# Graphical game theory

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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.

Game theory is the study of mathematical models of strategic interaction between rational decision-makers. It has applications in all fields of social science, as well as in logic and computer science. Originally, it addressed zero-sum games, in which one person's gains result in losses for the other participants. Today, game theory applies to a wide range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals, and computers.

In game theory, normal form is a description of a game. Unlike extensive form, normal-form representations are not graphical per se, but rather represent the game by way of a matrix. While this approach can be of greater use in identifying strictly dominated strategies and Nash equilibria, some information is lost as compared to extensive-form representations. The normal-form representation of a game includes all perceptible and conceivable strategies, and their corresponding payoffs, for each player.

An extensive-form game is a specification of a game in game theory, allowing for the explicit representation of a number of key aspects, like the sequencing of players' possible moves, their choices at every decision point, the information each player has about the other player's moves when they make a decision, and their payoffs for all possible game outcomes. Extensive-form games also allow for the representation of incomplete information in the form of chance events modeled as "moves by nature".

## Contents

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

## The size of the game's representation

For a general ${\displaystyle n}$ players game, in which each player has ${\displaystyle m}$ possible strategies, the size of a normal form representation would be ${\displaystyle O(m^{n})}$. The size of the graphical representation for this game is ${\displaystyle O(m^{d})}$ where ${\displaystyle d}$ is the maximal node degree in the graph. If ${\displaystyle d\ll n}$, 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 ${\displaystyle n}$ functions (tables) of size ${\displaystyle m^{2}}$. So, the total size of the input will be ${\displaystyle nm^{2}}$.

## 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.

Vijay Virkumar Vazirani is an Indian American distinguished professor of computer science in the Donald Bren School of Information and Computer Sciences at the University of California, Irvine.

Noam Nisan is an Israeli computer scientist, a professor of computer science at the Hebrew University of Jerusalem. He is known for his research in computational complexity theory and algorithmic game theory.

Timothy Avelin Roughgarden is a professor of Computer Science at Columbia University. Tim received his Ph.D. at Cornell University in 2002, under the supervision of Éva Tardos.

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