mini-max theorem

mathematics

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June 30, 2026

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The minimax theorem is a fundamental concept in game theory. It states that in a two-person constant-sum game, the lowest maximum expected loss for one player is equal to the highest minimum expected gain for the other player. This means there's a value, v, that represents the optimal outcome for both players when they employ their best strategies.12

The minimax principle, related to this theorem, suggests choosing an action that minimizes potential losses, even in the worst-case scenario. When no "saddlepoint" (a clear best strategy) exists, players use mixed strategies, choosing actions based on probability distributions to make themselves unpredictable and achieve the optimal outcome v.32

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game theory
Mixed strategies and the minimax theorem
When saddlepoints exist, the optimal strategies and outcomes can be easily determined, as was just illustrated. However, when there is no saddlepoint the calculation is more elaborate, as illustrated in Table 2.
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When saddlepoints exist, the optimal strategies and outcomes can be easily determined, as was just illustrated. However, when there is no saddlepoint the calculation is more elaborate, as illustrated in Table 2.
Read More

war
Diplomacy
…the basis of the “minimax” principle—the party in a position of advantage satisfies itself with the minimum acceptable gain because it realizes that the other party, in a position of disadvantage, would yield on the basis of its possible minimum loss but would violently oppose a distribution even more…
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…the basis of the “minimax” principle—the party in a position of advantage satisfies itself with the minimum acceptable gain because it realizes that the other party, in a position of disadvantage, would yield on the basis of its possible minimum loss but would violently oppose a distribution even more…
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John von Neumann
European career, 1921–30
…mathematical substance by proving the mini-max theorem. This asserts that for every finite, two-person zero-sum game, there is a rational outcome in the sense that two perfectly logical adversaries can arrive at a mutual choice of game strategies, confident that they could not expect to do better by choosing another…
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…mathematical substance by proving the mini-max theorem. This asserts that for every finite, two-person zero-sum game, there is a rational outcome in the sense that two perfectly logical adversaries can arrive at a mutual choice of game strategies, confident that they could not expect to do better by choosing another…
Read More