CONCEPT
The Minimax Theorem
Von Neumann's 1928 proof that in any two-person zero-sum game there exists an optimal strategy—the mathematics of rational agency in conflict that became the skeleton of game theory, AI alignment, and every account of how capable systems pursue goals against opposition.
In 1928, in a paper published in a German mathematical journal, the twenty-five-year-old
von Neumann proved a result that would eventually anchor both game theory and AI alignment: in any two-person zero-sum game, where one player's gain is exactly the other's loss, there exists a pair of strategies such that neither player can improve by deviating, and the expected payoff is the same from both players' perspectives. Each player seeks to minimize their maximum loss; equivalently, to maximize their minimum gain. Von Neumann showed that these two quantities—the minimax and the maximin—coincide. There is a single value to the game, and strategies that achieve it, possibly through deliberate randomization. The proof drew on a fixed-point theorem from topology, and it established that rational play in situations of pure conflict is not a matter of psychology or cunning but of mathematics: the optimal strategy can, in principle, be computed. This result became the load-bearing wall