If there is no pure strategy Nash equilibrium in a complete information game, there is a mixed strategy equilibrium, and if there is no mixed strategy equilibrium, there is a pure strategy equilibrium.

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This is a statement of the famous theorem by John Nash, which asserts that every finite game with complete information has at least one Nash equilibrium. If there is no pure strategy Nash equilibrium, it must mean that all strategies of all players are strictly dominated, which implies the existence of a mixed strategy Nash equilibrium. Conversely, if there is no mixed strategy Nash equilibrium, it must mean that every mixed strategy is also dominated by some pure strategy, which implies the existence of a pure strategy Nash equilibrium.

If There Is No Pure Strategy Nash Equilibrium in a Complete