Nash Equilibrium: Existence Proof

110 words

1 min read

Visual companion

Python

Type and operator map

Python Week 1: the first filter for runtime behavior

Open gallery Related surface

View

Revision summary

What this note is really saying

Short form

# 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

\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.

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). Join Discord PreviousNash EquilibriumNextMixed Strategy Equilibrium