Game theory

: This article discusses the mathematical modelling of incentive structures. For other games (and their theories) see Game (disambiguation). For the band named Game Theory, please see Game Theory (band).

Mathematical definitions

There are a few alternative definitions of the notion of a 'game'.

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Normal form game design

A game in normal or strategic form combines the set of possible strategies for each player and records the payoffs for each outcome. Let mathrm{N} be a set of players. For each player i in mathrm{N} there is given a set of strategies Sigma ^i . The game is then a function:

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: pi : prod_{iin mathrm{N}} Sigma ^i o mathbb{R}^mathrm{N}

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So that, if one knows the tuple of strategies that were chosen by the players, one is given the allocation payments, a real number assignment. A further generalization can be achieved by splitting the game into two functions: the normal form game, describing the way in which strategies define outcomes, and a second function depicting player's preferences on the set of outcomes. Hence:

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: pi : prod_{i in mathrm{N}} Sigma ^i o Gamma

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Where Gamma is the outcome set of the game. And for each player iin mathrm{N} there is a preference function

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: u ^i : Gamma o mathbb{R} .

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A reduced normal form exists as well. The reduced normal form combines strategies for which are associated with the same payoffs.

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Extensive form game

The normal form gives the mathematician an easy notation for the study of equilibria problems, because it bypasses the question of how strategies are calculated, i.e. how the game is actually played. The convenient notation for dealing with these questions, more relevant to combinatorial game theory, is the extensive form of the game. This is given by a tree, where at each vertex of the tree a different player has the choice of choosing an edge.

Related Topics:
Combinatorial game theory - Tree - Vertex - Edge

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Simple game

The normal form and the extensive form capture the essence of non-cooperative games. But in some games the formation of coalitions and the way cooperation is developed are more important. For dealing with questions of cooperation, the notion of a simple game was developed.

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