Cournot, Bertrand, and Modern Game Theory
Morrison, Clarence C., Atlantic Economic Journal
In 1838, Augustin Cournot published his now famous Recherches sur les Principes Mathematiques de la Theorie des Richesses. In this small volume and with mathematical precision, Cournot explicitly set forth much of the modern day theory of competition, monopoly, and oligopoly. In 1883, J. Bertrand undertook a joint review of Cournot's book and Leon Walrus' Theorie Mathematique de la Richesse Sociale  which had just appeared. In this review, Bertrand argued that Cournot's equilibrium for duopoly was not a true equilibrium because "whatever the common price adopted, if one of the owners, alone, reduces his price, he will, ignoring any minor exceptions, attract all of the buyers, and thus double his revenue if his rival lets him do so" [Bertrand, 1883].(1) It is now textbook-commonplace that, for homogeneous products, if each rival assumes that the other rival will let him do so, this type of rivalry would lead to the competitive result of price set equal to marginal cost.
As criticized by Bertrand, Cournot arrived at the equilibrium by assuming that each rival took the other rival's quantities as given and put his profit-maximizing quantity on the market. After stating each rival's profit function regarding the quantities that all rivals place on the market, Cournot partially differentiated each rival's profit function, with respect to that rival's own quantity and equated each of the resulting expressions to zero.(2)
For the duopoly case, Cournot plotted the resulting equations in rectangular coordinates and pointed out that it is evident that an equilibrium can only be established where the curves intersect [1838, p. 81]. Figure 2 gives the plotted curves and illustrates the sequential algorithm for finding this equilibrium.(3) In the more general case of n proprietors, equilibrium is given by the simultaneous solution of the equations [pp. 84-5].
In plotting the respective first-order conditions (for maximizing the profit of each rival given the other rival's quantities), Cournot implicitly solved for functions giving the reactions of each rival to the other rival's strategies. In modern game theory, these functions are called best-response functions. Where the curves intersect (for the two-dimensional case or for the simultaneous solution of Cournot's equations in general), it turns out that all of the rivals' conjectures about strategies are actually correct. No rival changes his strategy in reaction to the observed strategies of the other rivals. J. F. Nash [1950, 1951] extended this basic idea to noncooperative games in general and provided sufficient conditions for such equilibria to exist. In modern game theory, best-response solutions with mutually correct conjectures are referred to as Nash equilibria.
The above summary is provided as background for discussing the nomenclature that has evolved in the application of modern game theory to the analysis of market structures. Almost without exception in current industrial organization literature, market rivalry involving quantity strategies is referred to as Cournot competition and market rivalry involving price strategies is referred to as Bertrand competition. The corresponding equilibria are referred to as Cournot equilibria and Bertrand equilibria. Where the equilibria are best-response solutions with mutually correct conjectures, they are described as being Cournot-Nash and Bertrand-Nash, respectively. In light of the summary, this would seem to be convenient nomenclature that is firmly rooted in the historical evolution of economic ideas. In fact, this nomenclature actually does great violence to the history of economic thought.
What has been forgotten (or never learned) is that, in his 1838 classic, Cournot symmetrically treated both quantity rivalry and price rivalry (in the sense of analyzing both best-response functions with equilibrium given where conjectures are mutually correct). The most glaring example of the problem arises in the analysis of oligopoly with differentiated products. …