Decisions in Economics and Finance

, Volume 41, Issue 2, pp 313–333 | Cite as

An evolutionary model with best response and imitative rules

  • Lorenzo Cerboni BaiardiEmail author
  • Ahmad K. Naimzada


We formulate an evolutionary oligopoly model where quantity setting players produce following either the static expectation best response or a performance-proportional imitation rule. The choice on how to behave is driven by an evolutionary selection mechanism according to which the rule that brought the highest performance attracts more followers. The model has a stationary state that represents a heterogeneous population where rational and imitative rules coexist and where players produce at the Cournot–Nash level. We find that the intensity of choice, a parameter representing the evolutionary propensity to switch to the most profitable rule, the cost of the best response implementation as well as the number of players have ambiguous roles in determining the stability property of the Cournot–Nash equilibrium. This marks important differences with most of the results from evolutionary models and oligopoly competitions. Such differences should be referred to the particular imitative behavior we consider in the present modeling setup. Moreover, the global analysis of the model reveals that the above-mentioned parameters introduce further elements of complexity, conditioning the convergence toward an inner attractor. In particular, even when the Cournot–Nash equilibrium loses its stability, outputs of players little differ from the Cournot–Nash level and most of the dynamics is due to wide variations of imitators’ relative fraction. This describes dynamic scenarios where shares of players produce more or less at the same level alternating their decision mechanisms.


Imitation Heterogeneity Evolutionary game Logit dynamics Dynamic instability Dynamic systems 

JEL Classification

C62 C63 C72 C73 



This work has been developed in the framework of the research project on “Dynamic Models for behavioural economics” financed by DESP, University of Urbino. The authors thank two anonymous referees for their useful comments.


  1. Agiza, H.N., Elsadany, A.A.: Nonlinear dynamics in the cournot duopoly game with heterogeneous players. Physica A Stat. Mech. Appl. 320, 512–524 (2003)CrossRefGoogle Scholar
  2. Agiza, H.N., Elsadany, A.A.: Chaotic dynamics in nonlinear duopoly game with heterogeneous players. Appl. Math. Comput. 149(3), 843–860 (2004)Google Scholar
  3. Agiza, H.N., Hegazi, A.S., Elsadany, A.A.: Complex dynamics and synchronization of a duopoly game with bounded rationality. Math. Comput. Simul. 58(2), 133–146 (2002)CrossRefGoogle Scholar
  4. Andaluz, J., Jarne, G.: On the dynamics of economic games based on product differentiation. Math. Comput. Simul. 113, 16–27 (2015)CrossRefGoogle Scholar
  5. Andaluz, J., Elsadany, A.A., Jarne, G.: Nonlinear cournot and bertrand-type dynamic triopoly with differentiated products and heterogeneous expectations. Math. Comput. Simul. 132, 86–99 (2017)CrossRefGoogle Scholar
  6. Angelini, N., Dieci, R., Nardini, F.: Bifurcation analysis of a dynamic duopoly model with heterogeneous costs and behavioural rules. Math. Comput. Simul. 79(10), 3179–3196 (2009)CrossRefGoogle Scholar
  7. Apesteguia, J., Huck, S., Oechssler, J.: Imitation—theory and experimental evidence. J. Econ. Theory 136(1), 217–235 (2007)CrossRefGoogle Scholar
  8. Askar, S.: The rise of complex phenomena in cournot duopoly games due to demand functions without inflection points. Commun. Nonlinear Sci. Numer. Simul. 19(6), 1918–1925 (2014a)CrossRefGoogle Scholar
  9. Askar, S.S.: Complex dynamic properties of cournot duopoly games with convex and log-concave demand function. Oper. Res. Lett. 42(1), 85–90 (2014b)CrossRefGoogle Scholar
  10. Bigoni, M., Fort, M.: Information and learning in oligopoly: an experiment. Games Econ. Behav. 81, 192–214 (2013)CrossRefGoogle Scholar
  11. Bischi, G.I., Naimzada, A.: Global analysis of a dynamic duopoly game with bounded rationality. In: Filar, J.A., Gaitsgory, V., Mizukami, K. (eds.) Advances in Dynamic Games and Applications, pp. 361–385. Birkhäuser, Boston, MA (2000)CrossRefGoogle Scholar
  12. Bischi, G.I., Gallegati, M., Naimzada, A.: Symmetry-breaking bifurcations and representative firm in dynamic duopoly games. Ann. Oper. Res. 89, 252–271 (1999)CrossRefGoogle Scholar
  13. Bischi, G.I., Kopel, M., Naimzada, A.: On a rent-seeking game described by a non-invertible iterated map with denominator. Nonlinear Anal. Theory Methods Appl. 47(8), 5309–5324 (2001)CrossRefGoogle Scholar
  14. Bischi, G.I., Naimzada, A.K., Sbragia, L.: Oligopoly games with local monopolistic approximation. J. Econ. Behav. Org. 62(3), 371–388 (2007)CrossRefGoogle Scholar
  15. Bischi, G.I., Chiarella, C., Kopel, M., Szidarovszky, F.: Nonlinear Oligopolies: Stability and Bifurcations. Springer, Berlin (2009)Google Scholar
  16. Bischi, G.I., Lamantia, F., Radi, D.: An evolutionary cournot model with limited market knowledge. J. Econ. Behavior Org. 116, 219–238 (2015)CrossRefGoogle Scholar
  17. Bischi, G.I., Lamantia, F., Radi, D.: Evolutionary oligopoly games with heterogeneous adaptive players. In: Corchón, L., Marini, M. (eds.) Handbook of Game Theory and Industrial Organization, Volume I. Edward Elgar publishing (2018)Google Scholar
  18. Brock, W.A., Hommes, C.H.: A rational route to randomness. Econom. J. Econ. Soc. 65, 1059–1095 (1997)Google Scholar
  19. Cavalli, F., Naimzada, A.: A cournot duopoly game with heterogeneous players: nonlinear dynamics of the gradient rule versus local monopolistic approach. Appl. Math. Comput. 249, 382–388 (2014)Google Scholar
  20. Cavalli, F., Naimzada, A.: Nonlinear dynamics and convergence speed of heterogeneous cournot duopolies involving best response mechanisms with different degrees of rationality. Nonlinear Dyn. 81(1–2), 967–979 (2015)CrossRefGoogle Scholar
  21. Cavalli, F., Naimzada, A., Pireddu, M.: Effects of size, composition, and evolutionary pressure in heterogeneous cournot oligopolies with best response decisional mechanisms. Discrete Dyn. Nat. Soc. (2015a).
  22. Cavalli, F., Naimzada, A., Tramontana, F.: Nonlinear dynamics and global analysis of a heterogeneous cournot duopoly with a local monopolistic approach versus a gradient rule with endogenous reactivity. Commun. Nonlinear Sci. Numer. Simul. 23(1), 245–262 (2015b)CrossRefGoogle Scholar
  23. Cerboni Baiardi, L., Lamantia, F., Radi, D.: Evolutionary competition between boundedly rational behavioral rules in oligopoly games. Chaos Solitons Fractals 79, 204–225 (2015)CrossRefGoogle Scholar
  24. Cerboni Baiardi, L., Naimzada, A.K.: Experimental oligopolies modeling: a dynamic approach based on heterogeneous behaviors. Commun. Nonlinear Sci. Numer. Simul. 58, 47–61 (2018)CrossRefGoogle Scholar
  25. Cournot, A.-A.: Recherches sur les principes mathématiques de la théorie des richesses par Augustin Cournot. chez L. Hachette (1838)Google Scholar
  26. Den Haan, W.J.: The importance of the number of different agents in a heterogeneous asset-pricing model. J. Econ. Dyn. Control 25(5), 721–746 (2001)CrossRefGoogle Scholar
  27. Droste, E., Hommes, C., Tuinstra, J.: Endogenous fluctuations under evolutionary pressure in cournot competition. Games Econ. Behav. 40(2), 232–269 (2002)CrossRefGoogle Scholar
  28. Fanti, L., Gori, L., Sodini, M.: Nonlinear dynamics in a cournot duopoly with isoelastic demand. Math. Comput. Simul. 108, 129–143 (2015)CrossRefGoogle Scholar
  29. Hofbauer, J., Sigmund, K.: Evolutionary game dynamics. Bull. Am. Math. Soc. 40(4), 479–519 (2003)CrossRefGoogle Scholar
  30. Hommes, C.: Behavioral Rationality and Heterogeneous Expectations in Complex Economic Systems. Cambridge University Press, Cambridge (2013)CrossRefGoogle Scholar
  31. Huck, S., Normann, H.-T., Oechssler, J.: Two are few and four are many: number effects in experimental oligopolies. J. Econ. Behav. Org. 53(4), 435–446 (2004)CrossRefGoogle Scholar
  32. Kopel, M., Lamantia, F., Szidarovszky, F.: Evolutionary competition in a mixed market with socially concerned firms. J. Econ. Dyn. Control 48, 394–409 (2014)CrossRefGoogle Scholar
  33. Lamantia, F., Radi, D.: Evolutionary technology adoption in an oligopoly market with forward-looking firms. Chaos Interdiscipl. J. Nonlinear Sci. 28(5), 055904 (2018)CrossRefGoogle Scholar
  34. Leonard, D., Nishimura, K.: Nonlinear dynamics in the Cournot model without full information. Ann. Oper. Res. 89, 165–173 (1999)CrossRefGoogle Scholar
  35. McFadden, D.: Conditional logit analysis of qualitative choice behavior. In: Zarembka, P. (ed.) Frontiers in Econometrics, pp. 105–142. Academic Press, New York (1973)Google Scholar
  36. Naimzada, A., Ricchiuti, G.: Monopoly with local knowledge of demand function. Econ. Model. 28(1), 299–307 (2011)CrossRefGoogle Scholar
  37. Naimzada, A.K., Sbragia, L.: Oligopoly games with nonlinear demand and cost functions: two boundedly rational adjustment processes. Chaos Solitons Fractals 29(3), 707–722 (2006)CrossRefGoogle Scholar
  38. Naimzada, A.K., Tramontana, F.: Controlling chaos through local knowledge. Chaos Solitons Fractals 42(4), 2439–2449 (2009)CrossRefGoogle Scholar
  39. Naimzada, A., Tramontana, F.: Two different routes to complex dynamics in an heterogeneous triopoly game. J. Differ. Equ. Appl. 21(7), 553–563 (2015)CrossRefGoogle Scholar
  40. Oechssler, J., Roomets, A., Roth, S.: From imitation to collusion: a replication. J. Econ. Sci. Assoc. 2, 1–9 (2016)CrossRefGoogle Scholar
  41. Offerman, T., Potters, J., Sonnemans, J.: Imitation and belief learning in an oligopoly experiment. Rev. Econ. Stud. 69(4), 973–997 (2002)CrossRefGoogle Scholar
  42. Pireddu, M.: Chaotic dynamics in three dimensions: a topological proof for a triopoly game model. Nonlinear Anal. Real World Appl. 25, 79–95 (2015)CrossRefGoogle Scholar
  43. Radi, D.: Walrasian versus cournot behavior in an oligopoly of boundedly rational firms. J. Evolut. Econ. 27, 1–29 (2017)CrossRefGoogle Scholar
  44. Theocharis, R.D.: On the stability of the cournot solution on the oligopoly problem. Rev. Econ. Stud. 27(2), 133–134 (1960)CrossRefGoogle Scholar
  45. Tramontana, F.: Heterogeneous duopoly with isoelastic demand function. Econ. Model. 27(1), 350–357 (2010)CrossRefGoogle Scholar
  46. Tramontana, F., Elsadany, A.A., Xin, B., Agiza, H.N.: Local stability of the cournot solution with increasing heterogeneous competitors. Nonlinear Anal. Real World Appl. 26, 150–160 (2015)CrossRefGoogle Scholar
  47. Tuinstra, J.: A price adjustment process in a model of monopolistic competition. Int. Game Theory Rev. 6(03), 417–442 (2004)CrossRefGoogle Scholar
  48. Vega-Redondo, F.: The evolution of walrasian behavior. Econometrica 65(2), 375–384 (1997)CrossRefGoogle Scholar

Copyright information

© Associazione per la Matematica Applicata alle Scienze Economiche e Sociali (AMASES) 2018

Authors and Affiliations

  1. 1.Department of Economics, Quantitative Methods and ManagementUniversity of Milano - BicoccaMilanoItaly

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