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 profiles, and $\phi(\sigma)$ is the set of profiles where each player best-responds to $\sigma_{-i}$. By Kakutani's theorem, $\phi$ has a fixed point $\sigma^* \in \phi(\sigma^*)$, which is a Nash equilibrium.

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). Join Discord PreviousNash EquilibriumNextMixed Strategy Equilibrium