Game Theory

Game theory uses mathematics to model the interaction between two or more parties when they make decisions that will affect their outcomes. It sets up rules that determine each player's possible moves, the available information, and the various payoffs. Their actions are strategic in the sense that the actions of all parties are not fixed. The players have to make their best assessment of what to do in the face of actions of the other party and other variables to achieve an optimal outcome. Game theory has been used extensively in various fields, including economics, business, politics, social sciences, psychology, military planning, and more recently, biology.

Game theory originated shortly after World War II from work done by the mathematicians John von Neumann and Oskar Morgenstern for the RAND Corporation. It was further developed by John Nash and John Harsanyi in the 1950s.

The standard terminology calls the strategic interaction a game, and the participants are players. The best strategy is known as the solution, and the outcome is [Page 968]the value of the game. Games in this sense need not be trivial; for example, they have been used to work out nuclear strategy, where quite literally the payoffs involve life and death.

The game will specify who the players are, what moves are allowed, and the payoffs for each. A very simple example of a game used by von Neumann had two players who simultaneously present a coin face up or face down. If they are the same, then Player 1 wins by taking both coins; if they are different the coins go to Player 2. Games may be represented either by decision trees known as the extensive form that plot all the available information or more typically as a simple matrix (see Table 1).

The rows typically correspond to the strategic possibilities for the first player, and the payoffs are typically rendered with the row player's result, followed by those for the column player. Thus in this case, if both play heads, the top left cell, then the row player will win the coin at Player 2's expense. This is a zerosum game since a win for one side represents a loss for the other: +1, −1 = 0. We assume the players to be self-interested, and so they will seek to maximize their gains and minimize their losses.

Either before they begin or after a few rounds, players will develop a strategy. If a strategy were written down it would provide a set of directions that specify how to act and react to various payoff options. Here, the first player recognizes that if she is consistent in showing heads, then the other player will work out what is going on and play to his advantage by always selecting tails. Both sides impute similar rational thinking to the other and develop a strategy that denies any advantage to the opponent. In this case, the first player is best off choosing randomly and the best response by the second will be to play randomly as well. Thus in this game, and in fact for any similar game, the sides will come to a point where they have achieved the best possible response given the constraints and payoffs involved.

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