Nash equilibrium

The Nash equilibrium describes a particular state in game theory. This state can be achieved in a non-cooperative game in which each player tries to pick the best possible strategy to gain the best possible outcome for themselves, based on what they expect the other players to do. Eventually, all the players reach a point at which they have all picked the best possible strategy for themselves based on the decisions made by the other players. At this point in the game, they would gain no benefit from changing their strategy. This state is the Nash equilibrium.

A famous example of how the Nash equilibrium can be reached is with the Prisoner's Dilemma. In this example, two criminals (A and B) have been arrested for committing a crime. Both have been placed in separate cells with no way of communicating with each other. The prosecutor only has enough evidence to convict them for a smaller offense and not the principal crime, which would see them go to jail for a long time. To get a conviction, the prosecutor gives them an offer:

• If A and B both implicate each other in the principal crime, they both serve 2 years in jail.
• If A implicates B but B remains silent, A will be set free and B will serve 3 years in jail (and vice versa).
• If A and B both keep quiet, they both serve only 1 year in jail on the lesser charge.

From these three scenarios, it is obvious that the best possible outcome for A and B is to keep quiet and serve 1 year in jail. However, the risk of keeping quiet is 3 years as neither A nor B have any way of knowing that the other will also keep quiet. Thus, they would reach a state where their actual optimum strategy would be to confess as it is the choice that provides the highest reward and lowest penalty. When this state has been reached, neither criminal would gain any advantage by changing their strategy; thus, they would have reached a Nash equilibrium.