Mixed Strategies & Zero-Sum

Mixed Strategies and Zero-Sum · Open board

Every game so far had one move that was simply correct to settle on. This board is the other kind: pick anything, and a sharp opponent can read it and beat it, so the only way to stay ahead is to turn your own choice into a coin toss.

Before you start. The last lesson (Repeated Games & Cooperation) showed that when you expect to face the same player again and again, being predictable pays: a rational player can be trusted to cooperate, because burning that trust costs every future round. This lesson is the mirror image - a game where facing the same player again is exactly what makes being predictable fatal, and the only rational move is to stop being readable at all.

The idea in 30 seconds

Some games have no move that’s simply correct: whatever you pick, the other side has something that beats it.
In such a game you play a mixed strategy: a fixed set of odds for choosing among your moves, instead of one move every time.
The indifference principle sets those odds: pick them so every choice your opponent makes earns them the same, leaving nothing to exploit.
In a zero-sum game, whatever one player wins, the other loses, and the minimax theorem (1928) proved every such game has an exact number you can guarantee by mixing correctly.
Favouring your strong option is usually right; committing to it completely is what gets you read and beaten.
Randomise against someone trying to predict you; stay predictable when you and the other side both want to agree.

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