# Symmetric game

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In game theory, a symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. If one can change the identities of the players without changing the payoff to the strategies, then a game is symmetric. Symmetry can come in different varieties. Ordinally symmetric games are games that are symmetric with respect to the ordinal structure of the payoffs. A game is quantitatively symmetric if and only if it is symmetric with respect to the exact payoffs.

## Symmetry in 2x2 games

E F a, a b, c c, b d, d

Only 12 out the 144 ordinally distinct 2x2 games are symmetric. However, many of the commonly studied 2x2 games are at least ordinally symmetric. The standard representations of chicken, the Prisoner's Dilemma, and the Stag hunt are all symmetric games. Formally, in order for a 2x2 game to be symmetric, its payoff matrix must conform to the schema pictured to the right.

The requirements for a game to be ordinally symmetric are weaker, there it need only be the case that the ordinal ranking of the payoffs conform to the schema on the right.

## Symmetry and equilibria

Nash (1951) shows that every finite symmetric game has a symmetric mixed strategy Nash equilibrium. Cheng et al. (2004) show that every two-strategy symmetric game has a (not necessarily symmetric) pure strategy Nash equilibrium.

## Uncorrelated asymmetries: payoff neutral asymmetries

Symmetries here refer to symmetries in payoffs. Biologists often refer to asymmetries in payoffs between players in a game as correlated asymmetries. These are in contrast to uncorrelated asymmetries which are purely informational and have no effect on payoffs (e.g. see Hawk-dove game).

## The general case

Dasgupta and Maskin consider games ${\displaystyle (A_{i},U_{i})}$ where ${\displaystyle U_{i}:A_{i}\longrightarrow {\mathbb {R}}}$ where ${\displaystyle U_{i},i=1,\ldots N}$ is the payoff function for player ${\displaystyle i}$ and ${\displaystyle A_{1}=A_{2}=\ldots =A_{N}}$ is player ${\displaystyle i}$'s strategy set. Then the game is defined to be symmetric if for any permutation ${\displaystyle \pi }$,

${\displaystyle U_{i}(a_{1},\ldots ,a_{i},\ldots ,a_{N})=U_{\pi (i)}(a_{\pi (1)},\ldots ,a_{\pi (i)},\ldots ,a_{\pi (N)}).}$

## References

• Shih-Fen Cheng, Daniel M. Reeves, Yevgeniy Vorobeychik and Michael P. Wellman. Notes on Equilibria in Symmetric Games, International Joint Conference on Autonomous Agents & Multi Agent Systems, 6th Workshop On Game Theoretic And Decision Theoretic Agents, New York City, NY, August 2004. [1]
• Symmetric Game at Gametheory.net
• P. Dasgupta and E. Maskin 1986. "The existence of equilibrium in discontinuous economic games, I: Theory". The Review of Economic Studies, 53(1):1-26
• John Nash. "Non-cooperative Games". "The Annals of Mathematics", 2nd Ser., 54(2):286-295, September 1951.