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“Zero-sum game” describes a situation in which two “players” with strictly opposed interests each make a decision that results in one player’s winning equaling the opposing player’s loss. Many recreational games, such as chess, poker, and tic-tac-toe, are zero-sum because for one player to win, the opposing player(s) must lose.
The notion of zero-sum games originated in a branch of applied mathematics known as game theory, which has enjoyed extensive application in the social sciences. John von Neumann (1903-1957), a mathematician, is usually credited with creating game theory, and he first explicated the theory of zero-sum games in his seminal work with Oskar Morgenstern, Theory of Games and Economic Behavior (1944). Game theory is essentially a study of conflict situations between two or more opponents or players. Each player in the game situation must decide on a course of action, or strategy, and the strategy each player chooses affects the outcome for all players in the game. The outcome, or solution, to a zero-sum game specifies how each player should move, and if each player moves accordingly, then the resulting payoff is known as the value of the game (Kelly 2003).
The easiest class of games to analyze is two-person zero-sum games, and these games typically receive the most scholarly attention among those who study zero-sum games. In Theory of Games, von Neumann and Morgenstern focus their attention on two-person zero-sum games and show that in this type of game situation there always exists a solution that allows each player to avoid the worst possible outcome. To arrive at this solution, both players base their course of action on what they expect their opponent’s action will probably be. Keeping their opponent’s likely course of action in mind, both players attempt to minimize the opponent’s maximum payoff, thereby maximizing their own minimum payoff. In doing so, the outcome of the game ends up being that both players obtain the best payoff they possibly can, given the nature of the game, and neither is able to do any better. This outcome is known as the equilibrium of the game, and this point can be thought of as the outcome in which neither player has any regrets about the course of action chosen. This method of play is known as the minimax theorem, and von Neumann and Morgenstern showed that all two-person zero-sum games have a minimax solution.
All zero-sum games can be classified as having either perfect information or imperfect information. In a game with perfect information, each player in the game is fully aware of all previous moves in the game, meaning that each player knows what actions the opponent has already taken. In tic-tac-toe, for example, after the “X” player’s move, the “O” player knows exactly where the “X” player has placed an “X.” In games of perfect information, there is always at least one optimal or best possible strategy for each player. However, the existence of a best possible strategy does not guarantee that a player will win or even be able to identify that strategy. Using the best possible strategy only guarantees that both players will minimize their losses, regardless of whether they win. But there may also be so many viable strategies to choose from that it becomes impossible to determine what the best strategy is.
When applying the minimax theorem to zero-sum games with perfect information, it is possible to achieve the equilibrium point, or the point that represents the outcome that results from both players using their best possible strategy, also known as the saddle point. All zero-sum games with perfect information have at least one saddle point, and the saddle points can be determined using the minimax theorem. However, on some occasions the minimax theorem does not necessarily have to be used to determine a game’s saddle points. Occasionally, one player has strategies available that dominate the other strategies. A strategy is considered dominant if it yields a player a better outcome than any other strategy, despite the actions taken by the opponent. When a strategy is dominated by another, then the dominated strategy is said to be inadmissible because, if players are trying to get the best possible outcome, then it cannot make sense to choose a dominated strategy (Kelly 2003).
In games with imperfect information, the players are not fully aware of their opponent’s prior moves. This means that each player must choose an action without knowing what action the opponent has taken or may be taking simultaneously. A simple example of this would be the game rock-paper-scissors. While there may not be one best possible strategy, it is still possible to find a minimax solution to two-person games of imperfect information. This solution can be obtained by using mixed strategies. Using a mixed strategy means that a player uses one strategy sometimes, another strategy at other times. The player assigns each strategy a particular probability of being used and chooses a strategy based on these probabilities. When mixed strategies are in equilibrium, meaning that neither player can do better by deviating from these strategies, the strategies are sometimes called minimax mixed strategies (Kelly 2003).
Analysis of zero-sum games has been applied to a variety of social science disciplines, but it has probably enjoyed most extensive application in the fields of economics and political science. In political science, for example, most elections can be thought of as zero-sum games given that for one candidate to win, the opposing candidate must lose. Also, when considering the distribution of political resources, some scholars believe that for one group to gain political resources, others must lose resources, thus implying a zero-sum nature to political competition. However, the application of zero-sum games to political and economic phenomena is necessarily limited given that most conflict situations are not zero-sum. In many conflict situations, competitors do not have strictly opposed interests; it is often possible for both players in a game to win, as sometimes is the case with economic competition, or for both players to lose, as can happen with pollution or arms races (McCain 2004). Because of the dearth of real-world zero-sum situations, and thus zero-sum’s limited applicability, most game theoretic applications in the social sciences are not zero-sum.
Bibliography
- Dixit, A. K., and B. Nalebuff. 1991. Thinking Strategically: The Competitive Edge in Business, Politics, and Everyday Life. New York: Norton.
- Kelly, A. 2003. Decision Making Using Game Theory: An Introduction for Managers. Cambridge, U.K., and New York: Cambridge University Press.
- McCain, R. A. 2004. Game Theory: A Non-Technical Introduction to the Analysis of Strategy. Mason, OH: Thomson/South-Western.
- Von Neumann, J., and O. Morgenstern. 1944. Theory of Games and Economic Behavior. Princeton, NJ: Princeton University Press.
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