# Graphical game theory

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[edit]

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 depended 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[edit]

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[edit]

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[edit]

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[edit]

- 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".