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Zero-Sum Games and Minimax

A game is zero-sum when the payoffs in every cell add to nothing: whatever one player gains, the other loses. There is no joint outcome to protect and no cooperation to be had, which makes these the cleanest games to analyse and the harshest to play.

The security level

Against an opponent who might be actively working against you, a natural question is what you can guarantee. For each of your actions, look at the worst column, then pick the action whose worst case is best.

Maximin, your security level
v‾=max⁡ai  min⁡a−i  πi(ai,a−i)\underline{v} = \max_{a_i} \; \min_{a_{-i}} \; \pi_i(a_i, a_{-i})

The payoff you can guarantee yourself without knowing anything about the opponent, including whether they are competent.

The mirror calculation, vˉ=min⁡max⁡\bar{v} = \min \max, is the most the opponent can hold you to. In a zero-sum game with mixed strategies allowed these two are equal, which is the minimax theorem, and their common value is the value of the game. That equality is why zero-sum games have a defensible single answer in a way that general games do not.

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