The Nash Equilibrium (due to John Nash
) is one way (the most popular
) to predict
the outcome of a particular game. It's the central concept in noncooperative game theory
, and what most people mean when they refer to game theory
. Informally, the Nash equilibrium
is the situation in a game when your move is the best possible one given everyone else's moves, and everyone else's moves are the best possible
given your move, etc.
More formally, a Nash equilibrium is the ordered set of strategies—one for each player in the game—such that no single player can get a higher utility by choosing a different strategy.
There’s two kinds of Nash equilibria. A pure strategy Nash equilibrium is one in which everyone chooses a simple action as their strategy. A mixed strategy Nash equilibrium is one in which the players choose probabilities with which each of their strategies will be played and then back off and let chance make the actual move.
In the latter case, no player can choose a better probability function. The pure strategy case is technically a special case of the mixed strategy, with a probability of one on a single action.
The Nash equilibrium is applied to games in strategic form or normal form, which means to games in which everyone makes exactly one move, and does so simultaneously. For dynamic games (frequently meaning extensive form games), where players make moves sequentially, there are a large number of equilibrium refinements.
Nash proved that a mixed strategy equilibrium exists for any game which has a finite number of players, each with a finite set of possible strategies. There can often be multiple equilibria for a game. There isn’t always a pure strategy equilibrium. Reportedly, John von Neumann wasn’t very impressed with the proof, saying, "That's trivial, you know. That's just a fixed-point theorem."
A particularly strong kind of Nash equilibrium is an equilibrium in dominant strategies. That’s when every player has a single best move regardless of the other players’ moves. The Prisoner's Dilemma is an example of a Nash equilibrium in dominant strategies.