Editing Nash equilibrium (section)

== Variants ==

=== Pure/mixed equilibrium ===
A game can have a [[Pure strategy|pure-strategy]] or a [[Mixed strategy|mixed-strategy]] Nash equilibrium. In the latter, not every player always plays the same strategy. Instead, there is a [[probability distribution]] over different strategies.

=== Strict/non-strict equilibrium ===
Suppose that in the Nash equilibrium, each player asks themselves: "Knowing the strategies of the other players, and treating the strategies of the other players as set in stone, would I suffer a loss by changing my strategy?"

If every player's answer is "Yes", then the equilibrium is classified as a ''strict Nash equilibrium''.<ref>{{Cite web |url=http://hoylab.cornell.edu/nash.html |title=Nash Equilibria |author=Robert Wyttenbach |website=hoylab.cornell.edu |access-date=2019-12-08 |url-status=dead |archive-url= https://web.archive.org/web/20190616093922/http://hoylab.cornell.edu/nash.html |archive-date= Jun 16, 2019 }}</ref>

If instead, for some player, there is exact equality between the strategy in Nash equilibrium and some other strategy that gives exactly the same payout (i.e. the player is indifferent between switching and not), then the equilibrium is classified as a ''weak''{{NoteTag|This term is dispreferred, as it can also mean the opposite of a "strong" Nash equilibrium (i.e. a Nash equilibrium that is vulnerable to manipulation by groups).}} or ''non-strict Nash equilibrium''{{Citation needed|date=February 2024}}{{Clarify|reason=Is the term used exclusively to mean the equilibrium is non-strict for some players, or is it sometimes used to mean the Nash equilibrium is non-strict for every player?|date=February 2024}}.

=== Equilibria for coalitions ===
The Nash equilibrium defines stability only in terms of individual player deviations. In cooperative games such a concept is not convincing enough. [[Strong Nash equilibrium]] allows for deviations by every conceivable coalition.<ref name="CoalitionProof">{{Citation |author1=B. D. Bernheim |title=Coalition-Proof Equilibria I. Concepts |journal=Journal of Economic Theory |volume=42 |issue=1 |pages=1&ndash;12 |year=1987 |postscript=. |doi=10.1016/0022-0531(87)90099-8 |author2=B. Peleg |author3=M. D. Whinston|doi-access=free }}</ref> Formally, 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.<ref name="SNE">{{Cite book |last=Aumann |first=R. |title=Contributions to the Theory of Games |publisher=Princeton University Press |year=1959 |isbn=978-1-4008-8216-8 |volume=IV |location=Princeton, N.J. |chapter=Acceptable points in general cooperative n-person games}}</ref> However, the strong Nash concept is sometimes perceived as too "strong" in that the environment allows for unlimited private communication. In fact, strong Nash equilibrium has to be weakly [[Pareto efficient]]. As a result of these requirements, strong Nash is too rare to be useful in many branches of game theory. However, in games such as elections with many more players than possible outcomes, it can be more common than a stable equilibrium.

A refined Nash equilibrium known as [[coalition-proof Nash equilibrium]] (CPNE)<ref name="CoalitionProof" /> occurs when players cannot do better even if they are allowed to communicate and make "self-enforcing" agreement to deviate. Every correlated strategy supported by [[Dominance (game theory)|iterated strict dominance]] and on the [[Pareto frontier]] is a CPNE.<ref name="CPNE">{{Citation |author1=D. Moreno |title=Coalition-Proof Equilibrium |url=http://e-archivo.uc3m.es/bitstream/10016/4408/1/Coalition_GEB_1996_ps.PDF |journal=Games and Economic Behavior |volume=17 |issue=1 |pages=80&ndash;112 |year=1996 |postscript=. |doi=10.1006/game.1996.0095 |author2=J. Wooders |hdl=10016/4408 |hdl-access=free}}</ref> 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 [[Core (economics)|theory of the core]].