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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature 9 Asimov’s Vision Psychohistory, or sociophysics? “Humans are not numbers.” Wrong; we just do not want to be treated like numbers. —Dietrich Stauffer In 1951—the same year that John Nash published his famous paper on equilibrium in game theory—Isaac Asimov published the novel Foundation. It was the first in a series of three books (initially) telling the story of a decaying galactic empire and a new science of social behavior called psychohistory. Asimov’s books eventually became the most famous science fiction trilogy to appear between Lord of the Rings and Star Wars. His psychohistory became the model for the modern search for a Code of Nature, a science enabling a quantitative description and accurate predictions of collective human behavior.1 Mixing psychology with math, psychohistory hijacked the methods of physics to forecast—and influence—the future course of social and political events. Today, dozens of physicists and mathematicians around the world are following Asimov’s lead, seeking the equations that capture telling patterns in social behavior, trying to show that the madness of crowds has a method. As a result, Asimov’s vision is no longer wholly fiction. His psychohistory exists in a loose confederation of research enterprises that go by different names and treat different aspects of the
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature issue. At various schools and institutes around the world, collaborators from diverse departments are creating new hybrid disciplines, with names like econophysics, socionomics, evolutionary economics, social cognitive neuroscience, and experimental economic anthropology. At the Santa Fe Institute, a new behavioral sciences program focuses on economic behavior and cultural evolution. The National Science Foundation has identified “human and social dynamics” as a special funding initiative. Almost daily, research papers in this genre appear in scientific journals or on the Internet. Some examine voting patterns in diverse populations, how crowds behave when fleeing in panic, or why societies rise and fall. Others describe ways to forecast trends in the stock market or the likely effect of antiterrorist actions. Still others analyze how rumors, fads, or new technologies spread. Diverse as they are, all these enterprises share a common goal of better understanding the present in order to foresee the future, and possibly help shape it. Put them all together, and Asimov’s idea for a predictive science of human history no longer seems unthinkable. It may be inevitable. Among the newest of these enterprises—and closest to the spirit of Asimov’s psychohistory—is a field called sociophysics. The name has been around for decades, but only in the 21st century has it become more science than slogan. Like Asimov’s psychohistory, sociophysics is rooted in statistical mechanics, the math used by physicists to describe systems too complex to expose the intimate interactions of their smallest pieces. Just as physicists use statistical mechanics to show how the temperature of two chemicals influences how they react, sociophysicists believe they can use statistical mechanics to take the temperature of society, thereby quantifying and predicting social behavior. Taking society’s temperature isn’t quite as straightforward as it is with, say, gas molecules in a room. People usually don’t behave the same way as molecules bouncing off the walls, except during some major sporting events. To use statistical physics to take society’s temperature, physicists first have to figure out where to stick the thermometer.
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature Fortunately, the collisions of molecules have their counterpart in human interaction. While molecules collide, people connect, in various sorts of social networks. So while the basic idea behind sociophysics has been around for a while, it really didn’t take off until the new understanding of networks started grabbing headlines. Social networks have now provided physicists with the perfect playground for trying out their statistical math. Much of this work has paid little heed to game theory, but papers have begun to appear exploring the way that variants on Nash’s math become important in social network contexts. After all, von Neumann and Morgenstern themselves pointed out that statistical physics provided a model giving hope that game theory could describe large social groups. Nash saw his concept of game theory equilibrium in the same terms as equilibrium in chemical reactions, which is also described by statistical mechanics. And game theory provides the proper mathematical framework for describing how competitive interactions produce complex networks to begin with. So if the offspring of the marriage between statistical physics and networks is something like Asimov’s psychohistory, game theory could be the midwife. SOCIOCONDEMNATION Network math offers many obvious social uses. It’s just what the doctor ordered for tracking the spread of an infectious disease, for instance, or plotting vaccination strategies. And because ideas can spread like epidemics, similar math may govern the spread of opinions and social trends, or even voting behavior. This is not an entirely new idea, even within physics. Early attempts to apply statistical physics to such problems met with severe resistance, though, as Serge Galam has testified. Galam was a student at Tel-Aviv University during the 1970s, when statistical mechanics was the hottest topic in physics, thanks largely to some Nobel Prize–winning work by Kenneth Wilson at Cornell University. Galam pursued his education in statistical physics but with a
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature concern—its methods were so powerful that all the important problems of inert matter might soon be solved! So he began to advocate the use of statistical physics outside physics, especially for analyzing human phenomena, and published several papers along those lines. He even published one with “sociophysics” in the title in 1982. The response from other physicists was not enthusiastic. “Such an approach was strongly rejected by almost everyone,” he wrote, “leading and non-leading physicists, young and old. To suggest humans could behave like atoms was looked upon as a blasphemy to both hard science and human complexity, a total non-sense, something to be condemned.”2 My impression is that most physicists nowadays are not so hostile to such efforts (although some are) but are just mostly indifferent. There are some enthusiasts, though, and international conferences have been devoted to sociophysics and related topics. And thanks to the rapid advances in network math, the study of social networks has gained a certain respectability, diminishing the danger of instant condemnation for anyone pursuing it (although acceptance is clearly greater in Europe than in the United States). Part of this acceptance probably stems from the growing popularity of an analogous discipline known as econophysics, a much more developed field of study. Econophysics3 studies the interacting agents in an economy using statistical physics, and some prominent physicists have been attracted to it. Many young physicists have taken their skills in this field to Wall Street, where they can make money without the constant fear of government budget cuts. Sociophysics is much more ambitious. It should ultimately encompass econophysics within it, along with everything else in the realm of human interactions. Of course, it has a way to go. But whatever anybody thinks of this research, there is certainly now a lot of it. Galam himself remains a constant contributor to the field. Now working in France, he has studied such social topics as the spread of terrorism, for instance, trying to identify what drives the growth of terrorism networks. In other work, he has analyzed opinion transmission and voting behaviors, concluding that “hung
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature election scenarios,” like the 2000 U.S. presidential contest, “are predicted to become both inevitable and a common occurrence.”4 Other researchers have produced opinion-spreading papers that try to explain whether an extreme minority view can eventually split a society into two polarized opposite camps, or even eventually become an overwhelming majority. Most of this work is based on simple mathematical models that try to represent people and their opinions in a way that can be easily dealt with mathematically. There is no point in trying to be completely realistic—no amount of math could capture all the nuances in the process by which even a single individual formed his or her opinions, let alone an entire population. The idea is to find a simple way to represent opinions at their most basic and to identify a few factors that influence how opinions change—in a way that lends itself to mathematical manipulation. If the math then reproduces something recognizable about human behavior, it can be further refined in an attempt to inch closer to reality. It’s not hard to find people who think the whole enterprise is preposterous. Human beings are not particles—they bear not the slightest similarity to atoms or molecules. Why should you expect to learn anything about people from the math that describes molecular interactions? On the other hand, molecules are not billiard balls—yet Maxwell made spectacular progress for physics by analyzing them as though they were. In his paper introducing statistical considerations to the study of gases, Maxwell applied his math to a system containing “small, hard and perfectly elastic spheres acting on one another only during impact.” Molecules are small, to be sure, but otherwise that description is not very complete or accurate, as Maxwell knew full well. But he believed that insights into the behavior of real molecules might emerge by analyzing a simplified system. “If the properties of such a system of bodies are found to correspond to those of gases,” Maxwell wrote, “an important physical analogy will be established, which may lead to more accurate knowledge of the properties of matter.”5 Today, physicists hope
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature to find a similar analogy between particles and people that will lead to an improved knowledge of the functioning of society. SOCIOMAGNETISM One popular example of such an approach appeared in 2000 from Katarzyna Sznajd-Weron of the University of Wroclaw in Poland. She was interested in how opinions form and change among members of a society. She reasoned that the global distribution of opinions in a society must reflect the behavior and interactions of individuals—in physics terms, the macrostate of the system must reflect its microstate (like the overall temperature or pressure of a container of gas reflects the speed and collisions of individual molecules).6 “The question is if the laws on the microscopic scale of a social system can explain phenomena on the macroscopic scale, phenomena that sociologists deal with,” she wrote.7 Sznajd-Weron was well aware that people recoil when told they are just like atoms or electrons rather than individuals with feelings and free will. “Indeed, we are individuals,” she wrote, “but in many situations we behave like particles.” And one of those common properties that people share with particles is a tendency to be influenced by their neighbors. Sometimes what one person does or thinks depends on what someone else is doing, just as one particle’s behavior can be affected by other particles in its vicinity. Sznajd-Weron related an anecdote about a New Yorker staring upward at the sky one morning while other New Yorkers pass by, paying no attention. Then, the next morning, four people stare skyward, and soon others stop as well, all looking up for no reason other than to join in the behavior of the crowd. Such pack behavior suggested to Sznajd-Weron an analogy for crowd behavior as described by the statistical mechanics of phase transitions, the sudden changes in condition such as the freezing of water into ice. Another sort of phase transition, of the type that attracted her attention, is the sudden appearance of magnetism in some materials cooled below a certain temperature.
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature It makes sense to relate society to magnetism, since society reflects the collective behavior of people, and magnetism reflects the collective behavior of atoms. A material like iron can be magnetic because its atoms possess magnetic properties, thanks to the arrangement of their electrons, the electrically charged fuzzballs that shield each atom’s nucleus. Magnetism is related to the direction in which electrons spin. (You can view the spins as around an axis either pointing up or pointing down, corresponding to whether the electron spin is clockwise or counterclockwise.) Ordinarily a bar of iron is not magnetic, because its atoms are directing their magnetism in random directions, so they cancel out. If enough atoms align themselves in one particular direction, though, others will follow—kind of like the way if enough people look up to the sky, everybody else will, too. When all the atoms line up, the iron bar becomes a magnet. It’s as though each atom checks to see how its neighbor’s electrons are spinning. When two atoms are sitting next to each other, their partnerless electrons want to spin in the same direction—that confers a slightly lower energy on the system, and all physical systems seek the state of lowest possible energy. Consequently the spin of one iron electron can influence the spin of its neighbor, inducing it to take on the same orientation. (In most materials an atom’s electrons are mostly paired off with opposite spins. But iron and a few other materials possess some properly positioned electrons without partners. Magnetism is a little more complicated than this crude picture, of course, but the basic idea is good enough.) As scientists began to understand this aspect of magnetism, they wondered if such local interactions between neighbors could explain the global phase transition from the nonmagnetic to magnetic state. In the 1920s, the German physicist Ernst Ising tried to show how neighboring spins could induce a spontaneous phase transition in a system, but failed. The problem was not in the basic idea, though—it was that Ising analyzed only a one-dimensional system, like a string of spinning beads on a necklace. Soon other researchers showed that Ising’s approach did turn out to work when
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature applied to two-dimensional systems, like spinning balls arranged in a grid. Magnetism could thus be understood as a collective phenomenon stemming from the interactions of individuals—sort of like pack journalism. When one newspaper makes a big deal about a major story, all the other media jump in and beat the story to death—all the news is taken over by something like O.J. or Michael Jackson or some Runaway Bride. Similarly, rapid large-scale changes mimicking phase transitions occur in biology or the economy, such as mass extinctions or stock market crashes. In recent years it has occurred to physicists like Galam, Sznajd-Weron, and many others that the same principle could apply to social phenomena, such as the rapid spread of popular fads. Sznajd-Weron set out to devise an Ising-like model of social opinions, trying out a very simplified approach that would be easy to handle mathematically. Instead of up or down spins, people could take a yes or no stance with respect to some issue. If you start out with opinions at random, how would the system change over time? Sznajd-Weron proposed a model based on the idea of “social validation.” Just as the behavior of the New York skywatcher spread when others did the same, identical opinions between neighbors could cause their same opinion to spread socially, in a way similar to the way magnetism develops through Ising interactions. Sznajd-Weron’s model of society was pretty simple— something like one long street with houses on only one side. Each house is identified by a number (OK, that’s realistic), and each house gets one opinion (or spin): either yes (mathematically represented as +1), or no (–1). To start out, the houses all have opinions at random. Then, every day each house checks its neighbors and modifies its opinions based on some simple mathematical rules. Based on neighboring opinions, each house may (or may not) modify its own. In Sznajd-Weron’s model, you start by considering two neighbors— let’s say House 10 and House 11. Each of that pair has another
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature neighbor (House 9 and House 12). Sznajd-Weron’s rules say that if houses 11 and 12 have the same opinion, then houses 9 and 12 will adjust their opinions to match the common opinion of 10 and 11. If houses 10 and 11 disagree, though, House 9 will adjust its opinion to agree with House 11, and House 12 will change to agree with House 10. Mathematically, the rules look like this, with S representing a house and the subscript i representing the house number (in the above example, Si is House 10, Si+1 is House 11, etc.): In other words, when the two neighbors (10 and 11) agree, the two outside neighbors will share that opinion. If the first two neighbors disagree, then the one on the left will agree with the second and the one on the right will agree with the first. Why should that be? No reason, it’s just a model. In a variant on Sznajd-Weron’s original proposal, the second rule is switched: In the original model, Sznajd-Weron performed computer simulations on a street with 1,000 houses and watched as opinions changed over 10,000 days or so. No matter how the opinions started out, the neighborhood eventually reached one of three stable situations—either all the houses voting yes, all no, or a 50-50 split. (These conditions correspond, in Sznajd-Weron’s words, to either “dictatorship” or “stalemate.”) Since not all societies are dictatorships or stalemates, the model does not reflect the true complexity of the real world. But that doesn’t mean the model is dumb—it means that the model has told us something, namely that more than just local interaction between neighbors is involved in opinion formation. And you don’t need to know what all those other factors are to improve the model—you just need to know that they exist. In her 2000 paper,
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature Sznajd-Weron showed that such unknown factors (in technical terms, noise) could be described as a “social temperature” raising the probability that an individual would ignore the neighbor rules and choose an opinion apparently at random. With a sufficiently high social temperature, the system can stay in some disordered state, more like a democracy, rather than becoming a stalemate or dictatorship. Even so, as Sznajd-Weron pointed out, her one-dimensional model is not likely to be very useful for social systems, just as Ising’s one-dimensional model did not get the magnetism picture right, either. So in the years since her proposal, she and others have worked on extensions of the model. A similar model in two dimensions (with the “houses” occupying points on a grid) was developed by Dietrich Stauffer of the University of Cologne, probably today’s most prominent sociophysicist. With the people aligned on a grid, everybody has four neighbors, a pair has six neighbors, and a block of four has eight neighbors. In this case, one rule might be that a block of four changes its eight neighbors only if all four in the block have the same spin (or opinion). Or two neighbors paired with the same spin can change the spins of their six neighbors. A grid model can accommodate more complications and thus reproduce more of the real properties of society. SOCIONETWORKS Clearly, though, the way to get more social realism is to apply such rules not to simple strings or grids but to the complex social networks discovered in the real world. And much interesting work has begun to appear along these lines. One approach examines the general idea of “contagion”—the spread of anything through a population, whether infectious disease or ideas, fads, technological innovations, or social unrest. As it turns out, fads need not always spread the same way as a disease, as different scenarios may guide the course of different contagions. In some cases, a small starting “seed” (a literal virus, perhaps, or just a new idea) can eventually grow into an epidemic. In other
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature cases a seed infects too few people and the disease or idea dies out. Peter Dodds and Duncan Watts (of the Watts-Strogatz network paper) of Columbia University have shown that what happens can depend on how much more likely a second exposure is to infect an individual than a first exposure. Their analysis suggests that the spread of diseases or ideas depends less on “superspreaders” or opinion leaders than on how susceptible people are—how resistant they are to disease or how adamantly they hold their current opinion. Such results imply that the best way to hamper or advance contagion would be strategies that increase or reduce the odds of infection. Better health procedures, for instance, or financial incentives to change voting preferences, could tip the future one way or another. “Our results suggest that relatively minor manipulations … can have a dramatic impact on the ability of a small initial seed to trigger a global contagion event,” Dodds and Watts declared in their paper.8 It sounds like just the sort of thing that Hari Seldon incorporated into psychohistory, so that his followers could subtly alter the course of future political events. In real life, of course, people don’t necessarily transmit opinions or viruses in the simple ways that such analyses assume. So some experts question how useful the statistical mechanics approach to society will ultimately be. “I think in some limited domains it might be pretty powerful,” says Cornell’s Steven Strogatz. “It really is the right language for discussing enormous systems of whatever it is, whether it’s people or neurons or spins in a magnet.… But I worry that a lot of these physicist-style models of social dynamics are based on a real dopey view of human psychology.”9 Of course, that is precisely where game theory comes into play. Game theory has given economists and other social scientists the tool for quantifying human psychology in ways that Freud could only dream of. Neuroeconomics and behavioral game theory have already sculpted a much more realistic model of human psychology than the naive Homo economicus that lived only to maximize money. And once you have a better picture of human psychology—in particular, a picture that depicts the psychological
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature variations among individuals—you need game theory to tell you what happens when those individuals interact. SOCIOPHYSICS AND GAME THEORY After all, when you get to really complex social behaviors—not just yes or no votes, but the whole spectrum of human cultural behavior and all its variations—the complex interactions between individuals really do matter. It is yet again similar to the situation with molecules in a gas. In his original math describing gas molecules, Maxwell considered their only interaction to be bouncing off of each other (or the container’s walls), altering their direction and velocity. But atoms and molecules can interact in more complicated ways. Electrical forces can exert an attractive or repulsive force between molecules, and including those forces in the calculations can make statistical mechanical predictions more accurate. Similarly, the behavior of people depends on how they are affected by what other people are doing, and that’s what game theory is supposed to be able to describe. “Game theory was created,” Colin Camerer points out, “to provide a mathematical language for describing social interaction.”10 Numerous efforts have been made to apply game theory in just that way. One particularly popular game for analyzing social interaction is the minority game, based on an economist’s observations about a Santa Fe bar. Keep in mind that in game theory, a player’s choices should depend on what the other players are choosing. So the game as a whole reflects collective behavior, possibly described by a Nash equilibrium. In simple sociophysics models based on neighbors interacting, the global collective behavior results from purely local influences. But the Nash equilibrium idea suggests that individual behavior should be influenced by the totality of all the other behaviors. It may be, for instance, that the average choices of all the other players is the most important influence on any one individual’s choice (in physics terms, that would correspond to a “mean-field theory” version of statistical mechanics). In traditional game theory, each player supposedly is 100 per-
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature cent rational with total information and unlimited mental power to figure out what everybody else will do and then calculate the best move. But sometimes (actually, almost all the time) those conditions are not satisfied. People have limited calculating power and limited information. There are situations where the game is too complex and too many people are involved to choose a foolproof decision using game theory. And in fact, many simple situations can prove too complicated to calculate completely, even something as innocent as deciding whether to go to a bar on Friday night or stay home instead. This problem was made famous by Brian Arthur, an economist at the Santa Fe Institute, in the early 1990s. A Santa Fe bar called El Farol had become so popular that it was no longer always a pleasant place to go because of the crowds. (It was reminiscent of baseball player Yogi Berra’s famous comment about the New York City restaurant Toots Shor’s. “Toots Shor’s is so crowded,” said Yogi, “nobody goes there anymore.”) Arthur saw in the El Farol situation a problem of decision making with limited information. You don’t know in advance how many people will go to the bar, but you assume that everybody would like to go unless too many other people are going also. Above some level of attendance, it’s no fun. This situation can be framed as a game where the winners are those in the minority— you choose to go or stay home and hope that the majority of people make the opposite choice. In 1997, Damien Challet and Yi-Cheng Zhang developed the mathematics of the El Farol problem in detail, in the form of what they called the minority game. Since then it has been a favorite framework of many physicists for dealing with economic and social issues.11 In the basic version of the game, each would-be bar patron (in mathematical models, such customers are called “agents”) possesses a memory of how his or her last few bar-going decisions have turned out. (Players find out after every trial whether the stay-at-home or bar-going choices were the winners.) Suppose that Friday is your regular drinking night, and you can remember what hap-
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature pened three weeks back. Say, for instance, that on each of the last three Friday nights, a majority of people went to the bar. They were therefore the losers, as the minority of players avoided the crowd by staying at home. Your strategy for next Friday might be to go to the bar, figuring that after three loser trips in a row most people will decide to stay home and the bar will be less crowded. On the other hand, your strategy might be to go based only on the results of the past week, regardless of what happened the two weeks before. At the start of the game, each agent gets a set of possible strategies like these, and then keeps track of which strategies work better than others. Over time, the agents will learn to use the strategies that work the best most often. As a result, the behavior of all the players becomes coordinated, and eventually attendance at the bar will fluctuate around the 50-50 point—on some Fridays a minority will go to the bar, and on some a slight majority, but attendance will never be too far off from the 50-50 split. You don’t have to be a drinker to appreciate the usefulness of the minority game for describing social situations. It’s not just about going to bars—the same principles apply to all sorts of situations where people would prefer to be in a minority. You can imagine many such scenarios in economics, for instance, such as when it’s better to be a buyer or a seller. If there are more sellers than buyers, you have the advantage if you’re a buyer—in the minority. Further work on the minority game has shown that in some circumstances it is possible to predict which choice is likely to be in the minority on next Friday night. It depends on how many players there are and how good their memory is. As the number of players goes down (or their memory capacity goes up), at some point the outcome is no longer random and can be predicted with some degree of statistical confidence. MIXED CULTURES While the minority game provides a good example of using (modified) game theory to model group behaviors, it still leaves a lot to
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature be desired. And it certainly is a far cry from Asimov’s psychohistory. Psychohistory quantified not only the interactions between individuals in groups, but also the interactions among groups, exhibiting bewildering cultural diversity. Today’s nonfictional anthropologists have used game theory to demonstrate such cultural diversity, but it’s something else again to ask game theory to explain it. Yet if sociophysics is to become psychohistory, it must be able to cope with the global potpourri of human cultural behaviors, and achieving that goal will no doubt require game theory. At first glance, the prospects for game theory encompassing the totality of cultural diversity seem rather bleak. Especially in its most basic form, the ingredients for a science of human sociality seem to be missing. People are not totally rational beings acting purely out of self-interest as traditional game theory presumes, for example. Individuals playing games against other individuals make choices colored by emotion. And societies develop radically different cultural patterns of collective behavior. No Code of Nature dictates a universal psychology that guides civilizations along similar cultural paths. As Jenna Bednar and Scott Page of the University of Michigan have described it, game theory would seem hopeless as a way to account for the defining hallmarks of cultural behavior. “Game theory,” they write, “assumes isolated, context-free strategic environments and optimal behavior within them.”12 But human cultures aren’t like that. Within a culture, people behave in similar, fairly consistent ways. But behavior differs dramatically from one culture to the next. And whatever the culture, behavior is typically not optimal, in the sense of maximizing self-interest. When incentives change, behavior often remains stubbornly stuck to cultural norms. All these features of culture run counter to some basic notions of game theory. “Cultural differences—the rich fabric of religions, languages, art, law, morals, customs, and beliefs that diversifies societies—and their impact would seem to be at odds with the traditional game theoretic assumption of optimizing behavior,” say Bednar and Page. “Thus, game theory would seem to be at a loss to explain the
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature patterned, contextual, and sometimes suboptimal behavior we think of as culture.”13 But game theory has a remarkable resilience against charges of irrelevance. It’s explanatory power has not yet been exhausted, even by the demands of explaining the many versions of human culture. “Surprisingly,” Bednar and Page declare, “game theory is up to the task.”14 The individuals, or agents, within a society may very well possess rational impulses driving them to seek optimum behaviors, Bednar and Page note. But the effort to figure out optimal behaviors in a complicated situation is often considerable. In any given game, a player has to consider not only the payoff of the “best” strategy, but also the cost of calculating the best moves to achieve that payoff. With limited brain power (and everybody’s is), you can’t always afford the cost of calculating the most profitable response. Even more important, in real life you are never playing only one game. You are in fact engaging in an ensemble of many different games simultaneously, imposing an even greater drain on your brain power. “As a result,” write Bednar and Page, “an agent’s strategy in one game will be dependent upon the full ensemble of games it faces.” So Alice and Bob (remember them?) may be participating in a whole bunch of other games, requiring more complicated calculations than they needed back in Chapter 2. If they have only one game in common, the overall demand on their calculating powers could be very different. Even if they face identical situations in the one game they play together, their choices might differ, depending on the difficulty of all the other games they are playing at the same time. As Bednar and Page point out, “two agents facing different ensembles of games may choose distinct strategies on games that are common to both ensembles.” In other words, with limited brain power, and many games to play, the “rational” thing to do is not to calculate pure, ideal game theory predictions for your choices, but to adopt a system of general guidelines for behavior, like the Pirate’s Code in the Johnny
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature Depp movie Pirates of the Caribbean. And that’s what it means to behave culturally. Cultural patterns of behavior emerge as individuals tailor a toolkit of strategies to apply in various situations, without the need to calculate payoffs in detail. “Diverse cultures emerge not in spite of optimizing motivation,” Bednar and Page write, “but because of how those motivations are affected by incentives, cognitive constraints, and institutional precedents. Thus agents in different environments may play the same game differently.”15 The Michigan scientists tested this idea with computer simulations on a variety of games, giving the agents/players enough brain power to compute optimal strategies for any given game. In the various games, incentives for the self-interested agents differed, to simulate different environmental conditions. These multiple-game simulations show that game theory itself drives self-interested rational agents to adopt “cultural” patterns of behavior. This approach doesn’t explain everything about culture, of course, but it shows how playing games can illuminate aspects of society that at first glance seem utterly beyond game theory’s scope. And it suggests that the scope of sociophysics can be grandly expanded by incorporating game theory into its statistical physics formulations. In any event, recent developments in the use of statistical physics in describing networks and society—and game theory’s intimate relationship with both—instill a suspicion that game theory and physics are somehow related in more than a superficial way. As game theory has become a unifying language for the social sciences, attempts by physicists to shed light on social science inevitably must encounter game theory. In fact, that’s exactly what has already happened in economics. Just yesterday, the latest issue of Physics Today arrived in my mail, with an article suggesting that economics may be “the next physical science.” “The substantial contribution of physics to economics is still in an early stage, and we think it fanciful to predict what will ultimately be accomplished,” wrote the authors, Doyne Farner and Eric Smith of the Santa Fe Institute and Yale economist Martin
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature Shubik. “Almost certainly, ‘physical’ aspects of theories of social order will not simply recapitulate existing theories in physics.”16 Yet there are areas of overlap, they note, and “striking empirical regularities suggest that at least some social order … is perhaps predictable from first principles.” The role of markets in setting prices, allocating resources, and creating social institutions involves “concepts of efficiency or optimality in satisfying human desires.” In economics, the tool for gauging efficiency and optimality in satisfying human desires is game theory. In physics, analogous concepts correspond to physical systems treated with statistical mechanical math. The question now is whether that analogy is powerful enough to produce something like Asimov’s psychohistory, a statistical physics approach to forecasting human social interaction, a true Code of Nature. One possible weakness in the analogy between physics and game theory, though, is that physics is more than just statistical mechanics. Physics is supposed to be the science of physical reality, and physical reality is described by the weird (yet wonderful) mathematics of quantum mechanics. If the physics–game theory connection runs deep, there should be a quantum connection as well. And there is.
Representative terms from entire chapter: