A Beautiful Math

by Tom Siegfried

  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 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 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 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.

  * * *

  Epilogue

  Let the physical basis of a social economy be given— or, to take a broader view of the matter, of a society. According to all tradition and experience human beings have a characteristic way of adjusting themselves to such a background. This consists of not setting up one rigid system of apportionment … but rather a variety of alternatives, which will probably all express some general principles but nevertheless differ among themselves in many particular respects. This system … describes the ‘established order of society' or ‘accepted standard of behavior.'

  —Von Neumann and Morgenstern, Theory of Games and Economic Behavior

  Despite its title, the science fiction cult classic Ender's Game isn't really about game theory, at least not explicitly. But implicitly it is. It's all about choosing strategies to achieve goals—about adults plotting methods for manipulating young Ender Wiggin, Ender choosing among maneuvers to win on a simulated battlefield, and Ender's siblings' devising tactics for influencing public opinion. And two passages from Orson Scott Card's novel sound like they could have been quoted from a game theory textbook, as they illustrate aspects of human nature that game theory has evolved to explain. Ender's brother Peter, for instance, epitomizes the selfish rational agent of game theory's original naive formulation:

  Peter could delay any desire as long as he needed to; he could conceal any emotion. And so Valentine knew that he would never hurt her in a fit of rage. He would only do it if the advantages outweighed the risks…. He always, always acted out of intelligent self-interest.1

  Ender himself represents the social actor who plays games with a combination of calculation and intuition, more in line with the notion of game theory embraced by today's behavioral game theorists:

  "Every time, I've won because I could understand the way my enemy thought. From what they did. I could tell what they thought I was doing, how they wanted the battle to take shape. And I played off of that. I'm very good at that. Understanding how other people think."2

  That is, after all, what the modern science of game theory is all about—understanding how other pe
ople think. And consequently being able to figure out what they will choose to do. It is also what Isaac Asimov's fictional psychohistory was all about, and what the centuries-long quest by social scientists has been all about—discerning the drumbeat to which society dances. Discovering the Code of Nature.

  The modern search for a Code of Nature began in the century following Newton's Principia, which established the laws of motion and gravity as the rational underpinning of physical reality. Philosophers and political economists such as David Hume and Adam Smith sought a science of human behavior in the image of Newtonian physics, pursuing the dream that people could be described as precisely as planets. That dream persisted through the 19th century into the 20th, from Adolphe Quetelet's desire to describe society with numbers to Sigmund Freud's quest for a deterministic physics of the brain. Along the way, though, the physics model on which the dream was based itself changed, morphing from the rigid determinism of Newton into the statistical descriptions of Maxwell—the same sorts of statistics used, by Quetelet and his followers, to quantify society. By the end of the 20th century, the quest for a Code of Nature was taken up by physicists who wanted to use those statistics to bring the sciences of society and the natural world back together. Because after all, physics—just ask any physicist—is the science of everything.

  PHYSICS AND EVERYTHING

  Historically, the physicist's notion of everything has been a bit limited, though. For most of the past three centuries, physics concerned itself mostly with matter and the forces guiding its motion; eventually, the study of matter in motion incorporated energy and its transformations. In the century just gone by, Einstein added cosmic time and space to the mix. He even simplified reality's recipe by combining matter with energy and space with time. Through the 20th-century physicist's eyes, then, "everything" comprised mass-energy and space-time.

  Toward the end of that century, a number of physicists began to realize that one ingredient was missing. Awakened by the metaphorical power of the digital computer, astute observers realized that information was the glue connecting the outside world to its scientific description. From the second law of thermodynamics to the weirdness of quantum mechanics to the murky milieu of a black hole's interior, physicists found information to be an indispensable element in codifying and quantifying their understanding of nature.

  Information opened physicists' eyes to the rest of reality. Information encompassed biology. Biology included people. People created a new universe of realities for physics to contemplate—vast networks of economic, social, and cultural systems and institutions. So physicists began applying their favorite all-purpose tool—statistical mechanics—to everything from the stock market to flu epidemics. It was all very much in the spirit of Isaac Asimov's fictional mathematician, Hari Seldon, who adopted the principles of statistical mechanics to forecast the future. By the dawn of the 21st century, real-life physicists were trying to do almost exactly the same thing that Seldon had done, using statistical mechanics to build mathematical models of society for the purpose of making predictions.

  From its beginnings, game theory had expressed similar ambitions. Von Neumann and Morgenstern focused on economics, but clearly viewed economics as simply one (albeit a major) example of social science in general. They believed that their theory of games was a first step toward a mathematical representation of collective behavior, indeed a Code of Nature (their terms were "standard of behavior" or "order of society").

  A few years later, John Nash took a second major step toward a mathematics of society by introducing the Nash equilibrium into game theory's arsenal of ideas. If all the competitors in a game pursue their self-interest—attempting to maximize their expected payoff—there is always some combination of strategies that will produce the best deal that everybody can get (given that everybody plays their best). The existence of a Nash equilibrium in any game implied that societies could be stable—nobody having incentive to change their behavior, as any deviation would lower their payoff if everybody else continued to play the same way.

  In both von Neumann's and Nash's math, the essential feature was the need for "mixed strategies" to achieve the maximum payoff. Only rarely is one single "pure" strategy consistently your best bet. Your best strategy is typically to choose from among a range of possible choices, with specified probabilities for each choice.

  This idea of a mixed strategy is a recurring theme in game theory and its applications to various aspects of life and society. In evolution, nature plays a mixed strategy, generating complex ecosystems containing a wide range of species. The human race plays a mixed strategy, comprising cooperators, competitors, and punishers. Planet Earth's populations represent a mixed strategy of cultures, from the stingy and solitary Machiguenga in Peru to the generous and gregarious Orma in Kenya. Even in the physical realm, quantum physics shows that reality itself is a mixed strategy at the subatomic level, a feature that game theorists may be able to exploit to solve their thorniest dilemmas.

  Such a mixture of choices, with specific probabilities of each, is known in mathspeak as a probability distribution. And probability distributions, it just so happens, are what statistical physics deals with as well. Asimov's basis for psychohistory was applying the laws of probability to large numbers of individual humans to forecast collective human behavior, just as statistical physicists calculate probability distributions of large numbers of molecules to predict the properties of a gas or the course of chemical reactions. Like matter and energy, or space and time, game theory and physics are different sides of a coin. As Pat Benatar would say, they belong together. It's a neat, tight fit, and it's a mystery why it took so long for game theory and physics to mutually realize this underlying relationship.

  SEPARATED AT BIRTH

  Of course, game theory was conceived with fertilization from physical science, as both von Neumann and Nash applied reasoning rooted in statistical physics. Von Neumann referred to the usefulness of statistics in describing large numbers of interacting agents in an economy. Nash alluded to the statistical interactions of reacting molecules in his derivation of the Nash equilibrium. Nash, after all, studied chemical engineering and chemistry at Carnegie Tech before becoming a math major, and his dissertation at Princeton drew on the chemical concept of "mass action" in explaining the Nash equilibrium. Mass action refers to the way that amounts of reacting chemicals determine the reaction's equilibrium condition, a process described by the statistical mechanics of molecular energies. Borrowing the physical concept of equilibrium in chemical systems of molecules, Nash derived an analogous concept of equilibrium in social systems composed of people. Nash's math was about people, but it was based on molecules, and that math embodies the unification of game theory and social science with physics. The seed of the physics-society link resided within Nash's beautiful mind.

  That seed has sprouted and grown in unexpected ways, and its fruits are multiplying, feeding progress in a vast range of sciences, from economics, psychology, and sociology to evolutionary biology, anthropology, and neuroscience. Game theory provides the common mathematical language for unifying these sciences, sciences that represent the puzzle pieces that fit together to generate life, mind, and culture—the totality of collective human behavior. The fact that game theory math can also be translated into the mathematics of the physical sciences argues that it is the key to unlocking the real theory of everything, the science that unifies physics with life.

  After all, both physical and living systems seek stability, or equilibrium. If you want to predict the way a chemical reaction will proceed or how people will behave, and how the future will evolve, you need to know how to compute an equilibrium. Game theory shows why reaching an equilibrium point requires mixed strategies—and how this need for mixed strategies drives the creation of complexity. In other words, evolution. Game theory describes the evolutionary process that produces mixtures of different species, mixtures of different types of people, mixtures of different strategies that people
employ, mixtures of different cultures that arise in the mixture of environments found around the planet.

  Game theory describes the evolutionary process that produces complex networks. The brains that choose from a mix of strategies are networks of nerve cells; the societies that exhibit a mixture of cultures are networks of brains. Put it all together, and you get a framework for quantifying nature that really does encompass everything, a framework merging the game theory of the life and social sciences with the statistical physics describing the material world.

  Game theory is not, however, the same as the popular "Theory of Everything" that theoretical physicists have long sought. That quest is merely for the equations describing all of nature's basic particles and forces, the math describing the building blocks. Once you know how the pieces of atoms are put together, this view holds, you don't need to worry about everything else. Game theory, though, is precisely about everything else. It's about the realm of life that builds itself upon the universe's physical foundation. It's about how people carve civilization out of that jungle, and it's about the rules of conduct, the established order of society, the "Code of Nature" that results.

  DANGER

  There has always been a danger in seeking a Code of Nature—a risk that it would be regarded as a dogmatic deterministic view of human behavior, denying the freedom of the human spirit. Some people react very negatively to that sort of thing. The idea that a Code of Nature is inscribed into human genes, advanced in the 1970s under the label sociobiology, evoked a vitriolic response demonstrating how invective often overwhelms intellect. Sociobiology's intellectual descendant, evolutionary psychology, has produced a more elaborate web of evolution-based explanations for human behavior, but its implied prediction of hardwired brains that play pure strategies doesn't mesh well with the findings of modern neurobiology and behavioral anthropology.