A Nash equilibrium is a set of strategies, one for each player in a game, such that no player can improve their own outcome by changing only their own strategy while every other player's strategy stays fixed. John Nash proved in 1950 that every finite non cooperative game has at least one such equilibrium, using Kakutani's fixed point theorem, generalizing von Neumann's earlier minimax result beyond two player zero sum games.
Facts
StatementEvery finite non cooperative game with a finite number of players and finite pure strategy sets has at least one equilibrium point, possibly in mixed strategies. 1 Classification
Statement Form Statement Form Connections
Associated With
Zero-Sum Game, Concepts Von Neumann and Morgenstern first proved a mixed-strategy equilibrium exists for any zero-sum game; Nash's 1951 result generalized that same equilibrium concept beyond the zero-sum case.
Additional Source Zero-Sum Game (Wikipedia)Lead section
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Nash, Non-Cooperative Games (1951)John Nash
Proved By
Source Nash, Non-Cooperative Games (1951)John Nash
Sources
1. Nash, Non-Cooperative Games (1951)
John Nash, Annals of Mathematics, vol. 54, 1951p. 286
The notion of an equilibrium point is the basic ingredient in our theory.
- Proved By: John Nash
- In Branch: Game Theory
Zero-Sum Game (Wikipedia)
Wikimedia FoundationAssociated With: Zero-Sum Game, Lead sectionQuote, Associated With: Zero-Sum Game, Lead section
Zero-sum game is a mathematical representation in game theory and economic theory of a situation that involves two competing entities, where the result is an advantage for one side and an equivalent loss for the other
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