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 percent 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 happened 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 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 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.
Ev
en 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 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 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.
* * *
10
Meyer's Penny Quantum fun and games
Do games have anything deeper to say about physics, or vice versa? Maybe. Most surprisingly, the connection might arise at the most fundamental level of all: quantum physics.
—Chiu Fan Lee and Neil F. Johnson, Physics World
It's the 24th century, aboard the starship Enterprise.
Captain Jean-Luc Picard places a penny heads up in a box, so that it can be touched but not seen. His nemesis Q, an alien with mysterious powers, then chooses whether to flip the coin over or not. Without knowing what Q has done, Picard then must decide to flip, or not flip, the coin as well. Q then gets the last turn. He either flips the penny or leaves it alone. If the penny shows heads when the box is removed, Q wins; tails wins for Picard.
They play the game 10 times, and Q wins them all.
It's not a scene from any actual episode of Star Trek: The Next Generation, but rather a scenario from a physics journal introducing an entirely new way of thinking about game theory.
The penny-flipping game is an old game theory favorite. It appears in various disguises, such as the game of chicken. (Whether you flip the coin or leave it alone corresponds to veering out of the path of the oncoming car or continuing straight on.) If they were playing the original version of the penny game, Q and Picard should, in the long run, break even, no one player winning more often than the other. Ten wins in a row for one player defies any reasonable definition of luck.
So if this had really happened on the show, Commander Riker would have immediately accused Q of cheating. But the wiser Picard would have pondered the situation a little longer and eventually would have realized that Q's name must be short for quantum. Only someone possessing quantum powers can always win the penny game.
As it turns out, Earth's physicists did not need an alien to teach them about quantum games. They emerged three centuries early, on the eve of the 21st century, out of an interest in using the powers of quantum mechanics to perform difficult computations. It was an unexpected twist in the story of game theory, as quantum games disrupted the understanding of traditional "classical" games in much the way that quantum mechanics disturbed the complacency of classical physics. The invention of quantum game theory suggested that the bizarre world of quantum physics, once restricted to explaining atoms and molecules, might someday invade economics, biology, and psychology. And it may even be (though perhaps not until the 24th century) that quantum games will cement the merger of game theory and physics. In fact, if physics ever finds the recipe for forecasting and influencing the social future, it might be that quantum game theory will provide the essential ingredient.
Now, if you've been reading carefully all this time, it might seem a little unfair that, after coming to grips with the complexities of game theory, network math, and statistical mechanics, you must now face the bewildering weirdness of quantum physics on top of it. Fortunately, the space available here does not permit the presentation of a course in quantum mechanics. Besides, you don't need to know everything there is to know about quantum physics to see how quantum game theory works. But you do have to be willing to suspend your disbelief about some of quantum theory's strangest features—most importantly, the concept of multiple realities.
QUANTUM TV
I've described this quantum confusion before (in my book The Bit and the Pendulum) by relating it to television. In the old days, TV signals traveled through the air, all the possible channels passing through your living room at the same time. (Nowadays they usually arrive via cable.) By turning the dial on your TV set (or punching a button on the remote control), you can make one of those shows—one of
many possible realities—come to life on your screen. The realm of atoms, molecules, and particles even smaller works in a similar way. Left to themselves, particles buzz about like waves, and their properties are not sharply defined. In particular, you cannot say that a particle occupies any specific location. An atom can literally be in two places at once—until you look at it. An observation will find it located in one of the many possible positions that the quantum equations allow.
An important issue here, one that has occupied physicists for decades, involves defining just what constitutes an "observation." In recent years, it has become generally agreed that humans are not necessary to perform an observation or measurement on a particle. Other particles bouncing off it can accomplish the same effect. That is to say, an atom, on its own, cannot be said to occupy a specific location. But once other atoms start hitting it, the atom will become localized in a position consistent with the altered paths of the other atoms. This phenomenon is known as decoherence. As long as decoherence can be avoided (for example, by isolating a particle from other influences, maintained at very low temperatures), the weird multiplicity of quantum realities can be sustained.
This feature of quantum physics has been an endless source of controversy and consternation for physicists and nonphysicists alike. But experimental tests have left no room for doubt on this point. In the subatomic world, reality is fuzzy, encompassing a multiplicity of possibilities. And those possibilities all have a claim to being real. It's not just that you don't know where an atom is— it occupies no definite location, but rather occupies many locations simultaneously.
From a game theory point of view, there is a simple enough way of looking at this—reality itself is a mixed strategy.
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