There are two firms without capacity constraints in the market for diving eyeglasses in Atlantis. Customers in this market consider the products of these firms to be homogeneous. As it happens, the production cost for each eyeglass is independent of the number of produced eyeglasses, the same for both firms, and given by $15. Buyers prefer to buy from the firm that charges the lowest price. In case of equal prices, buyers flip a fair coin to decide where to buy. Does this scenario constitute a game? What is the market price in Atlantis? How many Nash equilibria are there? Does the outcome differ from that in a Cournot Duopoly?

For experts only: If you are up to a challenge, consider a market analogously to the one described but with three identical firms. How many Nash equilibria exist? Characterize all of them, including the prevailing market price!

Learning Goals:

-Is it a game? Why?

-What are the Nash equilibria?

-How many are there?

-What is the difference between a Cournot vs Bertrand oligopoly

First an explanation about the difference between the Cournot and Bertrand oligopoly

Cournot oligopoly:

-Cournot competition is an economic model used to describe an industry structure in which companies compete on the amount of output they will produce, which they decide on independently of each other and at the same time. It is named after Antoine Augustin Cournot (1801–1877) who was inspired by observing competition in a spring water duopoly.

It has the following features:

-There is more than one firm and all firms produce a homogeneous product, i.e. there is no product differentiation;

-Firms do not cooperate, i.e. there is no collusion;

-Firms have market power, i.e. each firm’s output decision affects the good’s price;

-The number of firms is fixed;

-Firms compete in quantities, and choose quantities simultaneously;

-The firms are economically rational and act strategically, usually seeking to maximize profit given their competitors’ decisions.

Bertrand oligopoly:

Bertrand competition is a model of competition used in economics, named after Joseph Louis François Bertrand (1822–1900). It describes interactions among firms (sellers) that set prices and their customers (buyers) that choose quantities at the prices set. The model was formulated in 1883 by Bertrand in a review of Antoine Augustin Cournot’s book in which Cournot had put forward the Cournot model.Cournot argued that when firms choose quantities, the equilibrium outcome involves firms pricing above marginal cost and hence the competitive price. In his review, Bertrand argued that if firms chose prices rather than quantities, then the competitive outcome would occur with price equal to marginal cost.

The Difference between Bertrand and Cournot:

Bertrand is a model that competes on price while Cournot is model that competes on quantities (sales volume).


Bertrand competitions a Model were firms compete on price, which naturally triggers the incentive to undercut competition by lowering price, thereby depleting profit until the product is selling at zero economic profit. This effectively is the pure-strategy Nash equilibrium.

Nash learning goals:

Why is the competitive price a Nash equilibrium in the Bertrand model? First, if both firms set the competitive price with price equal to marginal cost (unit cost), neither firm will earn any profits. However, if one firm sets price equal to marginal cost, then if the other firm raises its price above unit cost, then it will earn nothing, since all consumers will buy from the firm still setting the competitive price (recall that it is willing to meet unlimited demand at price equals unit cost even though it earns no profit). No other price is an equilibrium. If both firms set the same price above unit cost and share the market, then each firm has an incentive to undercut the other by an arbitrarily small amount and capture the whole market and almost double its profits. So there can be no equilibrium with both firms setting the same price above marginal cost. Also, there can be no equilibrium with firms setting different prices. The firms setting the higher price will earn nothing (the lower priced firm serves all of the customers). Hence the higher priced firm will want to lower its price to undercut the lower-priced firm. Hence the only equilibrium in the Bertrand model occurs when both firms set price equal to unit cost (the competitive price).