Nash Equilibrium

Core Idea

A Nash equilibrium is a strategy profile where no player can improve their payoff by unilaterally changing their strategy, given what everyone else is doing. It generalizes dominant strategy equilibrium and applies to a much broader class of games. Nash equilibria can be in pure strategies (deterministic choices) or mixed strategies (probability distributions over strategies). Nash's theorem guarantees that every finite game has at least one Nash equilibrium in mixed strategies. Nash equilibrium is the central solution concept in non-cooperative game theory.

How It's Best Learned

Find Nash equilibria by underlining best responses in payoff matrices — cells where both players have underlined payoffs are Nash equilibria. Practice with coordination games, Battle of the Sexes, and Chicken to see that games can have zero, one, or multiple Nash equilibria.

Common Misconceptions

• A Nash equilibrium is not necessarily a dominant strategy equilibrium; dominant strategy equilibrium is a stronger condition.
• Multiple Nash equilibria create a coordination problem that Nash equilibrium analysis alone cannot resolve — it predicts *possible* outcomes, not which one will occur.

Explainer

You already know from game theory basics that strategic interaction means your best move depends on what others do. Nash equilibrium formalizes the natural stopping point of this reasoning: a strategy profile where every player is already doing the best they can, given what everyone else is doing. No one has an incentive to deviate unilaterally.

The cleanest way to find Nash equilibria in a payoff matrix is the best-response underline method. For each column (Player 2's strategy), find Player 1's best response and underline their payoff. For each row (Player 1's strategy), find Player 2's best response and underline their payoff. Any cell where both payoffs are underlined is a Nash equilibrium — both players are simultaneously best-responding.

It is important to distinguish Nash equilibrium from stronger solution concepts you may encounter. In a dominant strategy equilibrium, each player's strategy is best regardless of what others do — Nash equilibrium only requires best-response *to what others are actually doing*. This means every dominant strategy equilibrium is also a Nash equilibrium, but not vice versa. The Prisoner's Dilemma has a dominant strategy equilibrium; the Coordination Game has Nash equilibria but no dominant strategies.

Games can have zero, one, or many pure strategy Nash equilibria. When there are multiple equilibria, Nash equilibrium analysis alone cannot predict which one players will reach — this is the coordination problem. In Matching Pennies, there is no pure strategy Nash equilibrium at all: whatever one player does, the other wants to switch. Nash's theorem rescues predictive power by guaranteeing that every finite game has at least one Nash equilibrium in mixed strategies (probability distributions over pure strategies). A mixed equilibrium typically involves each player randomizing in a way that makes opponents indifferent between their pure strategy options.

Nash equilibrium captures a form of rational consistency, not optimality. Players at a Nash equilibrium may all be doing quite poorly — the Prisoner's Dilemma is a famous example where the Nash equilibrium is worse for both players than the cooperative outcome. Understanding Nash equilibrium as a stability concept (no one wants to deviate) rather than an optimality concept (everyone is doing well) is the key to applying it correctly.