COURNOT, BERTRAND, AND MODERN GAME THEORY
ДАТА ПУБЛИКАЦИИ: 23 декабря 2004ОПУБЛИКОВАЛ:
Morrison, Clarence C. .
COURNOT, BERTRAND, AND MODERN GAME THEORY
Although it is not generally remembered or known, Cournot applied his equilibrium concept to both quantity rivalry and price rivalry. This makes some of the nomenclature in modern game theory seriously inappropriate. Several critics, past and present, have treated Cournot's quantity rivalry case as only conveniently veiled price rivalry. If Cournot's mathematics are pursued far enough, it is clear that he had a method that symmetrically applies to both quantity rivalry and price rivalry. (JEL B10, C72, L13)
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].(n1) 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.(n2)
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.(n3) 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. With differentiated products, it is no longer the case that if one rival were able to get his price lower than all other rivals, then that rival would take the entire market (at his price). In this case, there is a best-response solution concerning prices with mutually correct conjectures. In modern industrial organization literature, this solution is uniformly referred to as the Nash or the Bertrand-Nash equilibrium. However, where the duopoly best-response curves for the quantity rivalry case with homogeneous goods are negatively sloped, the duopoly best-response curves for price rivalry with heterogeneous goods are positively sloped. In effect, when one rival raises his price, the profit opportunities of the other rivals are enhanced.
The differing slopes of the best-response curves might be taken to justify not attaching Cournot's name to the modern solution for differentiated oligopoly, but this would be false justification. For a duopoly case, Cournot [pp. 100-1] states the proprietor profit functions concerning prices for both rivals and partially differentiates each of the profit functions with respect to their own prices. He then equates both of the resulting expressions to zero [p. 101] and indicates that the same method of reasoning applies as in the quantity rivalry case. Then, positively sloped and intersecting best-response curves are given in Figure 7 with prices measured on the coordinate axes. Thus, the derivation of Figure 7 is identical with the derivation of the modern theory of differentiated oligopoly. It then follows that if proper names are used to identify this case, then one of the names must be Cournot rather than Bertrand. There is nothing in Bertrand's review that would justify the use of his name in this context.(n4)
When proper names are used to identify concepts, the names used should be of those individuals who originated the concepts. Where another individual has significantly extended an idea or made especially significant applications of an idea, it is legitimate to add that name with a hyphen. For example, the sum of elasticities theorem is the theorem that goes by the name of Marshall-Lerner. Thus, the equilibrium of competitive pricing for homogeneous goods under oligopoly is properly identified as being Bertrand or Bertrand-Nash. However, these labels should never be applied to the case of oligopolistic price rivalry with heterogeneous goods. The equilibria in both this case and the usual quantity rivalry case should be identified as Cournot or Cournot-Nash. Justification of the present practice for convenience sake is spurious. Quantity competition and price competition trip off the tongue just as easily as Cournot competition and Bertrand competition. Further, just because a current usage is firmly entrenched, resisting the required change in nomenclature would be an unseemly surrender to the status quo.
(n1.) This quotation is from the translation by Margaret Chevaillier. This translation was published as the Appendix to Magna de Bornier .
(n2.) Cournot used the d notation for both partial and ordinary differentiation. See the Bacon [1897, pp. 79-86] translation of Cournot  as reprinted by Augustus M. Kelly . All page citations for Cournot will be for this volume.
(n3.) All of the figures are given on a foldout sheet at the back of Cournot's book.
(n4.) There is one small caveat that must be entered here. On pages 100-1, Cournot is not discussing differentiated oligopoly but rather the case of a composite commodity whose components are supplied by rival monopolists. The analytical structures are essentially similar, nonetheless.
Bertrand, J. "Theorie Mathematique de la Richesse Sociale," Journal des Savants, 67, 1883, pp. 499-508.
Cournot, Augustin. Recherches sur les Principes Mathematiques de la Theorie des Richesses, Paris, France: Hachette, 1838. (Italian translation in Biblioteca Dell'Econ., 1875. English translation by N. T. Bacon in Economic Classics, New York, NY: Macmillan, 1897; reprinted by Augustus M. Kelly, 1960.)
Magnan de Bornier, Jean. "The Cournot-Bertrand Debate: A Historical Perspective," History of Political Economy, 24, 3, 1992, pp. 623-54. Nash, Jr., J. F. "Non-Cooperative Games," Annals of Mathematics, 54, 1951, pp. 289-95.
------. "Equilibrium Points in n-Person Games," Proceeding of the National Academy of Science U.S.A., 36, 1950, pp. 48-9.
------. Walrus, Leon. Theorie Mathematiques de la Richesse Sociale, Lausanne, France: Corbaz,