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In game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it) is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his or her own strategy (i.e., by changing unilaterally). If each player has chosen a strategy and no player can benefit by changing his or her strategy while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium. In other words, to be a Nash equilibrium, each player must answer negatively to the question "Knowing the strategies of the other players, and treating the strategies of the other players as set in stone, can I benefit by changing my strategy?" Stated simply, Amy and Bill are in Nash equilibrium if Amy is making the best decision she can, taking into account Bill's decision, and Bill is making the best decision he can, taking into account Amy's decision. Likewise, many players are in Nash equilibrium if each one is making the best decision that they can, taking into account the decisions of the others. However, Nash equilibrium does not necessarily mean the best cumulative payoff for all the players involved; in many cases all the players might improve their payoffs if they could somehow agree on strategies different from the Nash equilibrium (eg. competing businessmen forming a cartel in order to increase their profits). The concept of the Nash equilibrium (NE) in pure strategies was first developed by Antoine Augustin Cournot in his theory of oligopoly (1838). Firms choose a quantity of output to maximize their own profit. However, the best output for one firm depends on the outputs of others. A Cournot equilibrium occurs when each firm's output maximizes its profits given the output of the other firms, which is a pure-strategy NE. However, the modern game-theoretic concept of NE is defined in terms of mixed-strategies, where players choose a probability distribution over possible actions. The concept of the mixed strategy NE was introduced by John von Neumann and Oskar Morgenstern in their 1944 book The Theory of Games and Economic Behavior. However, their analysis was restricted to the very special case of zero-sum games. They showed that a mixed-strategy NE will exist for any zero-sum game with a finite set of actions. The contribution of John Forbes Nash in his 1951 article Non-Cooperative Games was to define a mixed strategy NE for any game with a finite set of actions and prove that at least one (mixed strategy) NE must exist. Informally, a set of strategies is a Nash equilibrium if no player can do better by unilaterally changing his or her strategy. As a heuristic, one can imagine that each player is told the strategies of the other players. If any player would want to do something different after being informed about the others' strategies, then that set of strategies is not a Nash equilibrium. If, however, the player does not want to switch (or is indifferent between switching and not) then the set of strategies is a Nash equilibrium.
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