In game theory, the common ways to describe a game are the normal form and the extensive form. The graphical form is an alternate compact representation of a game using the interaction among participants.

Consider a game with players with strategies each. We will represent the players as nodes in a graph in which each player has a utility function that depends only on him and his neighbors. As the utility function depends on fewer other players, the graphical representation would be smaller.

Formal definition

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A graphical game is represented by a graph  , in which each player is represented by a node, and there is an edge between two nodes   and   iff their utility functions are dependent on the strategy which the other player will choose. Each node   in   has a function  , where   is the degree of vertex  .   specifies the utility of player   as a function of his strategy as well as those of his neighbors.

The size of the game's representation

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For a general   players game, in which each player has   possible strategies, the size of a normal form representation would be  . The size of the graphical representation for this game is   where   is the maximal node degree in the graph. If  , then the graphical game representation is much smaller.

An example

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In case where each player's utility function depends only on one other player:

The maximal degree of the graph is 1, and the game can be described as   functions (tables) of size  . So, the total size of the input will be  .

Nash equilibrium

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Finding Nash equilibrium in a game takes exponential time in the size of the representation. If the graphical representation of the game is a tree, we can find the equilibrium in polynomial time. In the general case, where the maximal degree of a node is 3 or more, the problem is NP-complete.

Further reading

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  • Michael Kearns (2007) "Graphical Games". In Vazirani, Vijay V.; Nisan, Noam; Roughgarden, Tim; Tardos, Éva (2007). Algorithmic Game Theory (PDF). Cambridge, UK: Cambridge University Press. ISBN 0-521-87282-0.
  • Michael Kearns, Michael L. Littman and Satinder Singh (2001) "Graphical Models for Game Theory".