# Download e-book for kindle: Introducing game theory and its applications by Lucrecia Crider

By Lucrecia Crider

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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 Nash Equilibrium. The modern game-theoretic concept of Nash Equilibrium is instead defined in terms of mixed strategies, where players choose a probability distribution over possible actions. The concept of the mixed strategy Nash Equilibrium was introduced by John von Neumann and Oskar Morgenstern in their 1944 book The Theory of Games and Economic Behavior.

Applications The Nash equilibrium concept is used to analyze the outcome of the strategic interaction of several decision makers. In other words, it is a way of predicting what will happen if several people or several institutions are making decisions at the same time, and if the outcome depends on the decisions of the others. The simple insight underlying John Nash's idea is that we cannot predict the result of the choices of multiple decision makers if we analyze those decisions in isolation.

Later, (Aumann and Brandenburger 1995) re-interpreted Nash equilibrium as an equilibrium in beliefs, rather than actions. For instance, in Rock-paper-scissors an equilibrium in beliefs would have each player believing the other was equally likely to play each strategy. This interpretation weakens the predictive power of Nash equilibrium, however, since it is possible in such an equilibrium for each player to actually play a pure strategy of Rock. Ever since, game theorists' attitude towards mixed strategies-based results have been ambivalent.