THE MINIMAX THEOREM — a dart

THE MINIMAX THEOREM every zero-sum game has a value

In a two-player zero-sum game, the most you can guarantee yourself (maximin) equals the least your opponent can hold you to (minimax). That shared number is the value of the game. When a pure saddle point exists it is visible directly; when not, von Neumann showed a mixed strategy always closes the gap. It is the theorem that founded game theory.

THE TECHNIQUE maximin = minimax = value

The demo takes a 2×2 payoff matrix with a saddle point and shows maximin = minimax: live demo

HISTORY & CREDIT von Neumann · 1928

“Von Neumann and Morgenstern proved the minimax theorem in 1944.” — von Neumann proved it alone in 1928 (Zur Theorie der Gesellschaftsspiele); the 1944 book applied it. cited

maximin · the row player’s best guaranteed floor.
minimax · the column player’s best guaranteed ceiling.
1928 · John von Neumann proved they always meet (with mixed strategies) — “there could be no theory of games… without it.”

Two opposed optimizations, meeting at one number. theorem

RECOMMEND FOR I-13 the saddle, on the compiler

On i-13, the matrix [[4,3],[2,1]] has maximin = minimax = 3 — a saddle:

$ i13 run gm_minimax-theorem.i13 RUN OK · 85 step(s) maximin = 3 -- max of row minimums minimax = 3 -- min of column maximums value = 3 saddle = 1

Recommend as a NULL — a theorem (B39). The value of a zero-sum game is forall-pinned by the payoff matrix; every correct solver returns the same number. NULL — every zero-sum game has a value.