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Imitation in Cournot oligopolies with multiple markets

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Abstract

This paper analyzes imitation dynamics in Cournot oligopolies when firms imitate both rivaling firms and firms in other markets. The resulting tension between relative and absolute performance leads to a unique prediction strictly between the Nash equilibrium and perfectly competitive outcomes, which is fully characterized by a simple formula. The outcome becomes less competitive as the number of markets increases, i.e., as firms receive more information about firms in other markets. A link with relative payoff maximization is provided. An extension of the benchmark model reveals that sophisticated firms imitating across asymmetric markets converge to a related but somewhat less competitive outcome.

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Notes

  1. Some empirical evidence suggests that strategic decisions are sometimes guided by observing firms in other markets. For example, Greve (1998) finds that local US radio stations imitate the choice of market position of stations serving other regions.

  2. Ellison and Fudenberg (1993), Eshel et al. (1998), Apesteguia et al. (2007) and Bergin and Bernhardt (2009), among others, consider similar imitation rules. While focusing on IBA allows a clear exposition, one can of course think of alternative rules. For example, more optimistically biased firms may imitate the quantity with the highest maximum profit. Section 3.5 discusses such behavior and argues that it leads to results closely paralleling Alós-Ferrer’s (2004) analysis of imitation among firms with finite memory.

  3. The “Cournot quantity” is the quantity produced by each firm in the symmetric Nash equilibrium.

  4. Some, but not all, results hold under log-concave inverse demands. For example, Lemma 2 below can be shown to hold under this assumption, but counterexamples reveal that Lemma 3 does not.

  5. Henceforth, time indexes are omitted whenever possible.

  6. Notice that the informational environment bears some resemblance to the multiple locations model in Anwar (2002) and Ely (2002), see also Alós-Ferrer and Weidenholzer (2008).

  7. For some values of \(\mu \), convergence to an absorbing state is fast. For example, if \(\mu =1\), the process immediately enters an absorbing state from any state at which there are no ties.

  8. When possible, the arguments of \(q^{s}(k,n)\) will be omitted in order to reduce notation.

  9. Apesteguia et al. (2010) find such preferences to be behaviorally relevant.

  10. While it is perhaps most natural to think of \(K(q_{i},q_{-i})\) as a simple average, it could, e.g., also be that \(K(q_{i},q_{-i})=\max _{j\in N\backslash \{i\}}\pi (q_{j},Q)\), i.e., each firm wants to maximize the distance to the runner-up.

  11. Alós-Ferrer (2004), Apesteguia et al. (2007), Oyarzun and Ruf (2009) and Matros (2012), among others, consider similar imitation rules.

  12. Hedlund and Oyarzun (2014) argue that social comparison can motivate a similar approach in an environment of technology adoption.

  13. Notice that since firms are not assumed to know the inverse demand or cost function, more sophisticated behavior, such as best responding behavior, is still unfeasible here.

  14. Assumptions of no inertia are quite common in the literature, see, e.g., Eshel et al. (1998), Alós-Ferrer (2004) and Bergin and Bernhardt (2009).

  15. While, as in the benchmark model, it can be shown that any absorbing set is an absorbing state, the argument is somewhat lengthy and is omitted for reasons of space. The proof of the main result of this section, however, explicitly shows that no non-singleton absorbing set is stochastically stable.

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Authors and Affiliations

  1. Department of Economics, University of Heidelberg, Bergheimer Str. 58, 69115, Heidelberg, Germany

    Jonas Hedlund

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  1. Jonas Hedlund
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Correspondence to Jonas Hedlund.

Additional information

I thank an associate editor and two referees for very constructive feedback, and Carlos Alós-Ferrer, José Apesteguia, Antonio Morales, Joerg Oechssler, Carlos Oyarzun, Adam Sanjurjo, Luis Ubeda and Amparo Urbano, as well as seminar participants at the University of Alicante, Stockholm School of Economics, ADRES 2010 and ASSET 2010, for many helpful comments and suggestions. Financial support from the Spanish Ministry of Innovation and Science, with reference BES-2008-008040, is gratefully acknowledged.

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Hedlund, J. Imitation in Cournot oligopolies with multiple markets. Econ Theory 60, 567–587 (2015). https://doi.org/10.1007/s00199-015-0878-7

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