Nash's Math Gets More Beautiful: Even with Infinite Choices, Games May Have Stable Strategies
Barry, Patrick, Science News
Life's a game, or at least treating it like a game mathematically can be a powerful way to explain the choices people make. John Nash, the mentally troubled mathematician depicted in the book and movie A Beautiful Mind, discovered one of the bedrock theories for understanding competitive interactions (genetically called "games") in which players have a limited set of choices.
Now mathematicians are expanding Nash's idea to cases when the players' options are infinite. Under certain conditions, infinite-choice games are guaranteed to have at least one scenario in which each player's choice gets that player the best deal possible (given everyone else's choices), according to a proof to be published in the February 2009 Nonlinear Analysis.
Such a scenario is called a Nash equilibrium. It is stable because no player can do any better by changing strategy. Like a rock at the bottom of a valley, a game reaching this stable scenario should tend to stay that way. In a sense, it's the fate of the game to end up at a Nash equilibrium, and this predictive power is why Nash's ideas have become widely used in economics and other social sciences.
Nash proved that there is always at least one such equilibrium for games with a finite number of strategic choices. But not all possible games are so limited.
"There are many economically important games in which the sets of pure strategies are infinite," comments Andy McLennan, a mathematician and economist who studies game theory at the University of Queensland campus in St. …