Strong Nash equilibrium

Strong Nash equilibrium
A solution concept in game theory
Relationships
Subset of	Evolutionarily stable strategy (if the strong Nash equilibrium is not also weak)
Significance
Used for	All non-cooperative games of more than 2 players

A strong Nash equilibrium is a Nash equilibrium in which no coalition, taking the actions of its complements as given, can cooperatively deviate in a way that benefits all of its members.^[1] While the Nash concept of stability defines equilibrium only in terms of unilateral deviations, strong Nash equilibrium allows for deviations by every conceivable coalition.^[2] This equilibrium concept is particularly useful in areas such as the study of voting systems, in which there are typically many more players than possible outcomes, and so plain Nash equilibria are far too abundant.

The strong Nash concept is criticized as too "strong" in that the environment allows for unlimited private communication. In fact, strong Nash equilibrium has to be Pareto-efficient. As a result of these requirements, Strong Nash rarely exists in games interesting enough to deserve study. Nevertheless, it is possible for there to be multiple strong Nash equilibria. For instance, in Approval voting, there is always a strong Nash equilibrium for any Condorcet winner that exists, but this is only unique (apart from inconsequential changes) when there is a majority Condorcet winner.

A relatively weaker yet refined Nash stability concept is called coalition-proof Nash equilibrium (CPNE) ^[2] in which the equilibria are immune to multilateral deviations that are self-enforcing. Every correlated strategy supported by iterated strict dominance and on the Pareto frontier is a CPNE.^[3] Further, it is possible for a game to have a Nash equilibrium that is resilient against coalitions less than a specified size k. CPNE is related to the theory of the core.

Confusingly, the concept of a strong Nash equilibrium is unrelated to that of a weak Nash equilibrium. That is, a Nash equilibrium can be both strong and weak, either, or neither.

References

↑ R. Aumann (1959), Acceptable points in general cooperative n-person games in "Contributions to the Theory of Games IV", Princeton Univ. Press, Princeton, N.J..
1 2 B. D. Bernheim; B. Peleg; M. D. Whinston (1987), "Coalition-Proof Equilibria I. Concepts", Journal of Economic Theory, 42: 1–12, doi:10.1016/0022-0531(87)90099-8.
↑ D. Moreno; J. Wooders (1996), "Coalition-Proof Equilibrium", Games and Economic Behavior, 17: 80–112, doi:10.1006/game.1996.0095.

Topics in game theory

Definitions	Normal-form game Extensive-form game Escalation of commitment Graphical game Cooperative game Succinct game Information set Hierarchy of beliefs Preference

Equilibrium concepts	Nash equilibrium Subgame perfection Mertens-stable equilibrium Bayesian Nash equilibrium Perfect Bayesian equilibrium Trembling hand Proper equilibrium Epsilon-equilibrium Correlated equilibrium Sequential equilibrium Quasi-perfect equilibrium Evolutionarily stable strategy Risk dominance Core Shapley value Pareto efficiency Gibbs equilibrium Quantal response equilibrium Self-confirming equilibrium Strong Nash equilibrium Markov perfect equilibrium

Strategies	Dominant strategies Pure strategy Mixed strategy Tit for tat Grim trigger Collusion Backward induction Forward induction Markov strategy

Classes of games	Symmetric game Perfect information Simultaneous game Sequential game Repeated game Signaling game Screening game Cheap talk Zero-sum game Mechanism design Bargaining problem Stochastic game n-player game Large Poisson game Nontransitive game Global games Strictly determined game Potential game

Games	Prisoner's dilemma Traveler's dilemma Coordination game Chicken Centipede game Volunteer's dilemma Dollar auction Battle of the sexes Stag hunt Matching pennies Ultimatum game Rock-paper-scissors Pirate game Dictator game Public goods game Blotto games War of attrition El Farol Bar problem Fair division Fair cake-cutting Cournot game Deadlock Diner's dilemma Guess 2/3 of the average Kuhn poker Nash bargaining game Prisoners and hats puzzle Trust game Princess and monster game Monty Hall problem Rendezvous problem

Theorems	Minimax theorem Nash's theorem Purification theorem Folk theorem Revelation principle Arrow's impossibility theorem

Key figures	Albert W. Tucker Amos Tversky Ariel Rubinstein Daniel Kahneman David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Hervé Moulin Jean Tirole Jean-François Mertens John Harsanyi John Maynard Smith John Nash John von Neumann Kenneth Arrow Kenneth Binmore Leonid Hurwicz Lloyd Shapley Melvin Dresher Merrill M. Flood Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Aumann Robert B. Wilson Roger Myerson Samuel Bowles Thomas Schelling William Vickrey

See also	All-pay auction Alpha–beta pruning Bertrand paradox Bounded rationality Combinatorial game theory Confrontation analysis Coopetition List of game theorists List of games in game theory No-win situation Topological game Tragedy of the commons Tyranny of small decisions

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