Symmetric game

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. 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.

Symmetry in 2x2 games

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

Only 12 out of 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. 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 uncorrelated asymmetries which are purely informational and have no effect on payoffs (e.g. see Hawk-dove game).

The general case

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

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

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]

References

^ ^a ^b Ham, Nicholas (18 November 2013). "Notions of Anonymity, Fairness and Symmetry for Finite Strategic-Form Games". arXiv:1311.4766 [math.CO].

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
Dasgupta, Partha; Maskin, Eric (1986). "The existence of equilibrium in discontinuous economic games, I: Theory". Review of Economic Studies. 53 (1): 1–26. doi:10.2307/2297588. JSTOR 2297588.
Nash, John (September 1951). "Non-cooperative games". Annals of Mathematics. 2nd Ser. 54 (2): 286–295. doi:10.2307/1969529. 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.

v t e Topics in game theory
Definitions	Congestion game Cooperative game Determinacy Escalation of commitment Extensive-form game First-player and second-player win Game complexity Graphical game Hierarchy of beliefs Information set Normal-form game Preference Sequential game Simultaneous game Simultaneous action selection Solved game Succinct game
Equilibrium concepts	Bayes correlated equilibrium Bayesian Nash equilibrium Berge equilibrium Core Correlated equilibrium Epsilon-equilibrium Evolutionarily stable strategy Gibbs equilibrium Mertens-stable equilibrium Markov perfect equilibrium Nash equilibrium Pareto efficiency Perfect Bayesian equilibrium Proper equilibrium Quantal response equilibrium Quasi-perfect equilibrium Risk dominance Satisfaction equilibrium Self-confirming equilibrium Sequential equilibrium Shapley value Strong Nash equilibrium Subgame perfection Trembling hand
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Classes of games	Bargaining problem Cheap talk Global game Intransitive game Mean-field game Mechanism design n-player game Perfect information Large Poisson game Potential game Repeated game Screening game Signaling game Stackelberg competition Strictly determined game Stochastic game Symmetric game Zero-sum game
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Theorems	Arrow's impossibility theorem Aumann's agreement theorem Folk theorem Minimax theorem Nash's theorem Negamax theorem Purification theorem Revelation principle Sprague–Grundy theorem Zermelo's theorem
Key figures	Albert W. Tucker Amos Tversky Antoine Augustin Cournot Ariel Rubinstein Claude Shannon Daniel Kahneman David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Hervé Moulin John Conway Jean Tirole Jean-François Mertens Jennifer Tour Chayes John Harsanyi John Maynard Smith John Nash John von Neumann Kenneth Arrow Kenneth Binmore Leonid Hurwicz Lloyd Shapley Melvin Dresher Merrill M. Flood Olga Bondareva Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Axelrod Robert Aumann Robert B. Wilson Roger Myerson Samuel Bowles Suzanne Scotchmer Thomas Schelling William Vickrey
Miscellaneous	All-pay auction Alpha–beta pruning Bertrand paradox Bounded rationality Combinatorial game theory Confrontation analysis Coopetition Evolutionary game theory First-move advantage in chess Glossary of game theory List of game theorists List of games in game theory No-win situation Solving chess Topological game Tragedy of the commons Tyranny of small decisions

This page was last edited on 21 April 2024, at 20:11

From Wikipedia, the free encyclopedia

Symmetry in 2x2 games

Symmetry and equilibria

Uncorrelated asymmetries: payoff neutral asymmetries

The general case

References

Further reading