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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature Introduction Could not mind, as well as mindless motion, have an underlying order? —Emperor Cleon to Hari Seldon, Prelude to Foundation Isaac Asimov excelled at predicting the future. In one of his early science fiction stories, he introduced pocket calculators decades before you could buy them at Radio Shack. In a later book, he described a digital camera transmitting photos directly to a computer via WiFi.1 He just forgot to mention that you could also use the same device to make phone calls. And in his most celebrated work, a series of 1950s science fiction novels known as the Foundation Trilogy, Asimov foresaw a new kind of science called psychohistory, capable itself of forecasting political, economic, and social events. Psychohistory, as Asimov envisioned it, was “the science of human behavior reduced to mathematical equations.”2 Real-life psychohistory does not yet exist—not now, not really, and not for a long time. But there are many research enterprises under way in the world today that share the goal of better understanding human behavior in order to foresee the future. At the foundation of these enterprises are mathematical methods closely resembling Asimov’s psychohistory. And in the midst of it all is the work of a mathematician named John Forbes Nash.
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature Brilliant but odd, intellectually sophisticated but socially awkward, Nash dazzled the world of mathematics in the 1950s with astounding and original results in several arenas. He rattled the routines at Princeton University and the Rand Corporation in California with both his mental magnificence and his disruptive behavior. By now, the subsequent tragic aspects of Nash’s life story are familiar to millions of people, thanks to the Oscar-winning movie starring Russell Crowe, and Sylvia Nasar’s A Beautiful Mind, the acclaimed book on which the movie was based. Yet while book and movie probed the conflicting complexities in Nash the man, neither delved deeply into Nash’s math. So for most people today, his accomplishments remain obscure. Within the world of science, though, Nash’s math now touches more disciplines than Newton’s or Einstein’s. What Newton’s and Einstein’s math did for the physical universe, Nash’s math may now be accomplishing for the biological and social universe. Indeed, had mental illness not intervened, Nash’s name might today be commonly uttered in the same breath with those scientific giants of the past. As it is, he made important contributions to a few mathematical specialties. But he achieved his greatest fame in economics, the field in which he shared the 1994 Nobel Prize with John Harsanyi and Reinhard Selten for their seminal work on the theory of games—the math that analyzes how people make choices in contests of strategy. Game theory originated in efforts to understand parlor games like poker and chess, and was first fully formulated as a mathematical tool for describing economic behavior. But in principle, game theory encompasses any situation involving strategic interaction—from playing tennis to waging war. Game theory provides the mathematical means of computing the payoffs to be expected from various possible choices of strategies. So game theory’s math specifies the formulas for making sound decisions in any competitive arena. As such, it is “a tool for investigating the world,” as the economist Herbert Gintis points out. But it is much more than a mere tool. “Game theory is about how people cooperate as much as how they compete,” Gintis writes. “Game theory is about the
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature emergence, transformation, diffusion and stabilization of forms of behavior.”3 Nash did not invent game theory, but he expanded its scope and provided it with more powerful tools for tackling real-world problems. At first, though, the depth of his accomplishment was little appreciated. When his revolutionary papers appeared, in the early 1950s, game theory briefly became popular among Cold War analysts who saw similarities between international aggression and maximizing profits. But within economics, game theory remained mainly a curiosity. “It didn’t take off,” the economist Samuel Bowles told me. “Like a lot of good ideas in economics, it just fell by the wayside.”4 In the 1970s, though, evolutionary biologists adopted game theory to study the competition for survival among animals and plants. And in the 1980s, economists finally began to use game theory in various ways, finding it especially helpful in designing actual experiments to test economic theory. By the late 1980s game theory had re-emerged in economics in a big way, leading to Nash’s 1994 Nobel. Even before then, game theory had already migrated into the curricula of many scientific disciplines. You could find it taught in departments not only of mathematics and economics and biology but also political science, psychology, and sociology. By the opening years of the 21st century, game theory’s uses had spread even wider, to fields ranging from anthropology to neurobiology. Today, economists continue to use game theory to analyze how people make choices about money. Biologists apply it to scenarios explaining the survival of the fittest or the origin of altruism. Anthropologists play games with people from primitive cultures to reveal the diversity of human nature. And neuroscientists have joined the fun, peering inside the brains of game-playing people to discover how their strategies reflect different motives and emotions. In fact, a whole new field of study, called neuroeconomics, has taken shape, mixing game theory’s methods with brain-scanning technology to detect and measure neural activity corresponding to human judgments and behavior. “We’re
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature quantifying human experience,” says neuroscientist Read Montague, “in the same way we quantify airflow over the wings of a Boeing 777.”5 In short, Nash’s math—with the rest of modern game theory built around it—is now the weapon of choice in the scientist’s arsenal on a wide range of research frontiers related to human behavior. In fact, Herbert Gintis contends, game theory has become “a universal language for the unification of the behavioral sciences.”6 I think it might go even farther than that. Game theory may become the language not just of the behavioral sciences, but of all the sciences. As science stands today, that claim is rather bold. It might even be wrong. But game theory already has conquered the social sciences and invaded biology. And it is now, in the works of a few pioneering scientists, forming a powerful alliance with physics. Physicists, of course, have always sought a unity in the ultimate description of nature, and game theory may have the potential to be a great unifier. That realization hit me in early 2004, when I read a paper by physicist-mathematician David Wolpert, who works at NASA’s Ames Research Center in California. Wolpert’s paper disclosed a deep connection between the math of game theory and statistical mechanics, one of the most powerful all-purpose tools used by physicists for describing the complexities of the world. Physicists have used statistical mechanics for more than a century to describe such things as gases, chemical reactions, and the properties of magnetic materials—essentially to quantify the behavior of matter in all sorts of circumstances. It’s a way to describe the big picture when lacking data about the details. You can’t track every one of the trillion trillion molecules of air zipping around in a room, for instance, but statistical mechanics can tell you how an air conditioner will affect the overall temperature. It’s no coincidence that statistical mechanics (which encompasses the kinetic theory of gases) is the math that inspired Asimov’s heroic mathematician, Hari Seldon, to invent psycho-
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature history. As Janov Pelorat, a character in the later novels of the Foundation series, explained: Hari Seldon devised psychohistory by modeling it upon the kinetic theory of gases. Each atom or molecule in a gas moves randomly so that we can’t know the position or velocity of any one of them. Nevertheless, using statistics, we can work out the rules governing their overall behavior with great precision. In the same way, Seldon intended to work out the overall behavior of human societies even though the solutions would not apply to the behavior of individual human beings.7 In other words, put enough people together and the laws of human interaction will produce predictable patterns—just as the interactions and motion of molecules determine the temperature and pressure of a gas. And describing people as though they were molecules is just what many physicists are doing today—in effect, they’re taking the temperature of society. One of the best ways to take that temperature, it turns out, is to view society in terms of networks. In much the same way that “temperature” captures an essential property of a jumble of gas molecules, network math quantifies how “connected” the members of a social group are. Today’s new network math applies statistical mechanics to all sorts of social phenomena, from fashion trends and voting behavior to the growth of terrorist cells. So just as Asimov envisioned, statistical physics has been enlisted to describe human society in a mathematically precise way. For the most part, this merger of network math and statistical mechanics has been exploring human behavior without recourse to the modern views of game theory built on Nash’s math. After all, Nash’s original formulation had its limits; what works on paper does not always play out the way his math predicts in real-world games. But the latest research has begun to show ways that game theory can help make sense out of the intricate pattern of links in complicated networks. The game theory approach may be able to induce the world of complex networks to more readily surrender its secrets.
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature Wolpert’s insight suggests that game theory itself can be elevated to a new level by exploiting its link to statistical mechanics. His work shows that the math of game theory can be recast in equations that mimic those used by statistical physicists to describe all sorts of physical systems. In other words, at some deep level statistical mechanics and game theory are, in a sense, two versions of the same underlying idea. And that may end up making game theory an especially sensitive social thermometer. This new realization—that game theory and statistical mechanics share a deep mathematical unity—enhances game theory’s status as the preferred tool for merging the life sciences and physical sciences into a unified description of nature. After all, there’s a reason why game theory has been embraced by so many disciplines. Game theory could someday become the glue that holds all of science’s puzzle pieces together. Some people (particularly many physicists) will scoff at this contention. But pause to consider how much sense it makes. Nature encompasses so many complex networks for a reason: complexity evolves. “Intelligent” design produces simple, predictable systems that are easy to understand. The complex systems that baffle science—like bodies, brains, and societies—arise not from any plan, but from interactions among agents like cells or people, all (more or less) out for themselves. And such competitive interaction is precisely what game theory is all about. So it should not be surprising that game theory has been so useful in evolutionary biology. Game theory is about competition, and evolution is the ultimate never-ending Olympic event. And if evolution followed game theory’s rules in generating complicated life, it no doubt also observed the same rules in developing the human brain. So it’s perfectly natural that game theory has become popular today in efforts to understand how the brain works, as brain scientists explore the neural physiology behind economic choices. In turn, the brain underlies all the rest of human behavior—personal and interpersonal, social and political, as well as economic. All that behavior directs the evolution of all those networks of
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature personal, social, political, or economic activity. Just as the complexities of life arose through eons of survival of the fittest, human culture evolves as societies or governments rise and fall; economies evolve as companies are founded and go bankrupt; even the World Wide Web evolves as pages are added and links expire. So Nash’s math does seem capable of catalyzing a merger of methods for understanding individual behavior, biology, and society. What about chemistry and physics? At first glance there doesn’t seem to be any struggle for survival among the molecules engaged in chemical reactions. But in a way there is, and the connections between game theory and statistical mechanics promise to reveal ways in which game theory still applies. Reacting molecules, for instance, always seek a stable condition, in which their energy is at a minimum. The “desire” for minimum energy in molecules is not so different from the “desire” for maximum fitness in organisms. They can be treated mathematically in a similar way. True, there’s much more to physics than statistical mechanics. At first glance, game theory does not seem to touch some of the grander arenas of physical science, such as astrophysics and cosmology, or the subatomic realm ruled by quantum physics. But guess what? In the past few years physicists and mathematicians have developed quantum versions of game theory. So far, quantum theory seems to be enriching game theory, but that enrichment just might turn out to be mutual.8 Furthermore, Wolpert forges the link between statistical mechanics and game theory with help from the mathematical theory of information. As I wrote in my book The Bit and the Pendulum (Wiley, 2000), modern science has become enamored of information theory, using both its math and its metaphor to describe all sorts of science, from the contents of black holes to the computational activity in the brain. Quantum physics itself has been illuminated over the past decade by new insights emerging from quantum information theory. And some theorists have pursued the notion that information ideas hold the key to unifying quantum physics with gravity, perhaps paving the way to the ultimate “theory of everything.” It’s possible, Wolpert speculates, that game
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature theory is the ingredient that could enhance the prospects for success in finding such a theory. In any case, it’s already clear that Nash’s math shows an unexpectedly powerful way of mirroring the regularities of the real world that make all science possible. As I described in my book Strange Matters (Joseph Henry, 2002), there is something strange about the human brain’s ability to produce math that captures deep and true aspects of reality, enabling scientists to predict the existence of exotic things like antimatter and black holes before any observer finds them. Part of the solution to that mystery, I suggested, is the fact that the brain evolved in the physical world, its development constrained by the laws of physics as much as by the laws of biology. I failed then to realize that game theory offers a tool for describing how the laws of physics and biology are related. It’s clear now that game theory’s math describes the capability of the universe to produce brains that can invent math. And math in turn, as Asimov envisioned, can be used to describe the behavior guided by those brains—including the social collective behavior that creates civilization, culture, economics, and politics. While seeking the secrets of that math, we can along the way watch people play games as neuroscientists monitor the activity in their brains; we can follow anthropologists to the jungle where they test the game-playing strategies of different cultures; we can track the efforts of physicists to devise equations that capture the essence of human behavior. And just maybe we’ll see how Nash’s math can broker the merger of economics and psychology, anthropology and sociology, with biology and physics—producing a grand synthesis of the sciences of life in general, human behavior in particular, and maybe even, someday, the entire physical world. In the process, we should at least begin to appreciate the scope of a burgeoning research field, merging the insights of Nash’s 1950s math with 21st-century neuroscience and 19th-century physics to pursue the realization of Asimov’s 1950s science fiction dream. It would be wrong, though, to suggest that Asimov was the first to articulate that dream. In a very real sense, psychohistory
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature was the reincarnation of the old Roman notion of a “Code of Nature” (fitting, since Asimov’s Foundation series was modeled on the Roman Empire’s decline and fall). As interpreted much later, that code supposedly captured the essence of human nature, providing a sort of rule book for behavior. It was not a rule book in the sense of prescribing behavior, but rather a book revealing how humans naturally behave. With the arrival of the Age of Reason in the 18th century, philosophers and the forerunners of social scientists sought in earnest to discover that code of codes—the key to understanding the natural order of human interaction. One of the earliest and most influential of those efforts was the economic system described in The Wealth of Nations by Adam Smith.
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