Nash equilibrium

The concept of the Nash equilibrium was originated by Nash in his dissertation, Non-cooperative games (1950). Nash showed that the various solutions for games that had been given earlier all yield Nash equilibria.

A game may have many Nash equilibria, or none. Nash was able to prove that, if we allow mixed strategies (players choose strategies randomly according to preassigned probabilities), then every n-player game in which every player can choose from finitely many strategies admits at least one Nash equilibrium of mixed strategies.

If a game has a unique Nash equilibrium and is played among completely rational players, then the players will choose the strategies that form the equilibrium.

Examples

Competition game

Coordination game

This game is a coordination game for driving. The choices are either to drive on the left or to drive on the right, with 100 meaning no crash and 0 meaning a crash.

In this case there are two pure-strategy Nash equilibria, when both choose to either drive on the left or on the right.

If we admit mixed-strategies (where a pure strategy is chosen at random, subject to some fixed probability), then there are three Nash Equillibria for the same case: two we have seen from the pure-strategy form, where the probabilities are (0%,100%) for player one, (100%, 0%) for player two; and (100%, 0%) for player one, (0%, 100%) for player two respectively. We add another where the probabilities for each player is (50%, 50%).

Prisoner's dilemma

The Prisoner's dilemma has one Nash equilibrium: when both players defect. However, "both defect" is inferior to "both cooperate", in the sense that the total jail time served by the two prisoners is greater if both defect. The strategy "both cooperate" is unstable, as a player could do better by defecting while their opponent still cooperates. Thus, "both cooperate" is not an equilibrium. As Ian Stewart put it, ‘sometimes rational decisions aren't sensible!’

Stability

The concept of stability, useful in the analysis of many kinds of equilibrium can also be applied to Nash equillibria.

A Nash equilibrium for a mixed strategy game is stable if a small change (specifically a infinitesimal change) in probabilities for one player leads to a situation where two conditions hold:

1. the player who did not change has no better strategy in the new circumstance
2. the player who did change is now playing with a strictly worse strategy

If these cases are both met, then a player with the small change in their mixed-strategy will return imediately to the Nash equilibrium. The equilibrium is said to be stable. If condition one does not hold then the equilibrium is unstable. If only condition one holds then there are likely to be an infinite number of optimal strategies for the player who changed. John Nash showed that the latter situation could not arise in a range of well-defined games.

We have both stable and unstable equilibria in the Coordination game example above.

The equilibria involving mixed-strategies with 100% probabilities are stable. If either player changes their probabilities slightly, they will be both at a disadvantage, and their opponent will have no reason to change their strategy in turn.

In the case of the (50%,50%) equilibrium, there is instability. If either player changes their probabilities, then the other player immediately has a better strategy at either (0%, 100%) or (100%, 0%).

Stability is crucial in practical applications of Nash equilibria, since the mixed-strategy of each player is not perfectly known, but has to be inferred from statistical distribution of their actions in the game. In this case unstable equilibria are very unlikely to arise in practice, since any minute change in the proportions of each strategy seen will lead to a change in strategy and the breakdown of the equilibrium.

Note that stability of the equilibrium is connected to, but not the same thing as the stability of a strategy.

See also

• Prisoner's dilemma
• Evolutionarily stable strategy
• Wardrop's Principle