Two-population evolutionary oligopoly with partial cooperation and partial hostility

Abstract

In this paper, we reconsider the model in Bischi and Lamantia (J Econ Interact Coord 17:3-27, 2022) and reformulate it in a two-population context. There, the Cournot duopoly market examined is in equilibrium (Cournot-Nash-equilibrium quantities are produced) conditionally to the players' (heterogeneous) attitudes toward cooperation. To accommodate players' attitudes, their objective functions partly include the opponent's profit, resulting in greater (partial) cooperation or hostility toward the opponent than in the standard duopoly setting. An evolutionary selection mechanism determines the survival of cooperative or competitive strategies in the duopoly. The game is symmetric and Bischi and Lamantia (J Econ Interact Coord 17:3-27, 2022) assumes that the two players involved start the game by choosing the same strategic profile. In this way, the full-fledged two-population game simplifies in a one-dimensional map. In this paper, we relax this assumption. On one hand, this approach allows us to investigate entirely the dynamics of the model and the evolutionary stability of the Nash equilibria of the static game that is implicit in the evolutionary setup. In fact, the model with only one population partially represents the system dynamics occurring in an invariant subset of the phase space. As a remarkable result, this extension shows that the steady state of the evolutionary model where all players are cooperative can be an attractor, although only in the weak sense, even when it is not a Nash equilibrium. This occurs when firms have a very high propensity to change strategies to the one that performs better. On the other hand, this approach allows us to accommodate players' heterogeneity (non-symmetric version of the game), whose analysis confirms the main insights attained in the homogeneous setting.