Nash equilibrium

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In game theory a Nash equilibrium is a set of decisions by the players such that they maximize there own payoff, taking the other player's strategies as given. The "equilibrium" is the set of decisions such that no individual participant, acting alone, can gain anything by altering his decision. The equilibrium can be determined by examining each players "best responses". Each player will have a set of actions that yield him the best possible payoff for each of the potential actions of other players. If each player is playing their best response to the other players' actions, (all the actions are best responses to each other) it is an equlibrium. This is a similar concept to a chemical equilibrium or a steady-state solution in physics, whereby each individual maximizes his own gain based on the actions of other participants.

The Nash equilibrium is named after John Nash, a mathematician. It is a topic in game theory, and its significance was recognized by award of the Nobel Prize in Economics.

The Nash equilibrium is a modification to the strict self-interest predicted by Adam Smith, because the Nash equilibrium takes into account the influence of competitors' decisions on the primary decision-maker. It also provides counter-examples to the idea that individuals working in their own self-interest work in the best interest of society as a whole.

Application

The Nash equilibrium is used to describe situations when several people or companies (usually known as players) have benefits (usually known as payoffs) that depend on the decisions of other players. The Nash equilibrium predicts the choices players will make to maximize their payoff.

In economics, the Nash equilibrium describes pricing decisions by an oligopoly. The set of selling prices will be such that no seller can benefit by changing his price if the other sellers keep their prices unchanged. If the cost structures are the same for each seller in an oligopoly, then the Nash equilibrium is where the price may equal the marginal cost, or P=MC. Whether or not this is true depends on whether it is assumed that the sellers choose the price they want to sell at, market forces determine the quantity sold [Bertrand oligopoly) or if sellers choose the quantity to sell and find the price on the market (Cournot oligopoly).

Nash equilibrium and intuition

There are cases in which the Nash equilibrium is a counter-intuitive outcome (and where the intuitive outcome is not a Nash equilibrium). The reason for this is the assumption that a participant assumes that nobody except for him will change strategies. If two very unlikely or disadvantageous strategies are best responses to each other, it is a Nash Equilibrium, but it may not be the most desirable outcome. For example, if you prefer a Republican government, but more people are voting Democratic, voting Democratic or abstaining are among your best responses, since your vote will not be able to affect the outcome. This may be part of the reason why countries sometimes elect liberal governments, even though most rational and logical people are conservative. If the media is able to convince people that most others are voting for a left-wing party, some would-be conservatives might get discouraged and not vote, or even vote for the left-wing party because they think it is what everyone else is doing, with disastrous implications.

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