16.9 Nash Equilibrium

A Nash equilibrium is used to predict the outcome of a game. By a game, we mean the interaction of a few individuals, called players. Each player chooses an action and receives a payoff that depends on the actions chosen by everyone in the game.

A Nash equilibrium is an action for each player that satisfies two conditions:

  1. The action yields the highest payoff for that player given her predictions about the other players’ actions.
  2. The player’s predictions of others’ actions are correct.

Thus a Nash equilibrium has two dimensions. Players make decisions that are in their own self-interests, and players make accurate predictions about the actions of others.

Consider the games in Table 16.5 "Prisoners’ Dilemma", Table 16.6 "Dictator Game", Table 16.7 "Ultimatum Game", and Table 16.8 "Coordination Game". The numbers in the tables give the payoff to each player from the actions that can be taken, with the payoff of the row player listed first.

Table 16.5 Prisoners’ Dilemma

Left Right
Up 5, 5 0, 10
Down 10, 0 2, 2

Table 16.6 Dictator Game

Number of dollars (x) 100 − x, x

Table 16.7 Ultimatum Game

Accept Reject
Number of dollars (x) 100 − x, x 0, 0

Table 16.8 Coordination Game

Left Right
Up 5, 5 0, 1
Down 1, 0 4, 4

Key Insights

  • A Nash equilibrium is used to predict the outcome of games.
  • In real life, payoffs may be more complicated than these games suggest. Players may be motivated by fairness or spite.

More Formally

We describe a game with three players (1, 2, 3), but the idea generalizes straightforwardly to situations with any number of players. Each player chooses a strategy (s1, s2, s3). Suppose σ1(s1, s2, s3) is the payoff to player 1 if (s1, s2, s3) is the list of strategies chosen by the players (and similarly for players 2 and 3). We put an asterisk (*) to denote the best strategy chosen by a player. Then a list of strategies (s*1, s*2, s*3) is a Nash equilibrium if the following statements are true:

σ1(s*1, s*2, s*3) ≥ σ1(s1, s*2, s*3) σ2(s*1, s*2, s*3) ≥ σ2(s*1, s2, s*3) σ3(s*1, s*2, s*3) ≥ σ3(s*1, s*2, s3)

In words, the first condition says that, given that players 2 and 3 are choosing their best strategies (s*2, s*3), then player 1 can do no better than to choose strategy s*1. If a similar condition holds for every player, then we have a Nash equilibrium.