Symmetry‐breaking bifurcations and representativefirm in dynamic duopoly games

Abstract

In this paper, we investigate the question of whether the assumption of the “representativeagent”, often made in economic modeling, is innocuous or whether it may be misleadingunder certain circumstances. In order to obtain some insight into this question, two dynamicCournot duopoly games are considered, whose dynamics are represented by discrete‐timedynamical systems. For each of these models, the dynamical behavior of the duopoly systemwith identical producers is compared to that with quasi‐identical ones, in order to study theeffects of small heterogeneities between the players. In the case of identical players, suchdynamical systems become symmetric, and this implies that synchronized dynamics can beobtained, governed by a simpler one‐dimensional model whose dynamics summarizes thecommon behavior of the two identical players. In both the examples, we show that a negligibledifference between the parameters that characterize the two producers can give dynamicevolutions that are qualitatively different from that of the symmetric game, i.e. a breakingof the symmetry can cause a noticeable effect. The presence of such bifurcations suggeststhat economic systems with quasi‐identical agents may evolve quite differently from systemswith truly identical agents. This contrasts with the assumption, very common in the economicliterature, that small heterogeneities of agents do not matter too much.