Game Theory

Theory and Concepts

The assumptions of common knowledge and strategic decision making define a game. Decisions are strategy choices. The strategy si of player i is the plan of how to act at various stages of the game when it is i's turn to make a move. The strategy choices of all players determine the outcome. Each player evaluates alternative outcomes. Player i's evaluation is expressed in terms of payoffs that derive from a utility function ui mapping the outcomes into the space of real numbers. To account for uncertain outcomes, we assume that ui satisfies the expected utility hypothesis.

The strategic form of a game is expressed by a payoff matrix that summarizes the strategy sets of the n players and assigns to each available strategy n-vector an n-dimensional payoff vector. One implicitly assumes that decisions are made simultaneously, that is, without knowledge of the decisions of the other players. Such a game is one of imperfect information.

The concept of Nash equilibrium describes the combination of strategies the players are expected to choose. A strategy n-vector s represents an equilibrium if,ceteris paribus, none of the players can choose a strategy different from the one prescribed by s and achieve a larger payoff. Some games have a mixed strategy equilibrium, that is,s* contains strategies that assume that players choose pure strategies with probabilities smaller than 1. Unfortunately, it is often the case that there is more than one Nash equilibrium, and an equilibrium selection concept has to be applied.

If players can observe moves of other players and information is to some degree perfect, then the game is sequential and a decision tree that expresses the information the players have can represent it. One can apply the concept of subgame perfectness to select appropriate Nash equilibria for games represented in this form.

This basic structure is noncooperative game theory because one assumes that players cannot form binding agreements (on their strategy choices). By dropping this assumption, one enters the realm of cooperative game theory. Here the problem of the players is not to select strategies but to agree on an outcome. One sometimes identifies cooperative games with bargaining games. Another variety is the coalitional form: it takes care of the fact that a group of players can form an agreement to interact, as an entity, with other groups of players. In general, the agreement is about the distribution of the coalition value. A prominent solution concept for these games is the “core.”

Illustration: The Stag-Hunt Game

The game matrix in Figure 1 describes the decision problem of two players, 1 and 2. One assumes that if they coordinate their behavior, such that they both choose to go stag hunting, they will shoot a stag. If, however, player 1 chooses to hunt hares, he will be successful, while player 2, choosing to hunt a stag, will fail. If both choose to hunt hares, both will be successful. The assumption here is that the number of hares each player can catch is the same and independent of whether the players hunt alone or together.

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