Simple and complex adjustment dynamics in Cournot duopoly models. (English) Zbl 1080.91541

Summary: We are investigating microeconomic foundations of Cournot duopoly games such that the reaction functions are unimodal. We demonstrate that cost functions incorporating an interfirm externality lead to a system of coupled logistic equations. In the situation where agents take turns, we observe periodic and complex behavior. A closer analysis reveals some well-known local bifurcations. In a more general situation, where agents move simultaneously, we observe global bifurcations which typically occur in two-parameter families of two-dimensional endomorphisms.


91B62 Economic growth models
91A40 Other game-theoretic models
Full Text: DOI


[1] Fudenberg, D.; Tirole, J., Game theory, (1991) · Zbl 1339.91001
[2] Rand, D., Exotic phenomena in games and duopoly models, J. math. econ., 5, 173-184, (1978) · Zbl 0393.90014
[3] Poston, T.; Stewart, I., Catastrophe theory and its applications, (1978) · Zbl 0382.58006
[4] Schaffer, S., Chaos, naivete, and consistent conjectures, Econ. lett., 14, 155-162, (1984) · Zbl 1273.91306
[5] Halligan, W.; Joerding, W., Polymorphic equilibrium in advertising, Bell J. econ., 14, 191-201, (1983)
[6] Puu, T., Chaos in duopoly pricing, Chaos, solitons & fractals, 1, 573-581, (1991) · Zbl 0754.90015
[7] Puu, T., The chaotic duopolists revisited. department of economics, (1995), Umea University, and Center for Economic Studies, University of Munich
[8] Dana, R.-A.; Montrucchio, L., Dynamic complexity in duopoly games, J. econ. theory, 40, 40-56, (1986) · Zbl 0617.90104
[9] Furth, D., Stability and instability in oligopoly, J. econ. theory, 40, 197-228, (1986) · Zbl 0627.90011
[10] Lopez-Ruiz, R.; Perez-Garcia, C., Dynamics of maps with a global multiplicative coupling, Chaos, solitons & fractals, 1, 511-528, (1991) · Zbl 0810.58022
[11] Lopez-Ruiz, R.; Perez-Garcia, C., Dynamics of two logistic maps with a multiplicative coupling, Int. J. bifur. chaos, 2, 421-425, (1992) · Zbl 0874.58015
[12] Yuan, J.-M.; Tung, M.; Feng, D.H.; Narducci, L.M., Instability and irregular behavior of coupled logistic equations, Phys. rev. A, 28, 1662-1666, (1983)
[13] Hogg, T.; Huberman, B.A., Generic behavior of coupled oscillators, Phys. rev. A, 29, 275-281, (1984)
[14] Schult, R.L.; Creamer, D.B.; Henyey, F.S.; Wright, J.A., Symmetric and nonsymmetric coupled logistic maps, Phys. rev. A, 35, 3115-3118, (1987)
[15] Frank, R.H., ()
[16] Cournot, A.; Cournot, A., Recherches sur LES principes mathematiques de la theorie des richesses, Researches into the mathematical principles of the theory of wealth, (1963), Translated as · Zbl 0174.51801
[17] Gibbons, R., ()
[18] Gardner, R., Games for business and economics, (1995)
[19] Theocharis, R.D., On the stability of the Cournot solution on the oligopoly problem, Rev. econ. stud., 27, 133-134, (1959)
[20] Okuguchi, K., Expectations and stability in oligopoly models, Lecture notes in economics and mathematical systems, Vol. 138, (1976) · Zbl 0339.90010
[21] Friedman, J.W., Oligopoly and the theory of games, (1977) · Zbl 0385.90001
[22] Milgrom, P.; Roberts, J., Adaptive and sophisticated learning in normal form games, Games econ. behav., 3, 82-100, (1991) · Zbl 0751.90093
[23] Dana, R.-A.; Montrucchio, L., On rational strategies in infinite horizon models where agents discount the future, J. econ. behav. organiz., 8, 497-511, (1987)
[24] Maskin, E.; Tirole, J., A theory of dynamic oligopoly III Cournot competition, Eur. econ. rev., 31, 947-968, (1987)
[25] Maskin, E.; Tirole, J., A theory of dynamic oligopoly I: overview and quantity competition with large fixed costs, Econometrica, 56, 549-569, (1988) · Zbl 0657.90029
[26] Maskin, E.; Tirole, J., A theory of dynamic oligopoly II: price competition, kinked demand curve, and Edgeworth cycles, Econometrica, 56, 571-599, (1988) · Zbl 0664.90023
[27] Kopel, M., Improving the performance of an economic system: controlling chaos, (1995), Department of Economics, Cornell University
[28] Loskutov, A.Y.; Shishmarev, A.I., Control of dynamical systems behavior by parametric perturbations, Chaos, 4, 391-395, (1992) · Zbl 1055.37543
[29] Kopel, M., Periodic and chaotic behavior of a simple R&D model, Ricerche economique, 50, 235-265, (1996) · Zbl 0870.90037
[30] Aronson, D.G.; Chory, M.A.; Hall, G.R.; McGehee, R.P., Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study, Commun. math. phys., 83, 303-354, (1982) · Zbl 0499.70034
[31] Gardini, L.; Abraham, R.; Record, R.J.; Fournier-Prunaret, D., A double logistic map, Int. J. bifur. chaos, 4, 145-176, (1994) · Zbl 0870.58020
[32] Cox, J.C.; Walker, M., Learning to play Cournot duopoly strategies, (1994), Department of Economics, University of Arizona
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.