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:
- The action yields the highest payoff for that player given her predictions about the other players’ actions.
- 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
|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
|Number of dollars (x)||100 − x, x||0, 0|
Table 16.8 Coordination Game
|Up||5, 5||0, 1|
|Down||1, 0||4, 4|
- Prisoners’ dilemma. The row player chooses between the action labeled Up and the one labeled Down. The column player chooses between the action labeled Left and the one labeled Right. For example, if row chooses Up and column chooses Right, then the row player has a payoff of 0, and the column player has a payoff of 10. If the row player predicts that the column player will choose Left, then the row player should choose Down (that is, down for the row player is her best response to left by the column player). From the column player’s perspective, if he predicts that the row player will choose Up, then the column player should choose Right. The Nash equilibrium occurs when the row player chooses Down and the column player chooses Right. Our two conditions for a Nash equilibrium of making optimal choices and predictions being right both hold.
- Social dilemma. This is a version of the prisoners’ dilemma in which there are a large number of players, all of whom face the same payoffs.
- Dictator game. The row player is called the dictator. She is given $100 and is asked to choose how many dollars (x) to give to the column player. Then the game ends. Because the column player does not move in this game, the dictator game is simple to analyze: if the dictator is interested in maximizing her payoff, she should offer nothing (x = 0).
- Ultimatum game. This is like the dictator game except there is a second stage. In the first stage, the row player is given $100 and told to choose how much to give to the column player. In the second stage, the column player accepts or rejects the offer. If the column player rejects the offer, neither player receives any money. The best choice of the row player is then to offer a penny (the smallest amount of money there is). The best choice of the column player is to accept. This is the Nash equilibrium.
- Coordination game. The coordination game has two Nash equilibria. If the column player plays Left, then the row player plays Up; if the row player plays Up, then the column player plays Left. This is an equilibrium. But Down/Right is also a Nash equilibrium. Both players prefer Up/Left, but it is possible to get stuck in a bad equilibrium.
- 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.
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.