Nash Equilibrium

Nash equilibrium (in economics and game theory) is a stable state of a system involving the interaction of different participants, in which no participant can gain by a unilateral change of strategy if the strategies of the others remain unchanged.

Game theorists use the Nash equilibrium concept to analyze the outcome of the strategic interaction of several decision makers. In other words, it provides 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 one cannot predict the result of the choices of multiple decision makers if one analyzes those decisions in isolation. Instead, one must ask what each player would do, taking into account the decision-making of the others.

Stated simply, Amy and Will are in Nash equilibrium if Amy is making the best decision she can, taking into account Will’s decision while Will’s decision remains unchanged, and Will is making the best decision he can, taking into account Amy’s decision while Amy’s decision remains unchanged. Likewise, a group of players are in Nash equilibrium if each one is making the best decision possible, taking into account the decisions of the others in the game as long as the other party’s decision remains unchanged.

The reality of the Nash equilibrium of a game can be tested using experimental economics methods.

Nash equilibrium has been used to analyze hostile situations like war and arms races and also how conflict may be mitigated by repeated interaction (tit-for-tat). It has also been used to study to what extent people with different preferences can cooperate , and whether they will take risks to achieve a cooperative outcome. It has been used to study the adoption of technical standards, and also the occurrence of bank runs and currency crises. Other applications include traffic flow, how to organize auctions , the outcome of efforts exerted by multiple parties in the education process, regulatory legislation such as environmental regulations, analysing strategies in marketing and even penalty kicks in football.

Formal Definition

Formal Definition of Nash equilibrium

Informal Definition

Informally, a set of strategies is a Nash equilibrium if no player can do better by unilaterally changing their strategy. To see what this means, imagine that each player is told the strategies of the others. Suppose then that each player asks themselves: “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?”

If any player could answer “Yes”, then that set of strategies is not a Nash equilibrium. But if every player prefers not to switch (or is indifferent between switching and not) then the set of strategies is a Nash equilibrium. Thus, each strategy in a Nash equilibrium is a best response to all other strategies in that equilibrium.

The Nash equilibrium may sometimes appear non-rational in a third-person perspective. This is because it may happen that a Nash equilibrium is not Pareto optimal.

The Nash equilibrium may also have non-rational consequences in sequential games because players may “threaten” each other with non-rational moves. For such games the subgame perfect Nash equilibrium may be more meaningful as a tool of analysis.

 

The Nash equilibrium was named after John Forbes Nash, Jr. A version of the Nash equilibrium concept was first known to be used in 1838 by Antoine Augustin Cournot in his theory of oligopoly. In Cournot’s theory, firms choose how much output to produce 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 Nash equilibrium. Cournot also introduced the concept of best response dynamics in his analysis of the stability of equilibrium. However, Nash’s definition of equilibrium is broader than Cournot’s. It is also broader than the definition of a Pareto-efficient equilibrium, since the Nash definition makes no judgments about the optimality of the equilibrium being generated.

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