The minimax theorem states that in a two person zero sum game with finitely many pure strategies, there is a value and a pair of mixed strategies, one for each player, such that each player can guarantee at least that value regardless of what the other does. John von Neumann proved it in 1928, founding mathematical game theory.
Facts
StatementFor every two person zero sum game with finitely many strategies, the maximum of the minimum outcomes equals the minimum of the maximum outcomes: max-min equals min-max. 2 Classification
Statement Form Connections
Associated With
Zero-Sum Game, Concepts Von Neumann's 1928 minimax theorem, about two-player zero-sum games and considered the starting point of game theory, proves optimal strategies exist for exactly this kind of game.
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.
In Branch
Source von Neumann, Zur Theorie der Gesellschaftsspiele (1928)John von Neumann
Additional Source Wikipedia: Minimax TheoremLead section
Proved By
Source von Neumann, Zur Theorie der Gesellschaftsspiele (1928)John von Neumann
Sources
1. Wikipedia: Minimax Theorem
Wikimedia FoundationLead section, statement-form reference
It is always true that the left-hand side is at most the right-hand side (max-min inequality) but equality only holds under certain conditions identified by minimax theorems.
In Branch: Optimization, Lead section
In the mathematical area of game theory and of convex optimization
View the Source 2. von Neumann, Zur Theorie der Gesellschaftsspiele (1928)
John von Neumann, Mathematische Annalen, vol. 100, 1928- Mathematische Annalen 100 (1928), pp. 295-320
- Proved By: John von Neumann
- 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
Dissenting Readings (1 dissenting reading)
Proved By: John von Neumann
In a 1953 Econometrica article, Maurice Frechet argued that Emile Borel, in papers published between 1921 and 1927, should be credited as the true initiator of game theory ahead of von Neumann, even though Borel himself never proved the minimax theorem. Von Neumann replied in the same issue, defending his own 1928 paper as the first rigorous proof and the founding result of the mathematical theory. The credit for founding game theory is genuinely contested in this narrow sense: Borel raised the questions first, von Neumann proved the theorem first.
A dissenting reading, from Maurice Frechet (1953)Maurice Frechet, Frechet, Emile Borel, Initiator of the Theory of Psychological Games and Its Application (1953), Econometrica, vol. 21, no. 1, 1953
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