# Nash Equilibrium

Written By
Paul Tracy
Updated March 16, 2021

## What is a Nash Equilibrium?

In economics, a Nash equilibrium occurs when two companies in a duopoly react to each other's production changes until their prices reach an equilibrium.

The term is named after John Nash, who is an American mathematician who won the Nobel Prize in Economics in 1994. The 2001 film A Beautiful Mind chronicles his life and struggles.

## How a Nash Equilibrium Works (Example)

For a simple example, imagine a strategic game between two hungry cats. Both cats can choose option A, to receive a mouse, or option B, to receive nothing. Obviously, both cats choose option A. When their strategies are revealed to one another, they still don't deviate from their original choice. Knowing each other's decision does not change their behavior. Therefore, option A represents a Nash equilibrium.

Now let's look at a Nash equilibrium in economics. A duopoly is a form of oligopoly occurring when two companies (or countries) control all or most of the market for a product or service.

There are two kinds of duopoly. In the first, the Cournot duopoly, competition between the two companies is based on the quantity of products supplied. The duopoly members essentially agree to split the market. The price each company receives for the product is based on the quantity of items produced, and the two companies react to each other's production changes until an equilibrium (called the Nash equilibrium) is achieved.

Nash equilibria do not occur in a Bertrand duopoly, in which two companies compete on price. Because consumers will generally purchase the cheaper of two identical products, this eventually leads to a zero-profit price as the two competitors attempt to attract more customers (and thus more profit) through price cuts. The threat of price undercutting means that Bertrand equilibrium prices and profits are generally lower (and quantities higher) than the Nash equilibrium in Cournot duopolies.

## Why the Nash Equilibrium Matters

A Nash equilibrium is important because it represents a scenario's outcome in which every participant wins because each one gets the outcome they desire.

The Nash equilibrium is actually a game theory that states no player can increase his or her payoff by choosing a different action given the other player's actions. In economics, a duopoly's small production base means that each producer must carefully consider its rival's potential reactions to certain business decisions. As with most oligopolies, when members of a duopoly compete on price, for example, they tend to drive the product's price down to the cost of production, thereby lowering profits for both members of the duopoly.

These circumstances give duopolists strong incentive to agree to charge a monopoly price and share the resulting profits. However, federal antitrust laws, most notably the Sherman Act, make collusive activity illegal in the United States. Additionally, each member of a duopoly still has incentive to compete, even while colluding with the competition: An undetected price cut (or increase in production) will attract customers who are buying from the competition and customers who are not buying the product at all. Price adjustments may be subtle, including better credit terms, faster delivery or related free services.

Duopolies are most effective when the demand for the duopoly's product is not very affected by price. This is why duopolies are more effective in the short term; over the long term, prices often become elastic as consumers find cheaper substitutes for the product. Also, demand volatility may lead to disagreements within a collusive duopoly regarding outputs and prices.