Existence of Nash equilibria
Statement
Every finite strategic-form game with a finite number of players, each having a finite set of pure strategies, has at least one Nash equilibrium in mixed strategies: a profile of (possibly randomized) strategies, one per player, such that no player can improve their expected payoff by unilaterally deviating, given the others' strategies.
Why is it true?
Think of each player continuously adjusting their (randomized) strategy to best respond to what everyone else is currently doing. This defines a map from the space of strategy profiles to itself. Because the strategy space (products of simplices, one per player) is compact and convex and the best-response map is well-behaved (convex-valued, upper semicontinuous), the map cannot avoid having a fixed point — a profile where every player is already best-responding to the others, so nobody wants to move.
Proof sketch
For each player , define the best-response correspondence mapping opponents' mixed-strategy profiles to the set of mixed strategies for that maximize 's expected payoff; has nonempty convex values and a closed graph because payoffs are multilinear and strategy simplices are compact convex sets. Combining all players gives a correspondence on the compact convex product of simplices satisfying the hypotheses of the Kakutani fixed-point theorem (a set-valued generalization of the Brouwer fixed-point theorem); any fixed point is by definition a Nash equilibrium.
Proved by
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Step-by-step proofs
No step-by-step proof yet for this theorem.
References
- John F. Nash Jr. (1950). Equilibrium points in n-person games
- John F. Nash Jr. (1951). Non-Cooperative Games