Nash Equilibrium

A Nash equilibrium is a profile of strategies, one for each player, such that no player can gain by changing their own strategy while the others keep theirs; equivalently, every player’s strategy is a best response to the strategies of the others. In brief: in the prisoner’s dilemma (Defect, Defect) is the only Nash equilibrium: against Defect, switching to Cooperate lowers a player’s payoff from 1 to 0. In a coordination game such as the stag hunt both (Stag, Stag) and (Hare, Hare) are equilibria.

Nash equilibrium is the central solution concept of non-cooperative game theory. It generalizes dominance: a dominance-solvable outcome is always a Nash equilibrium, but many games have equilibria without dominant strategies. This guide defines best responses, shows how to find pure-strategy equilibria in a payoff matrix by marking best replies, discusses games with several or no pure equilibria, efficiency and the existence theorem, with worked examples and a table.

What is a Nash equilibrium?

A best response of a player to the others’ strategies is a strategy that gives the highest payoff against them. A Nash equilibrium is a strategy profile in which every strategy is a best response to the others, so that no player has a profitable unilateral deviation. It can be read as a self-enforcing agreement or a stable convention: if everybody expects the profile, everybody is willing to play it. It does not say that the outcome is good for the players, nor that it will be reached from any starting point.

How do you find pure equilibria?

In a payoff matrix, mark the best response of the row player in each column (the largest first number in that column) and the best response of the column player in each row (the largest second number). A cell with both marks is a pure-strategy Nash equilibrium.

Example 1: the prisoner’s dilemma

With the payoffs (C, C) = (3, 3), (C, D) = (0, 5), (D, C) = (5, 0), (D, D) = (1, 1): the row player’s best reply to C is D (5 > 3), and to D it is D (1 > 0); the same for the column player. The only cell marked by both is (D, D), the unique equilibrium.

Example 2: the 3 × 3 game of the previous article

With the matrix of the preceding article on dominance: the row player’s best replies are B against L (5), T against C (4) and M against R (5). The column player’s best replies are C against T (3), L against M (3) and C against B (4). Only the cell (T, C) has both marks: the unique equilibrium, in agreement with the iterated elimination.

Example 3: a game without dominance

Payoffs (row, column): (A, X) = (2, 2), (A, Y) = (0, 0), (B, X) = (0, 0), (B, Y) = (1, 1). Row’s best replies: A against X, B against Y. Column’s: X against A, Y against B. Both (A, X) and (B, Y) are equilibria; no strategy is dominant. The payoff (2, 2) is better for both than (1, 1), so the players would like to coordinate on (A, X).

What if there are several equilibria or none?

Coordination games have several equilibria, and theory alone does not say which one will be played. Conventions, communication and history (focal points) decide: driving on the right or on the left are both equilibria.

Example 4: the stag hunt

Two hunters choose Stag or Hare. Stag succeeds only if both go: (Stag, Stag) = (4, 4); a hunter who goes alone for stag gets 0 while the other, taking hare, gets 3: (Stag, Hare) = (0, 3), (Hare, Stag) = (3, 0); both hare: (3, 3). The equilibria are (Stag, Stag) and (Hare, Hare). The first is better for both, but the second is safer, since Hare guarantees 3, which illustrates the tension between payoff dominance and risk dominance.

Example 5: no pure equilibrium

In matching pennies, for each cell one of the players wishes to deviate: if the coins match, the second player wants to change; if they differ, the first does. So there is no equilibrium in pure strategies. The equilibrium exists in mixed strategies, with each player showing heads with the probability 1/2.

Is a Nash equilibrium efficient?

Not necessarily. An outcome is Pareto efficient if no other outcome makes someone better off without making someone worse off. In the prisoner’s dilemma, the equilibrium (1, 1) is Pareto dominated by (3, 3). The reason is that each player ignores the effect of their choice on the other (an externality). In market models, the same effect explains why competing firms produce more and earn less than a cartel.

Does an equilibrium always exist?

Nash proved in 1950 that every finite game (finitely many players and strategies) has at least one equilibrium if mixed strategies are allowed, that is, if players can randomize. The proof uses a fixed-point theorem (Kakutani’s theorem, and in Nash’s later version Brouwer’s). The theorem guarantees existence but not uniqueness or an efficient way to find equilibria; computing a Nash equilibrium of a general game is a computationally hard problem. Many refinements (subgame perfection, trembling-hand perfection) select among equilibria, as in the market-entry game of the introduction, where the threat to fight was not credible.

Example 6: a check by deviation

To verify that a profile is an equilibrium, check each player’s deviations. In Example 3, at (B, Y) with the payoffs (1, 1): if the row player switches to A, the profile (A, Y) gives the row player 0 < 1; if the column player switches to X, the profile (B, X) gives 0 < 1. No profitable deviation exists, so (B, Y) is an equilibrium even though (A, X) is better for both.

Table of games and equilibria

Where does the concept come from?

Antoine Augustin Cournot described an equilibrium of this type for the competition of two firms in 1838, long before the general definition. John Forbes Nash Jr. introduced the general concept and proved the existence theorem in his 1950 doctoral thesis and a short paper in the same year, and shared the 1994 Nobel Prize in economics with John Harsanyi and Reinhard Selten.

Which mistakes are common?

Treating the equilibrium as the best joint outcome

Nash equilibrium only excludes unilateral gains. Coordinated changes by several players can make everyone better off.

Checking only one player’s deviations

All players must have no profitable deviation. A cell where only the row player is best-responding is not an equilibrium.

Expecting a unique answer

Games can have none (in pure strategies), one, or several equilibria. When several exist, extra considerations are needed to predict play.

Confusing equilibrium with dominance

A dominant-strategy profile is an equilibrium, but most equilibria arise without any dominant strategy.