A Beautiful Math John Nash, Game Theory, and the Modern Quest for a Code of Nature (2006) / Chapter Skim
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3 Nash’s Equilibrium--Game theory’s foundation
3 Nash’s Equilibrium--Game theory’s foundation
Pages 51-72
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From page 51... ...
Shortly before Nash's arrival at Princeton, von Neumann and Morgenstern had opened a whole new continent for mathematical exploration with the groundbreaking book Theory of Games and Economic Behavior. It was the Louisiana Purchase of economics.
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From page 52... ...
"Nash carried social science into a new world where a unified analytical structure can be found for studying all situations of conflict and cooperation," writes University of Chicago economist Roger Myerson. "The theory of noncooperative games that Nash founded has developed into a practical calculus of incentives that can help us to better understand the problems of conflict and cooperation in virtually any social, political, or economic institution."1 So it's not too outrageous to suggest that in a very real way, Nash's math provides the foundation for a modern-day Code of Nature.
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From page 53... ...
"But to a large extent they are only potentialities: results are still largely a matter of future developments."4 A more enthusiastic assessment appeared in a mathematics journal, where a reviewer wrote that "posterity may regard this book as one of the major scientific achievements of the first half of the twentieth century."5 The world at large also soon learned about game theory. In 1946, the von NeumannMorgenstern book rated a front page story in the New York Times; three years later a major piece appeared in Fortune magazine.
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From page 54... ...
BEAUTIFUL MATH The book A Beautiful Mind offers limited insight into Nash's math, particularly in regard to all the many areas of science where that math has lately become prominent.8 But the book reveals a lot about Nash's personal troubles. Sylvia Nasar's portrait of Nash is not very flattering, though.
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From page 55... ...
Von Neumann was at the Institute for Advanced Study, just a mile from the university, and Morgenstern was in the Princeton economics department. And at the university math department, a cadre of young game theory enthusiasts had begun exploring the new von Neumann Morgenstern continent in earnest.
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From page 56... ...
It was the paper introducing the "Nash equilibrium," eventually to become game theory's most prominent pillar. The idea of equilibrium is, of course, immensely important to many realms of science.
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From page 57... ...
Nash had just this sort of physical equilibrium in mind when he was contemplating stability in game theory. In his dissertation he refers to "the `mass-action' interpretation of equilibrium," and that such an equilibrium is approached in a game as players "accumulate empirical information" about the payoffs of their strategies.10 When equilibrium is reached in a chemical reaction, the quantities of the chemicals no longer change; when equilibrium is reached in a game, nobody has any incentive to change strategies -- so the choice of strategies should remain constant (the game situation is, in other words, stable)
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From page 58... ...
In a two-person zero-sum game, you can determine the equilibrium point using von Neumann's minimax solution. Whether using a pure strategy or a mixed strategy, neither player has anything to gain by deviating from the optimum strategy that game theory prescribes.
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From page 59... ...
you'll do worse. To put it more colloquially, says economist Robert Weber, you could say that "the Nash equilibrium tells us what we might expect to see in a world where no one does anything wrong."12 Or as Samuel Bowles described it to me, the Nash equilibrium "is a situation in which everybody is doing the best they can, given what everybody else is doing."13 Von Neumann was dismissive of Nash's result, as it did turn game theory in a different direction.
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From page 60... ...
In a dog-eat-dog world, the Nash equilibrium describes how every dog can have its best possible day. "The distinction between non-cooperative and cooperative games that Nash made is decisive to this day," wrote game theorist Harold Kuhn.16 To me, the really key point about the Nash equilibrium is that it cements the analogy between game theory math and the laws of physics -- game theory describing social systems, the laws of physics describing natural systems.
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From page 61... ...
Using Nash's math, you can figure out how people could reach stability in a social situation by comparing that situation to an appropriate game. So if you want to apply game theory to real life, you need to devise a game that captures the essential features of the real-life situation you're interested in.
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From page 62... ...
. Alice Keep Mum Rat Keep Mum 1, 1 5, 0 Bob Rat 0, 5 3, 3 Years in prison for Bob, Alice If you look at this game matrix, you can easily see where the Nash equilibrium is.
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From page 63... ...
But that doesn't detract from the importance of the Nash equilibrium, as economists Charles Holt and Alvin Roth point out. "The Nash equilibrium is useful not just when it is itself an accurate predictor of how people will behave in a game but also when it is not," they write, "because then it identifies a situation in which there is a tension between individual incentives and other motivations." So if people cooperate (at least at first)
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From page 64... ...
(And often there is more than one Nash equilibrium point, which makes things really messy.) Remember, each player's "strategy" will typically be a mixed strategy, drawn from maybe dozens or hundreds or thousands (or more)
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From page 65... ...
. Over time, the members of each group earned equal amounts of money, suggesting that something like a Nash equilibrium had been achieved -- they all won as much as they could, given the strategy of the others.
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From page 66... ...
GAME THEORY TODAY Together with his paper on the bargaining problem (which treats cooperative game situations) , Nash's work on equilibria in manyplayer games greatly expanded game theory's scope beyond von Neumann and Morgenstern's book, providing the foundation for much of the work in game theory going on today.
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From page 67... ...
The question is, are there ever any such circumstances? Early euphoria about game theory's potential to illuminate social issues soon dissipated, as a famous game theory text noted in 1957.
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From page 68... ...
There are very few insights from game theory that would improve one's game of chess or poker," Rubenstein wrote.25 He scoffed at theorists who believed game theory could actually predict behavior, or even improve performance in real-life strategic interactions. "I have never been persuaded that there is a solid foundation for this belief," he wrote.
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From page 69... ...
Just because game theory cannot predict human behavior infallibly today doesn't mean that its insights are worthless. In his book Behavioral Game Theory, Colin Camerer addresses these issues with exceptional insight and eloquence.
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From page 70... ...
Economist Thomas Schelling, of the University of Maryland, understood in the 1950s that game theory offered a mathematical language suitable for unifying the social sciences, a vision he articulated in his 1960 book The Strategy of Conflict. "Schelling's work prompted new developments in game theory and accelerated its use and application throughout the social sciences," the Royal Swedish Academy of Sciences remarked on awarding the prize.29 Schelling paid particular attention to game-theoretic analysis of international relations, specifically (not surprising for the time)
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From page 71... ...
Some of Schelling's later work applied game theory to the rapid change in some neighborhoods from a mixture of races to being largely segregated, and to limits on individual control over behavior -- why people do so many things they really don't want to do, like smoke or drink too much, while not doing things they really want to, like exercising. 2005's other economics Nobel winner, Robert Aumann, has long been a leading force in expanding the scope of game theory to many disciplines, from biology to mathematics.
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From page 72... ...
Gossip, in fact, turns out to be a crucial outcome of game theory in action, for it's at the heart of understanding human social behavior, the Code of Nature that made it possible for civilization to establish itself out of the selfish struggles to survive in the jungle. For it is in biology that game theory has demonstrated its power most dramatically, in explaining otherwise mysterious outcomes of Darwinian evolution.
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