Nash Equilibrium
Stable state in games
Nash Equilibrium is a concept in game theory that describes a stable state in which no player can improve their payoff by unilaterally changing their strategy, assuming all other players keep their strategies unchanged. It is a fundamental concept in game theory and is used to analyze the behavior of players in simultaneous-move games.
- No player can improve their payoff by unilaterally changing their strategy.
- All players must be rational and have complete knowledge of the game.
- The equilibrium is stable, meaning that no player has an incentive to deviate from it.
- Define the gameIdentify the players, their strategies, and the payoffs associated with each strategy.Pro tipUse a game tree or payoff matrix to visualize the game.WarningMake sure to include all possible strategies and payoffs.
- Find the best responsesFor each player, find the best response to each possible strategy of the other players.Pro tipUse the concept of best responses to narrow down the possible equilibria.WarningMake sure to consider all possible strategies and their associated payoffs.
- Identify the Nash EquilibriumFind the strategy profile in which no player can improve their payoff by unilaterally changing their strategy.Pro tipUse the concept of best responses to identify the equilibrium.WarningMake sure to check that the equilibrium is stable and that no player has an incentive to deviate from it.
Two firms, Rainbow's End and B. B. Lean, are competing in a market and must set their prices simultaneously. The firms have different costs and face different demand curves.
The concept of Nash Equilibrium was first introduced by John Nash in the 1950s. Nash showed that every finite game has at least one equilibrium, and this result has had a profound impact on the development of game theory.