In a pure contest — my gain is exactly your loss — the wise move is to make your BEST WORST-CASE: pick the plan whose worst outcome is least bad. Von Neumann proved that in such zero-sum games your maximin and the enemy’s minimax meet at a single value, the SADDLE POINT — a stable price of the battle both can compute. Slide across the strategies and find where the worst-cases meet.
In a two-player ZERO-SUM game (one’s gain is the other’s loss), the row player secures at least maximin = max over rows of (min in that row); the column player holds the loss to at most minimax = min over columns of (max in that column). Von Neumann’s minimax theorem (1928) says these are EQUAL — the game has a value, a SADDLE POINT where neither can do better. For the payoff matrix [[4,2],[3,1]] the maximin (2, row 1) equals the minimax (2, col 2): the saddle is 2. A fail-loud self-check throws unless maximin equals minimax. ◆ real game theory, node-verified.
A matrix with a pure saddle point (the exact maximin=minimax value); games without a pure saddle need MIXED strategies (the theorem still holds in expectation) — the best-worst-cases-meet result is exact.