Nash Existence Theorem

Game Theory · Axiom Academy

Unit 2 - Dominance and Nash Equilibrium Every finite game has at least one Nash equilibrium, possibly in mixed strategies. Finite number of players: The game must have finitely many participants (n players) Finite strategy sets: Each player must have a finite set of available pure strategies When these conditions are met, the theorem guarantees the existence of at least one Nash equilibrium. This equilibrium may be in pure strategies (where each player deterministically chooses one action) or in mixed strategies (where players randomize over their available actions). 2. The Role of Mixed Strategies While some games have Nash equilibria in pure strategies, not all games do. The power of Nash's theorem is that it extends to mixed strategies, guaranteeing that a solution always exists. When players use mixed strategies, they randomize their choices according to specific probability distributions. At a mixed strategy Nash equilibrium, each player's probability distribution is a best response to the other players' distributions. 3. Proof Intuition: Fixed Point Theorem Nash's proof relies on Brouwer's Fixed Point Theorem, which states that any continuous function mapping a compact, convex set to itself has at least one fixed point. The proof constructs a continuous mapping from the space of all mixed strategy profiles to itself, where each player's strategy is mapped to their best response. The fixed points of this mapping correspond exactly to Nash equilibria.

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