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Two-Population Evolutionary Oligopoly with Partial Cooperation and Partial Hostility

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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.

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Notes

  1. All the mentioned contributions, as well as most evolutionary games in economics, assume interactions among agents from infinite populations of players. A different approach to explaining the selection of cooperation in evolutionary games is based on strategic interaction from a finite population of players. Nowak and Sigmund (2005) report stronger forms of reciprocity that can lead to more cooperation with finite populations. In this work, however, we still consider infinite populations of agents.

  2. It is well known that a stable steady state that is locally asymptotically stable (or stable from a Lyapunov’s point of view) under a system of replicator dynamics is also a Nash equilibrium, see, e.g., Hofbauer and Weibull (1996), Hofbauer and Sigmund (1998) and Hofbauer and Sigmund (2003).

  3. In this context, optimality is referred to from the industry perspective while the consumer standpoint is left out. An evolutionary model in the spirit of this paper with the incorporation of shares of the consumer objective function into the firms’ objective has been proposed in Kopel et al. (2014). For an examination of possible alternative objectives to profit (without levels of aggression/cooperation) we refer to Fanti et al. (2017).

  4. Clearly a non-negativity constraint should be included in the price. However, we omit it because it is verified ex-post in the equilibrium quantities considered in the model.

  5. In Bischi and Lamantia (2022), to accommodate memory, the function F also accounts for past profits. Here, we neglect the role of memory and we instead focus on the two-population version of the evolutionary game.

  6. Consider map (21). If \(E_{0,0}\) is locally asymptotically stable, it must attract all points that lie in a neighborhood of it, which includes also points in the diagonal \(m_{1}=m_{2}\) where the dynamics is conjugated to the one of (23), therefore \(E_{0}\) is also locally asymptotically stable. Assume that \(E_{0}\) is also locally asymptotically stable for (23). Since the dynamics of (21) is conjugated to the one of (23) on \(m_{1}=m_{2}\), all points in a neighborhood of \(E_{0,0}\) that lie in the diagonal are attracted to \(E_{0,0}\). Then, \(E_{0,0}\) is either a saddle or a stable equilibrium. However, to be a saddle, it is required that \(m_{1}=m_{2}\) is an eigenvector (stable manifold) of the equilibrium. However, the eigenvectors of \(E_{0,0}\) are the left and right borders of \(\left[ 0,1\right] ^{2}\). It follows that \(E_{0,0}\) must be a stable equilibrium.

  7. To prove this analytically the sufficient conditions for the flip bifurcation need to be verified, see, e.g., Devaney (1989), Gardini et al. (2021) and references therein.

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Acknowledgements

We thank the Editors and two anonymous Referees for their valuable and constructive feedback on our paper. The authors acknowledge the financial support from the Czech Science Foundation (GACR) under project 23-06282S, and an SGS research project of VŠB-TUO (SP2023/19). The financial support of the European Union under the REFRESH – Research Excellence For REgion Sustainability and High-tech Industries project number CZ.10.03.01/00/22_003/0000048 via the Operational Programme Just Transition is acknowledged as well.

Funding

The authors acknowledge the financial support from the Czech Science Foundation (GACR) under project 23-06282S, and an SGS research project of VŠB-TUO (SP2023/19). The financial support of the European Union under the REFRESH – Research Excellence For REgion Sustainability and High-tech Industries project number CZ.10.03.01/00/22_003/0000048 via the Operational Programme Just Transition is acknowledged as well.

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Authors and Affiliations

  1. Department of Economics, Statistics and Finance, University of Calabria, Via P. Bucci 0-1C, 87036, Rende, CS, Italy

    F. Lamantia

  2. Department of Finance, VS̆B–Technical University of Ostrava, 17. listopadu 2172/15, 70800, Ostrava-Poruba, Czech Republic

    F. Lamantia, D. Radi & T. Tichy

  3. DiMSEFA, Catholic University of the Sacred Heart, Via Necchi 9, 20123, Milan, Italy

    D. Radi

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Lamantia, F., Radi, D. & Tichy, T. Two-Population Evolutionary Oligopoly with Partial Cooperation and Partial Hostility. Comput Econ (2024). https://doi.org/10.1007/s10614-023-10536-7

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