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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 <math>n</math> players with <math>m</math> 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

A graphical game is represented by a graph <math>G</math>, in which each player is represented by a node, and there is an edge between two nodes <math>i</math> and <math>j</math> iff their utility functions are depended on the strategy which the other player will choose. Each node <math>i</math> in <math>G</math> has a function <math>u_{i}:\{1\ldots m\}^{d_{i}+1}\rightarrow\mathbb{R}</math>, where <math>d_i</math> is the degree of vertex <math>i</math>. <math>u_{i}</math> specifies the utility of player <math>i</math> as a function of his strategy as well as those of his neighbors.

The size of the game's representation

For a general <math>n</math> players game, in which each player has <math>m</math> possible strategies, the size of a normal form representation would be <math>O(m^{n})</math>. The size of the graphical representation for this game is <math>O(m^{d})</math> where <math>d</math> is the maximal node degree in the graph. If <math>d\ll n</math>, then the graphical game representation is much smaller.

An example

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 <math>n</math> functions (tables) of size <math>m^{2}</math>. So, the total size of the input will be <math>nm^{2}</math>.

Nash equilibrium

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

  • Michael Kearns (2007) "Graphical Games". In Algorithmic Game Theory, N. Nisan, T. Roughgarden, E. Tardos and V. Vazirani, editors, Cambridge University Press, September, 2007.
  • Michael Kearns, Michael L. Littman and Satinder Singh (2001) "Graphical Models for Game Theory".