# What is game theory and what are some of its applications?

The Skin Game: Two players are each provided with an ace of diamonds and an ace of clubs. Player 1 is also given the two of diamonds and Player 2 the two of clubs. In a play of the game, Player 1 shows one card, and Player 2, ignorant of Player 1¿s choice, shows one card. Player 1 wins if the suits match, and Player 2 wins if they do not. The amount (payoff) that is won is the numerical value of the card of the winner. But, if the two deuces are shown, the payoff is zero. [Here, if the payoffs are in dollars, Player 1 can expect to win \$0.20. This game is a carnival hustler¿s (Player 1) favorite; his optimal mixed strategy is to never play the ace of diamonds, play the ace of clubs 60 percent of the time, and the two of diamonds 40 percent of the time.]

The power of game theory goes way beyond the analysis of such relatively simple games, but complications do arise. We can have many-person competitive situations in which the players can form coalitions and cooperate against the other players; many-person games that are nonzero-sum; games with an infinite number of strategies; and two-person nonzero sum games, to name a few. Mathematical analysis of such games has led to a generalization of von Neumann¿s optimal solution result for two-person zero-sum games called an equilibrium solution. An equilibrium solution is a set of mixed strategies, one for each player, such that each player has no reason to deviate from that strategy, assuming all the other players stick to their equilibrium strategy. We then have the important generalization of a solution for game theory: Any many-person non-cooperative finite strategy game has at least one equilibrium solution. This result was proven by John Nash and was pictured in the movie, A Beautiful Mind. The book (A Beautiful Mind, by Sylvia Nasar; Simon & Schuster, 1998) provides a more realistic and better-told story.

By now you have concluded that the answer to the opening question on competitive situations is "game theory." Aspects of all the cited areas have been subjected to analysis using the techniques of game theory. The web site www.gametheory.net lists about 200 fairly recent references organized into 20 categories. It is important to note, however, that for many competitive situations game theory does not really solve the problem at hand. Instead, it helps to illuminate the problem and offers us a different way of interpreting the competitive interactions and possible results. Game theory is a standard tool of analysis for professionals working in the fields of operations research, economics, finance, regulation, military, insurance, retail marketing, politics, conflict analysis, and energy, to name a few. For further information about game theory see the aforementioned web site and http://william-king.www.drexel.edu/top/eco/game/game.html.

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