Symmetric game

In

structure of the payoffs. A game is quantitatively symmetric if and only if it is symmetric with respect to the exact payoffs. A partnership game is a symmetric game where both players receive identical payoffs for any strategy set. That is, the payoff for playing strategy a against strategy b receives the same payoff as playing strategy b against strategy a.

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

Only 12 out of the 144 ordinally distinct

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.

Nash (1951) shows that every finite symmetric game has a symmetric

. Emmons et al. (2022) show that in every common-payoff game (a.k.a. team game) (that is, every game in which all players receive the same payoff), every optimal strategy profile is also a 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

Hawk-dove

game).

The general case

A game with a payoff of ${\displaystyle U_{i}\colon A_{1}\times A_{2}\times \cdots \times A_{n}\longrightarrow \mathbb {R} }$ for player ${\displaystyle i}$, where ${\displaystyle A_{i}}$ is player ${\displaystyle i}$'s strategy set and ${\displaystyle A_{1}=A_{2}=\ldots =A_{N}}$, is considered symmetric if for any permutation ${\displaystyle \pi }$,

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

Partha Dasgupta and Eric Maskin give the following definition, which has been repeated since in the economics literature

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

However, this is a stronger condition that implies the game is not only symmetric in the sense above, but is a common-interest game, in the sense that all players' payoffs are identical.^[1]

• 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
• JSTOR 2297588
  .
• JSTOR 1969529
  .
• Emmons, Scott; Oesterheld, Caspar; Critch, Andrew; Conitzer, Vincent; Russell, Stuart (2022). "Symmetry, Equilibria, and Robustness in Common-Payoff Games" (PDF). Proceedings of the International Conference on Machine Learning (ICML). PMLR 162. Retrieved 21 April 2024.

• David Robinson; David Goforth (2005). The topology of the 2x2 games: a new periodic table. Routledge.
  ISBN 978-0-415-33609-3
  .
• Notes on Equilibria in Symmetric Games