Graphical game theory

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

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

In case where each player's utility function depends only on one other player: Image:GraphicalGameExample.png|The graphical form of the described game The maximal degree of the graph is 1, and the game can be described as n functions (tables) of size m^{2}. So, the total size of the input will be nm^{2}. ==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. ==References==