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.
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
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.
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 .
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.