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In Cournot competition, each firm decides its production quantity simultaneously. The price is determined by the inverse demand function , where . The profit of firm is , where is the (marginal) cost of the production of firm . Thus, the profit of each firm depends on the production of the other firms. Cournot equilibrium is a vector that satisfies , for all and for all . In words, Cournot equilibrium is the strategy vector such that each firm chooses the quantity that maximizes its profit for given quantities of other firms.

Check that if , then in the equilibrium. Intuitively, since a firm with low marginal cost can produce efficiently, its production is larger than another firm with high marginal cost. The production of one of the firms can be zero if the difference of costs is sufficiently large.

To study how to calculate a Cournot equilibrium, see [1].

Incidentally, in many textbooks, Cournot competition is cited as the application of Nash equilibrium. In fact, however, Antoine Augustin Cournot (1801–1877) published his masterpiece, ''Recherches sur les principes mathématiques de la théorie des richesses," in 1838, over 100 years before the concept of Nash equilibrium was published.

Reference

[1] R. Gibbons, Game Theory for Applied Economists, Princeton: Princeton University Press, 1992.