Economics
Nash Equilibrium
No one wants to change strategy alone — fundamental solution concept in game theory
A Nash equilibrium is a solution concept in game theory: a strategy combination where no player wants to unilaterally change their strategy given the others' strategies. Named for John Nash (1950, Nobel 1994). Stable: no incentive to deviate. Multiple equilibria possible. Mixed strategies: assign probabilities to actions. Existence theorem: every finite game has at least one Nash equilibrium (possibly mixed). Important: doesn't guarantee best outcome — prisoner's dilemma equilibrium is bad for all. Used in: economics, biology, political science, computer science. Foundation of strategic analysis.
- AuthorJohn Nash (1950, Nobel Prize 1994)
- DefinitionNo player wants to unilaterally change strategy
- StabilityStable; no incentive to deviate
- Multiple equilibriaPossible; one game can have many
- Mixed strategiesProbability distributions over pure strategies
- Existence theoremAll finite games have NE (possibly mixed)
Interactive visualization
Press play, or step through manually. The visualization is yours to drive — try it before reading on.
Watch the 60-second explainer
A condensed visual walkthrough — narrated, captioned, under a minute.
Why Nash equilibrium matters
- Economics. Predicting market outcomes.
- Strategic analysis. Business strategy.
- Auctions. Bidding equilibria.
- Political science. Voting, conflict.
- Biology. Evolutionary stability.
- Computer science. Algorithm design, AI.
- Public policy. Mechanism design.
Common misconceptions
- Best outcome. Stable; not necessarily best.
- Always rational. Behavioral deviations.
- Single solution. Multiple often exist.
- Easy to compute. Some games complex.
- Pure strategies only. Mixed often necessary.
- Game-specific. Generalizes to many situations.
Frequently asked questions
What's Nash equilibrium?
In game with multiple players: strategy combination where each player playing best response given others' strategies. No incentive to change unilaterally. Stable: no one wants to deviate alone. Multiple equilibria possible. Provides: prediction of how rational players will play. Doesn't necessarily mean best outcome — just stable.
How is it found?
Several methods. (1) Best response analysis: for each player, identify best response to others' strategies. NE: where everyone playing best response. (2) Iterated elimination of dominated strategies: remove strategies dominated by others; repeat. Surviving strategies in NE. (3) Mathematical: solve fixed-point equations. (4) Software: numerical methods for large games.
What's a dominated strategy?
Strategy worse than another regardless of others. Always inferior. Example. Player has strategies A, B, C. If A always gives lower payoff than B regardless of others' choices: A is dominated by B. Rational player: never plays A. In NE: no dominated strategies played (player would deviate). Iterated elimination: removes layers of dominated strategies.
What's a mixed strategy?
Probability distribution over pure strategies. Example. Rock-paper-scissors: pure strategies are R, P, S. Mixed: play R with 1/3 prob, P with 1/3, S with 1/3. Mixed NE: each player's mixed strategy best response to others'. Some games: no pure-strategy NE exists; mixed NE always exists for finite games.
What's the prisoner's dilemma equilibrium?
Classic example. Two prisoners: cooperate (silent) or defect (talk). If both cooperate: both gain (light sentences). If both defect: both lose (medium sentences). If one cooperates, other defects: defector gains; cooperator loses heavily. Nash equilibrium: both defect (each is best response). But: both worse off than if both cooperated. Shows: NE not always best.
What about coordination games?
Multiple Nash equilibria exist. Example. Two people meeting; can choose location A or B. If choose same: high payoff. If choose different: zero. Two NE: (A, A) and (B, B). Both stable. Equilibrium selection: which one realized? Often: arbitrary, depends on focal points (Schelling), conventions, social norms. Common in real life.
How does it relate to dominant strategy?
Dominant strategy: best regardless of others. Stronger than NE. If player has dominant strategy, plays it in NE. But: NE doesn't require dominant strategies. Many NE exist without dominant strategies. Coordination games: no dominant strategies; multiple NE possible.