Game Theory

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Game Theory

Game theory is a branch of mathematics concerned with the analysis of conflict situations. It involves determining a strategy for a given situation and the costs or benefits realized by using the strategy. First developed in the early twentieth century, it was originally applied to parlor games such as bridge, chess, and poker. Now, game theory is applied to a wide range of subjects such as economics, behavioral sciences, sociology, military science, and political science.

The notion of game theory was first suggested by mathematician John von Neumann in 1928. The theory received little attention until 1944 when Neumann and economist Oskar Morgenstern wrote the classic treatise Theory of Games and Economic Behavior. Since then, many economists and operational research scientists have expanded and applied the theory.

Characteristics of games

An essential feature of any game is conflict between two or more players resulting in a win for some and a loss for others. Additionally, games have other characteristics which make them playable. There is a way to start the game. There are defined choices players can make for any situation that can arise in the game. During each move, single players are forced to make choices or the choices are assigned by random devices (such as dice). Finally, the game ends after a set number of moves and a winner is declared. Obviously, games such as chess or checkers have these characteristics, but other situations such as military battles or animal behavior also exhibit similar traits.

During any game, players make choices based on the information available. Games are therefore classified by the type of information that players have available when making choices. A game such as checkers or chess is called a "game of perfect information." In these games, each player makes choices with the full knowledge of every move made previously during the game, whether by herself or her opponent. Also, for these games there theoretically exists one optimal pure strategy for each player which guarantees the best outcome regardless of the strategy employed by the opponent. A game like poker is a "game of imperfect knowledge" because players make their decisions without knowing which cards are left in the deck. The best play in these types of games relies upon a probabilistic strategy and as such, the outcome can not be guaranteed.

Analysis of zero-sum, two-player games

In some games there are only two players and in the end, one wins while the other loses. This also means that the amount gained by the winner will be equal to the amount lost by the loser. The strategies suggested by game theory are particularly applicable to games such as these, known as zero-sum, two-player games.

Consider the game of matching pennies. Two players put down a penny each, either head or tail up, covered with their hands so the orientation remains unknown to their opponent. Then they simultaneously reveal their pennies and pay off accordingly; player A wins both pennies if the coins show the same side up, otherwise player B wins. This is a zero-sum, two-player game because each time A wins a penny, B loses a penny and visa versa.

To determine the best strategy for both players, it is convenient to construct a game payoff matrix, which shows all of the possible payments player A receives for any outcome of a play. Where outcomes match, player A gains a penny and where they do not, player A loses a penny. In this game it is impossible for either player to choose a move which guarantees a win, unless they know their opponent's move. For example, if B always played heads, then A could guarantee a win by also always playing heads. If this kept up, B might change her play to tails and begin winning. Player A could counter by playing tails and the game could cycle like this endlessly with neither player gaining an advantage. To improve their chances of winning, each player can devise a probabilistic (mixed) strategy. That is, to initially decide on the percentage of times they will put a head or tail, and then do so randomly.

According to the Minimax Theorem of game theory, in any zero-sum, two-player game there is an optimal probabilistic strategy for both players. By following the optimal strategy, each player can guarantee their maximum payoff regardless of the strategy employed by their opponent. The average payoff is known as the minimax value and the optimal strategy is known as the solution. In the matching pennies game, the optimal strategy for both players is to randomly select heads or tails 50% of the time. The expected payoff for both players would be 0.

Nonzero-sum games

Most conflict situations are not zero-sum games or limited to two players. A nonzero-sum game is one in which the amount won by the victor is not equal to the amount lost by the loser. The Minimax Theorem does not apply to either of these types of games, but various weaker forms of a solution have been proposed including noncooperative and cooperative solutions.

When more than two people are involved in a conflict, oftentimes players agree to form a coalition. These players act together, behaving as a single player in the game. There are two extremes of coalition formation; no formation and complete formation. When no coalitions are formed, games are said to be non-cooperative. In these games, each player is solely interested in her own payoff. A proposed solution to these types of conflicts is known as a non-cooperative equilibrium. This solution suggests that there is a point at which no player can gain an advantage by changing strategy. In a game when complete coalitions are formed, games are described as cooperative. Here, players join together to maximize the total payoff for the group. Various solutions have also been suggested for these cooperative games.

Application of game theory

Game theory is a powerful tool that can suggest the best strategy or outcome in many different situations. Economists, political scientists, the military, and sociologists have all used it to describe situations in their various fields. A recent application of game theory has been in the study of the behavior of animals in nature. Here, researchers are applying the notions of game theory to describe the effectiveness of many aspects of animal behavior including aggression, cooperation, hunting and many more. Data collected from these studies may someday result in a better understanding of our own human behaviors.