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# What is Nash Equilibrium?

Article Details
• Written By: M. K. McDonald
• Edited By: A. Joseph
2003-2018
Conjecture Corporation
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Named for economist and mathematician John Forbes Nash, Jr., a Nash equilibrium is a particular kind of solution in game theory. Game theory itself is a type of applied mathematics, common in economics and other fields, in which the strategic behavior of two or more individuals or entities is captured mathematically. A Nash equilibrium is a situation in which, given the actions taken by the other players involved in the competition, no player is better off by changing his or her own action. In economics, the applications of a Nash equilibrium include the setting of prices between competing companies.

A Nash equilibrium can be seen in the example of a simple market in which two companies sell the same product and have the same profit margin per unit sold. Essentially, in this example, their profits are determined by the number of products sold, which is determined by the price. If the companies can choose to set their price at \$1 US Dollar, \$2 USD or \$3 USD, the best price for each company will be affected by the price set by the other company. A Nash equilibrium will be reached when neither company will benefit from changing its price if the other company does not also change its price.

If, in this example, both companies set a price of \$3 USD, each company could have an incentive to decrease the price if the other company keeps its price at \$3 USD, as long as the lower revenue per unit is more than offset by an increase in sales. If so, the total profit for that company will increase, while the total profit for the company that kept the same price is likely to decrease because of a decrease in sales. If both companies set a price of \$2 USD, each one will have incentive to lower the price to \$1 USD for the same reason. If both companies set a price of \$1 USD in this example, a Nash equilibrium will be reached, because neither company will have an incentive to increase its price if the other company keeps its price at \$1 USD. In this example, if one company did raise its price to \$2 USD, its decrease in sales would more than offset the increased revenue per unit, and the company's total profit would decrease.

In the real world, the conditions are far more complex than in this simple example. In can be difficult to determine whether a change in price and the resulting change in the number of products sold will result in an increase or decrease in total profits. Other conditions that might come into play are things such as markets with more than two competing companies, overlapping markets and the effects of products that are similar but not identical.

When a Nash equilibrium is reached and neither competing company has an incentive to change its price, it often compels the companies to compete in other ways. For example, a company might be able to increase its profits by lowering its costs of operation and production. Companies also might be compelled to produce a better product or to come up with other innovations.

There are a few things to note about Nash equilibria. Any competitive market might have no Nash equilibria, one Nash equilibrium or multiple Nash equilibria. It is also important to note that while each company is making the best choice possible given its competitor's choices, not all outcomes in Nash equilibrium deliver the highest combined profits to the companies involved. There often are cases where the combined profits could be higher if the companies could agree to change their actions and cooperate, but this behavior often is prohibited by antitrust legislation designed to promote competition between companies.