the central role in the process of analyzing experimental data and testing theories. It’s what doing science is all about. You’d think they’d have it all worked out by now. But establishing rules for science is a little like framing a constitution for Iraq. There are different philosophies and approaches to science. The truth is that science (unlike mathematics) is not built on a rock-solid foundation of irreducible rules. Science is like grammar. Grammar arises from regularities that evolve in the way native speakers of a language form their words and string them together. A true grammarian does not tell people how they should speak, but codifies the way that people actually do speak. Science does not emanate from a cookbook that provides recipes for revealing nature’s secrets, but from a mix of methods that somehow succeed in rendering nature comprehensible. That’s why science is not all experiment, and not all theory, but a complex interplay of both. Ultimately, though, theory and experiment must mesh if the scientist’s picture of nature is to be meaningful and useful. And in most realms of science, you need math to test the mesh. Probability theory is the tool for performing that test. (Different ideas about how to perform the test, then, lead to different conceptions of probability.)

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature Before Maxwell, probability theory in science was mostly limited to quantifying things like measurement errors. Laplace and others showed the way to estimate how far off your measurement was likely to be from the true value for a particular degree of confidence. Laplace himself applied this approach to measuring the mass of Saturn. He concluded that there was only one chance in 11,000 that the true mass of Saturn would deviate from the then-current measurement by more than 1 percent. (As it turned out, today’s best measurement indeed differs from the one in Laplace’s day by only 0.6 percent.) Probability theory has developed into an amazingly precise way of making such estimates. But what does probability itself really mean? If you ask people who ought to know, you’ll get different answers. The “objective” school of thought insists that the probability of an event is a property of the event. You observe in what fraction of all cases that event happens and thereby measure its objective probability. The subjective view, on the other hand, argues that probability is a belief about how likely something is to happen. Measuring how often something happens gives you a frequency, not a probability, the subjectivists maintain. There is no point here delving into the debates about the relative merits of these two views. Dozens of books have been devoted to that controversy, which is largely irrelevant to game theory. The fact is that the prevailing view today, among physicists at least, is that the subjectivist approach contains elements that are essential for a sound assessment of scientific data. Subjective statistics often goes under the label of Bayesian, after Thomas Bayes, the English clergyman who discussed an approach of that nature in a paper published in 1763 (two years after his death). Today a formula known as Bayes’ theorem is at the heart of practicing the subjective statistics approach (although that precise theorem was actually worked out by Laplace). In any case, the Bayesian viewpoint today comes in a variety of flavors, and there is much disagreement about how it should be interpreted and applied (perhaps because it is, after all, subjective). From a practical point of view, though, the math of objective

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature and subjective probability theory does not really differ in any fundamental respect other than its interpretation. It’s just that in some cases it seems more convenient, or more appropriate, to use one rather than another, as Jaynes pointed out half a century ago. INFORMATION AND IGNORANCE In his 1957 paper,7 Jaynes championed the subjectivity side of the probability debate. He noted that both views, subjectivist and objectivist, were needed in physics, but that for some types of problems only the subjective approach would do. He argued that the subjective approach can be useful even when you know nothing about the system you are interested in to begin with. If you are given a box full of particles but know nothing about them—not their mass, not their composition, not their internal structure—there’s not much you can say about their behavior. You know the laws of physics, but you don’t have any knowledge about the system to apply the laws to. In other words, your ignorance about the behavior of the particles is at a maximum. Early pioneers of probability theory, such as Jacob Bernoulli and Laplace, said that in such circumstances you must simply assume that all the possibilities are equally likely—until you have some reason to assume otherwise. Well, that helps in doing the calculations, perhaps, but is there any real basis for assuming the probabilities are equal? Except for certain cases where an obvious symmetry is at play (say, a perfectly balanced two-sided coin), Jaynes said, many other assumptions might be equally well justified (or the way he phrased it, any other assumption would be equally arbitrary).8 Jaynes saw a way of coping with this situation, though, with the help of the then fairly new theory of information devised by Claude Shannon of Bell Labs. Shannon was interested in quantifying communication, the sending of messages, in a way that would help engineers find ways to communicate more efficiently (he worked for the telephone company, after all). He found math that

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature could quantify information quite nicely if you viewed communication as the reduction of uncertainty. Before communication begins, all messages are possible, so uncertainty is high; as messages are actually received, that uncertainty is reduced. Shannon’s math applied generally to any system of signaling, from Morse Code to smoke signals. But suppose, for example, that all you wanted to do was send someone a single English word (from among all the words in a standard unabridged dictionary, about half a million). If you tell the recipient of the message that the word is in the first half of the dictionary, you’ve reduced the number of possibilities from 500,000 to 250,000. In other words, you have reduced the uncertainty by half (which so happens to correspond to one bit of information). Shannon elaborated on this idea to show how all communication could be quantified based on the idea that messages reduce uncertainty. He found a formula for a quantity that measures that uncertainty precisely—the greater the uncertainty, the greater the quantity. Shannon called the quantity entropy, a conscious analogy to the entropy term used by physicists in statistical mechanics and thermodynamics. Physicists’ entropy is a measure of the disorder in a physical system. Suppose you have a chamber containing separate compartments, and you place a zillion molecules of oxygen in the left-side compartment and 4 zillion molecules of nitrogen in the right-side compartment. Then you remove the partition between the compartments. The molecules soon get all mixed up—more disordered—and so the entropy of the system has increased. But something else has happened—you no longer know where the molecules are. Your ignorance of their location has increased just as the entropy has increased. Shannon showed that his formula for entropy in communication—as a measure of ignorance or uncertainty—is precisely the same equation that is used in statistical mechanics to describe the increasing entropy of a collection of particles. Entropy, in other words, is the same thing as ignorance. En-

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature tropy is synonymous with uncertainty. Information theory therefore provides a precise new way of measuring uncertainty in a probability distribution. So here’s a clue about what to do when you know nothing about the probabilities in the system you want to study. Choose a probability distribution that maximizes the entropy! Maximum entropy means maximum ignorance, and if you know nothing, ignorance is by definition at a maximum. Assuming maximum entropy/ ignorance, then, is not just an assumption; it’s a factual statement about your situation. Jaynes proposed that this notion of maximum ignorance should be elevated to the status of a basic principle for describing anything scientifically. In his view, statistical mechanics itself just became a system of statistical inference about a system. By taking the maxent approach, you still get all the computational rules that statistical mechanics provides, without the need to assume anything at all about the underlying physics. In particular, you now can justify the notion that all the possibilities are equally possible. The whole idea is that no possibility (allowed by the laws of physics) gets left out. Everything not explicitly excluded by the information you’ve got has to be viewed as having some probability of occurring. (In standard statistical mechanics, that feature was simply assumed without evidence—probability distributions were based on the idea that molecules would explore all their possible behaviors.) And if you know nothing, you cannot say that any one possibility is more likely than any other—that would be knowledge. Of course, if you know something about the probabilities, you can factor that in to the probability distribution you use to make your predictions about what’s going to happen. But if you know nothing at all, there’s only one probability distribution that you can identify for making your bets: the one that has the maximum entropy, the maximum uncertainty, the maximum ignorance. It makes sense, after all, because knowing nothing is, in fact, being maximally ignorant. This is the magic that makes it possible to make a prediction

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature even when knowing nothing about the particles or people you’re making the prediction about. Of course, your prediction might not turn out to be right. But it’s still the best possible prediction you can make, the likeliest answer you can identify, when you know nothing to begin with. “The fact that a probability distribution maximizes the entropy subject to certain constraints becomes the essential fact which justifies use of that distribution for inference,” Jaynes wrote. “Whether or not the results agree with experiment, they still represent the best estimates that could have been made on the basis of the information available.”9 But what, exactly, does it mean to “maximize the entropy”? It simply means choosing the probability distribution that would result from adding up all the possibilities permitted by the laws of nature (since you know nothing, you cannot leave out anything that’s possible). Here’s a simple example. Suppose that you want to predict the average grade for a class of 100 students. All you know are the rules (that is, the laws of nature)—everybody gets a grade, and the grade has to be A, B, C, D, or F (no incompletes allowed). You don’t know anything about the caliber of the students or how hard the class is. What is your best prediction of the average grade for the kids in the class? In other words, how do you find a probability distribution for the grades that tells you which grade average is the most probable? Applying the maxent or maximum ignorance principle, you simply assume that the grades can be distributed in all possible ways—all possible combinations equally likely. For instance, one possible distribution is 100 A’s and nothing else. Another would be all F’s. There could be 20 students with each grade. You could have 50 C’s, 20 B’s and 20 D’s, 5 A’s and 5 F’s. All the combinations sum to an ensemble of possibilities that constitutes the probability distribution corresponding to no knowledge—maximum ignorance—about the class and the kids and their grades. In statistical physics, this sort of thing is called the “canonical ensemble”—the set of possible states for the molecules in a system. Each possible combination is a microstate. Many different possible

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature microstates (distributions of grades) can correspond to the same average (the macrostate). Don’t try to list all the possible combinations; it would take you a very long time. (You’re talking something close to 10 to the 70th power.) But you can calculate, or even see intuitively, that the most likely average grade will be C. Of all the possible microstate combinations, many more work out to a C average than to any other grade. There is only one way, for instance, to have a perfect A average—all 100 students getting A’s. But you can get a C average in many different ways—100 C’s, 50 A’s and 50 F’s, 20 students getting each of the five grades, and so on.10 It’s just like flipping pennies, four flips at a time, with the grade corresponding to the number of heads that turn up (0 = F, 4 = A). In 100 trials, many combinations give an average of 2, but only a few will give an average of 0 or 4. So your prediction, based on knowing nothing, will be an average grade of C. BACK TO THE GAME In game theory, a player’s mixed strategy is also a probability distribution, much like grades or penny flips. Game theory is all about how to figure out what each player’s best mixed strategy would be (for maximizing utility, or the payoff, of the game). In a multiplayer game, there is at least one mix of all players’ mixed strategies for which no one player could do any better by changing strategies— the Nash equilibrium, game theory’s most important foundational principle. But Nash’s foundation of modern game theory has its cracks. While it’s true that, as Nash showed, all games (with certain qualifications) have at least one Nash equilibrium, many games can have more than one. In those cases, game theory cannot predict which equilibrium point will be reached—you can’t say what sets of mixed strategies the players will actually adopt in a real-world situation. And even if there is only one Nash equilibrium in a complicated game, it is typically beyond the capability of a com-

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature mittee of supercomputers to calculate what all the players’ mixed strategies would have to be. In turn, that crack is exacerbated by a weakness in the cardinal assumption underlying traditional game theory—that the players are rational payoff maximizers with access to all the necessary information to calculate their payoffs. In a world where most people can’t calculate the sales tax on a cheeseburger, that’s a tall order. In real life, people are not “perfectly rational,” capable of figuring out the best money-maximizing strategy for any strategy combination used by all the other competitors. So game theory appears to assume that each player can do what supercomputers can’t. And in fact, almost everybody recognizes that such total rationality is unachievable. Modern approaches to game theory often assume, therefore, that rationality is limited or “bounded.” Game theorists have devised various ways to deal with these limitations on Nash’s original math. An enormous amount of research, of the highest caliber, has modified and elaborated game theory’s original formulations into a system that corrects many of these initial “flaws.” Much work has been done on understanding the limits of rationality, for instance. Nevertheless, many game theorists often cling to the idea that “solving a game” means finding an equilibrium—an outcome where all players achieve their maximum utility. Instead of thinking about what will happen when the players actually play a game, game theorists have been asking what the individual players should do to maximize their payoff. When I visited Wolpert at NASA Ames, a year after our conversation in Boston, he pointed out that the search for equilibrium amounts to viewing a game from the inside, from the viewpoint of one of the participants, instead of from the vantage point of an external scientist assessing the whole system. From the inside, there may be an optimal solution, but a scientist on the outside looking in should merely be predicting what will happen (not trying to win the game). If you look at it that way, you know you can never be sure how a game will end up. A science of game theory should therefore not be seeking a single answer, but a probability distribu-

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature tion of answers from which to make the best possible prediction of how the game will turn out, Wolpert insists. “It’s going to be the case that whenever you are given partial information about a system, what must pop out at the other end is a distribution over possibilities, not a single answer.”11 In other words, scientists in the past were not really thinking about the game players as particles, at least not in the right way. If you think about it, you realize that no physicist computing the thermodynamic properties of a gas worries about what an individual molecule is doing. The idea is to figure out the bulk features of the whole collection of molecules. You can’t know what each molecule is up to, but you can calculate, statistically, the macroscopic behavior of all the molecules combined. The parallel between games and gases should be clear. Statistical physicists studying gases don’t know what individual molecules are doing, and game theorists don’t know what individual players are thinking. But physicists do know how collections of molecules are likely to behave—statistically—and can make good predictions about the bulk properties of a gas. Similarly, game theorists ought to be able to make statistical predictions about what will happen in a game. This is, as Wolpert repeatedly emphasizes, the way science usually works. Scientists have limited information about the systems they are studying and try to make the best prediction possible given the information they have. And just as a player in a game has incomplete information about all the game’s possible strategy combinations, the scientist studying the game has incomplete information about what the player knows and how the player thinks (remember that different individuals play games in different ways). All sciences face this sort of problem—knowing something about a system and then, based on that limited knowledge, trying to predict what’s going to happen, Wolpert pointed out. “So how does science go about answering these questions? In every single scientific field of endeavor, what will come out of such an exercise is a probability distribution.”12 From this point of view, another sort of mixed strategy enters

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature game theory. It’s not just that the player has a mixed strategy, a probability distribution of possible moves from which to choose. The scientist describing the game also has a kind of “mixed strategy” of possible predictions about how the game will turn out. “When you think about it, it’s obvious,” Wolpert said. “If I give you a game of real human beings, no, you’re not going to always have the same outcome. You’re going to have more than one possible outcome…. It’s not going to be the case they are always going to come up with the exact same set of mixed strategies. There’s going to be a distribution over their mixed strategies, just like in any other scientific scenario.” This is clearly taking game theory to another level. While each player has a mixed strategy, a probability distribution of pure strategies, the scientist describing the game should compute a probability distribution of all the players’ mixed strategies. And how do you find those probability distributions of mixed strategies? By maximizing your ignorance, of course. If you want to treat game theory as though the people were particles, the best approach is to assume a probability distribution for their strategies that maximizes the uncertainty (or the entropy, in information theory terms). Using this approach, you don’t need to assume that the players in a game have limits on their rationality; such limits naturally appear in the formulas that information theory provides. Given a probability distribution of possible outcomes for the game, then, you can choose which outcome to bet on using the principles of decision theory. “When you need a prediction, a probability distribution won’t do,” said Wolpert. “You have to decide to fire the missile or don’t fire; turn left or right.” The underlying axioms for the mathematical basis for making such a decision were worked out in the 1950s by Leonard Savage13 in some detail, but they boil down to something like Pascal’s Wager. If you have a probability distribution of possible outcomes, but don’t know enough to distill the possibilities down to a single prediction, you need to consider how much you have to lose (or to gain) if your decision is wrong (or right). “If you predict X, but the truth is Y, how much are you hurt?

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature Or conversely, how much do you benefit?” Wolpert explained. “Certain kinds of mispredictions aren’t going to hurt you very much, depending on what the truth is. But in other instances … your prediction of truth might cause all sorts of problems—you’ve now launched World War III.” Decision theory dictates that you should make the prediction that minimizes your expected loss (“expected” signifying that the relative probabilities of the choices are taken into account—you average the magnitudes of loss over all the possibilities). Consequently, Wolpert observes, different individual observers would make different predictions about the outcome of a game, even if the probability distribution of possible outcomes is the same, because some people would have more to lose than others for certain incorrect predictions. “In other words, for the exact same game, your decision as the external person making the prediction is going to vary depending on your loss function,” he says. That means the best prediction about the outcome isn’t some equilibrium point established within the game, but rather depends on “the person external to the game who’s making the prediction about what’s going to come out of it.” And so sometimes the likeliest outcome of a game will not be a Nash equilibrium. But why not, if a Nash equilibrium represents the stable outcome where nobody has an incentive to change? It seems like people would keep changing their strategy until they had no incentive not to. But when game theory is cast in the information-theoretic equations of maximum entropy, the answer becomes clear. A term in the equations signifies the cost of computing the best strategy, and in a complicated game that cost is likely to be too high. In other words, a player attempting to achieve a maximum payoff must factor in the cost of computing what it takes to get that payoff. The player’s utility is not just the expected payoff, but the expected payoff minus the cost of computing it. What’s more, individual differences can influence the calculations. The math of the maximum ignorance approach (that is, maximizing the uncertainty) contains another term, one that can be

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature interpreted as a player’s temperature. Temperature relates ignorance (or uncertainty) to the cost of computing a strategy—more uncertainty about what to do means a higher cost of figuring out what to do. A low temperature signifies a player who focuses on finding the best strategy without regard to the cost of computing it; a higher-temperature player will explore more of the strategy possibilities. “So what that means,” Wolpert explained, “is that it is literally true that somebody who is purely rational, who always does the best possible thing, is cold—they are frozen. Whereas somebody who is doing all kinds of things all over the map, exploring, trying all kinds of possibilities, they are quite literally hot. That just falls out of the math. It’s not even a metaphor; it’s what it actually amounts to.”14 Temperature, in other words, represents a quantification of irrationality. In a gas, higher temperatures mean there’s a higher chance that the molecules are not in the arrangement that minimizes their energy. With game players, higher temperature means a greater chance that they won’t be maximizing their payoff. “The analogy is that you have some probability of being in a nonpurely rational state,” Wolpert said. “It’s the exact same thing. Lowering energy is raising utility.” You are still likely to play strategies that would increase your payoff, but just how much more likely depends on your temperature.15 Boiled down to the key point, the maximum entropy math tells you that game players will have limited rationality—it’s not something that you have to assume. It arises naturally from adopting the viewpoint of somebody looking from outside the game instead of being inside the game. “That is crucial,” Wolpert stressed. “Game theory has always had probability theory inside of it, because people play mixed strategies, but game theory has never actually applied probability theory to the game as a whole. That is the huge hole in conventional game theory.” Ultimately, the idea of a player’s temperature should allow better predictions of how real players will play real games. In the

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature probability distribution of grades in a class, the maximum entropy approach says all grade distributions are possible. But if you know something about the students—maybe all are honors students who’ve never scored below a B—you can adjust the probability distribution by adding that information into the equations. If you know something about a player’s temperature—the propensity to explore different possible strategies—you can add that information into the equations to improve your probability distribution. With collaborators at Berkeley and Purdue, Wolpert is beginning to test that idea on real people—or at least, college students. “We’ve just run through some experiments on undergrads where we’re actually looking at their temperatures, in a set of repeated games—voting games in this case—and seeing things like how does their temperature change with time. Do they actually get more rational or less rational? What are the correlations between different individuals’ temperatures? Do I get more rational as you get less rational?” If, for instance, one player is always playing the exact same move, that makes it easier for opponents to learn what to expect. “That suggests intuitively that if you drop your temperature, mine will go up,” Wolpert said. “So in these experiments our intention is to actually look for those kinds of effects.” VISIONS OF PSYCHOHISTORY Such experiments, it seemed to me, would add to the knowledge that behavioral game theorists and experimental economists had been accumulating (including inputs from psychology and neuroeconomics) about human behavior. It sounded like Wolpert was saying that all this knowledge could be fed into the probability distribution formulas to improve game theory’s predictive power. But before I could ask about what was really on my mind, he launched into an elaboration that took me precisely where I wanted to go. “Let’s say that you know something from psychology, and you’ve gotten some results from experiments,” he said. “Then you

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature actually have other stuff that goes in here [the equations] besides the knowledge that all human beings have temperatures. You also know something about their degree of being risk averse, and this, that, or the other…. You are not just a temperature; there are other aspects to you.” Adding such knowledge about real people into the equations reduces the ignorance that went into the original probability distribution. So instead of predictions based on all possible mixed strategies, you’ll get predictions that better reflect real people. “It’s a way of actually integrating game theory with psychology, formally,” Wolpert said. “You would have … quantification of individual human beings’ behavior integrated with an actual mathematical structure that deals with incentives and utility functions and payoffs.” Wolpert began talking about probability distributions of future states of the stock market and then, almost as an aside, disclosed a much grander vision. “This actually is a way of trying to get a mathematics of psychohistory in Isaac Asimov’s sense,” Wolpert said. “In other words, this is potentially—it’s not been done—this is potentially the physics of human behavior.”16 Just as I had suspected. The suggestive similarities between Asimov’s psychohistory and game theory’s behavioral science do, in fact, reflect a common underlying mathematics. It’s the math that merges game theory with statistical physics. So in pondering what Wolpert said, it occurred to me that there’s a better way to refer to the science of human behavior than psychohistory or sociophysics or Code of Nature. It should be called Game Physics. Alas, “game physics” is already taken—it’s a term used by computer programmers to describe how objects move and bounce around in computerized video games. But it captures the idea of psychohistory or sociophysics pretty well. Game theory combined with statistical physics, the physics of games, is the science of society.

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