game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...

a strong Nash equilibrium is a

Nash equilibrium In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equili ...

in which no coalition, taking the actions of its complements as given, can cooperatively deviate in a way that benefits all of its members. While the Nash concept of stability defines equilibrium only in terms of unilateral deviations, strong Nash equilibrium allows for deviations by every conceivable coalition. 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 Pareto efficiency or Pareto optimality is a situation where no action or allocation is available that makes one individual better off without making another worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian civil engin ...

. 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 Approval voting is an electoral system in which voters can select many candidates instead of selecting only one candidate. Description Approval voting ballots show a list of the options of candidates running. Approval voting lets each voter i ...

, there is always a strong Nash equilibrium for any

Condorcet winner An electoral system satisfies the Condorcet winner criterion () if it always chooses the Condorcet winner when one exists. The candidate who wins a majority of the vote in every head-to-head election against each of the other candidatesthat is, a ...

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 The concept of coalition-proof Nash equilibrium applies to certain "noncooperative" environments in which players can freely discuss their strategies but cannot make binding commitments. It emphasizes the immunization to deviations that are self-en ...

(CPNE) 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 In multi-objective optimization, the Pareto front (also called Pareto frontier or Pareto curve) is the set of all Pareto efficient solutions. The concept is widely used in engineering. It allows the designer to restrict attention to the set of effi ...

is a CPNE. 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

Game theory equilibrium concepts {{Gametheory-stub