Minimax: Minimizing the Worst-Case Loss

Minimax is a decision rule used in decision theory, game theory, statistics, and philosophy for minimizing the possible loss in the worst-case (maximum-loss) scenario. Rather than asking 'which option gives the best result on average,' it asks 'which option's worst outcome is least bad,' and chooses that one. The mirror image, when outcomes are framed as gains, is maximin: maximize the minimum guaranteed payoff. A defining feature is that minimax is non-probabilistic. It does not require any assumptions about how likely each outcome is — only an enumeration of what outcomes are possible under each choice. This makes it robust when probabilities are unknown or untrustworthy, in contrast to expected value reasoning, which weights outcomes by their probabilities and so depends on those estimates being right. Because of this, minimax is inherently risk-averse and conservative: it buys protection against catastrophe, sometimes at the cost of a lower average result. Minimax needs only ordinal information — a ranking of outcomes from worse to better — rather than a precise numeric scale, which lets it apply to qualitative decisions as well as quantitative ones. In two-player zero-sum games, minimax and maximin coincide and yield the minimax theorem of John von Neumann: each player can guarantee a certain value by assuming the opponent plays optimally against them. In John Rawls's political philosophy, a related maximin principle holds that social and economic inequalities are justified only if they benefit the least-advantaged members of society. The trade-off minimax makes — accepting a definite ongoing cost to rule out a rare disaster — is the same logic behind buying insurance, designing fail-safes, and many everyday precautions where the downside, not the average, is what people most want to control.