Nash Equilibrium: Existence Proof
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# Nash Equilibrium: Existence Proof ## Nash's Theorem **Theorem (Nash, 1950):** Every finite game has at least one Nash equilibrium (possibly in mixed strategies). **Proof sketch (using Kakutani fixed-point theorem):** Define a correspondence $\phi: \Sigma \to \Sigma$ where $\Sigma$ is the set of mixed strategy prof...
Nash Equilibrium: Existence Proof
Nash's Theorem
Theorem (Nash, 1950): Every finite game has at least one Nash equilibrium (possibly in mixed strategies).
Proof sketch (using Kakutani fixed-point theorem): Define a correspondence where is the set of mixed strategy profiles, and is the set of profiles where each player best-responds to . By Kakutani's theorem, has a fixed point , which is a Nash equilibrium.
Computing Nash Equilibria
For 2×2 games: Solve indifference conditions as we did in Week 3. For larger games: Lemke-Howson algorithm (complementary pivoting). General case: Linear complementarity problem (LCP).
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