The dynamics of a triopoly Cournot game Anna Agliari a,b , Laura Gardini b,c, * ,T onu Puu d a Catholic University in Milan, Italy b University of Parma, Parma, Italy c Istituto di Scienze Economiche, University of Urbino, 61029 Urbino, Italy d Department of Economics, Umea University, S-901 87 Umea, Sweden Accepted 13 August 1999 Abstract This paper reconsiders the Cournot oligopoly (noncooperative) game with iso-elastic demand and constant marginal costs, one of the rare cases where the reaction functions can be derived in closed form. It focuses the case of three competitors, and so also extends the critical line method for non-invertible maps to the study of critical surfaces in 3D. By this method the various bifurcations of the attractors and their basins are studied. As a special case the restriction of the map to an invariant plane when two of the three ®rms are identical is focused. Ó 2000 Elsevier Science Ltd. All rights reserved. 1. Introduction Oligopoly, though contextually an intermediate situation between monopoly and perfect competition, is analytically a more complex case. The reason for this is that the oligopolist must consider not only the behaviors of the consumers, but also those of the competitors and their reactions. It is well known that the ®rst formal theory of oligopoly goes back to A. Cournot, in 1838 [6], who treated the case with no re- taliation at all, so that in every step each oligopolist assumes the last values taken by the competitors without any estimation of their future reactions. The adjustment process was assumed to lead to a ®xed point, called Cournot equilibrium, independently of its stability character. More recent works have shown that the Cournot model may lead to cyclic behavior, and Rand proved in [28] that under suitable conditions the outcome may be chaotic. However his work does not include any economic assumption leading to this behavior. This was ®rst done in [24,26], where such substantial assumptions were supplied in terms of an `iso-elastic' demand function (i.e. re¯ecting a situation where the consumers always spend a constant sum on the commodity, regardless of price) and constant marginal costs. In the papers cited above only the duopoly case is considered, while in [25] a third producer is introduced (see also [27]), starting the study of a more complex situation, but also more interesting: a market with three oligopolists. The recent interests among researches for the dynamics associated with repeated games is documented by the wide production on this subject (see, among others, [2±4,12,19]). In particular, in [10,11] it is shown how the analysis of the only attractors of a Cournot game may not be enough in order to understand the dynamical behaviors, and the role played by the global basins of attraction (which may have complex structure) is evidenced. We shall follow the same local±global approach. The aim of our work is to study carefully the situation of a market with three oligopolists by an analysis of the local and global properties of the map describing the adjustment process, assuming the oligopolists www.elsevier.nl/locate/chaos Chaos, Solitons and Fractals 11 (2000) 2531±2560 * Corresponding author. Fax: +39-722-327-655. E-mail addresses: agliari@unipr.it (A. Agliari), gardini@econ.uniurb.it (L. Gardini), tonu.puu@econ.umu.se (T. Puu). 0960-0779/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 9 9 ) 0 0 1 6 0 - 5