The Nash Equilibrium

From gametheory.net the definition of a Nash Equilibrium, "A Nash equilibrium, named after John Nash, is a set of strategies, one for each player, such that no player has incentive to unilaterally change her action. Players are in equilibrium if a change in strategies by any one of them would lead that player to earn less than if she remained with her current strategy. For games in which players randomize (mixed strategies), the expected or average payoff must be at least as large as that obtainable by any other strategy.

Whereas John von Neumann studied cooperative games, John Nash studied noncooperative games. This definition is more easily understood by saying that Player 1 would be satisfied with his decision given that he knows what Player 2's decision is; neither player has any regrets. However, that does not necessarily mean that each player earned the maximum possible payoff. It just means that each player is willing to live with the outcome that was achieved. Actually, in most cases if one player realizes his maximum payoff, it probably is not the rational outcome (prisoner's dilemma is an excellent example of this). The reason for this, Nash argued, is that if either player has a reason to change strategy (and would if given the chance), then that outcome is unstable and irrational. And it makes sense, why would you let your opponent reach his maximum payoff while you don't? Obviously, you wouldn't, and Nash proved it. This built upon von Neumann's minimax principle that the solution to zero-sum games is the equilibrium point; Nash proved that non-zero-sum games have equilibrium points as well.

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