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 $$n$$ players with $$m$$ 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 $$G$$, 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:

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.