Abstract
Cournot and Bertrand oligopolies constitute the two most prevalent models of firm competition. The analysis of Nash equilibria in each model reveals a unique prediction about the stable state of the system. Quite alarmingly, despite the similarities of the two models, their projections expose a stark dichotomy. Under the Cournot model, where firms compete by strategically managing their output quantity, firms enjoy positive profits as the resulting market prices exceed that of the marginal costs. On the contrary, the Bertrand model, in which firms compete on price, predicts that a duopoly is enough to push prices down to the marginal cost level. This suggestion that duopoly will result in perfect competition, is commonly referred to in the economics literature as the “Bertrand paradox”.
In this paper, we move away from the safe haven of Nash equilibria as we analyze these models in disequilibrium under minimal behavioral hypotheses. Specifically, we assume that firms adapt their strategies over time, so that in hindsight their average payoffs are not exceeded by any single deviating strategy. Given this no-regret guarantee, we show that in the case of Cournot oligopolies, the unique Nash equilibrium fully captures the emergent behavior. Notably, we prove that under natural assumptions the daily market characteristics converge to the unique Nash. In contrast, in the case of Bertrand oligopolies, a wide range of positive average payoff profiles can be sustained. Hence, under the assumption that firms have no-regret the Bertrand paradox is resolved and both models arrive to the same conclusion that increased competition is necessary in order to achieve perfect pricing.
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References
Abernethy, J., Hazan, E., Rakhlin, A.: Competing in the dark: An efficient algorithm for bandit linear optimization. In: COLT (2008)
Amir, R.: Cournot oligopoly and the theory of supermodular games. Games and Economic Behavior (1996)
Auer, P., Cesa-Bianchi, N., Freund, Y., Schapire, R.E.: The nonstochastic multiarmed bandit problem. SIAM Journal on Computing (2002)
Baye, M., Morgan, J.: A folk theorem for one-shot bertrand games. Economic Letters 65(1), 59–65 (1999)
Bertrand, J.: Theorie mathematique de la richesse sociale. Journal des Savants 67, 499–508 (1883)
Cesa-Bianchi, N., Lugosi, G.: Prediction, Learning and Games. Cambridge University Press, Cambridge (2006)
Cournot, A.A.: Recherches sur les principes mathmatiques de la thorie des richesses (1838)
Dufwenberg, M., Gneezy, U.: Price competition and market concentration: An experimental study. International Journal of Industrial Organization 18 (2000)
Even-Dar, E., Mansour, Y., Nadav, U.: On the convergence of regret minimization dynamics in concave games. In: 41st ACM Symposium on Theory of Computing, STOC (2009)
Flaxman, A., Kalai, A.T., McMahan, B.: Online convex optimization in the bandit setting: Gradient descent without a gradient. In: SODA (2005)
Freund, Y., Schapire, R.E.: A decision-theoretic generalization of on-line learning and an application to boosting. Journal of Computer and System Sciences 55(1), 119–139 (1997)
Fudenberg, D., Levine, D.K.: The theory of learning in games. MIT Press, Cambridge (1998)
Hazan, E., Agarwal, A., Kale, S.: Logarithmic regret algorithms for online convex optimization. Machine Learning 69(2-3), 169–192 (2007)
Liu, L.: Correlated equilibrium of cournot oligopoly competition. Journal of Economic Theory (1996)
Mas-Colell, A., Winston, M., Green, J.: Microeconomic Theory. Oxford University Press, Oxford (1995)
Milgrom, P., Roberts, J.: Rationalizability, learning, and equilibrium in games with strategic complementarities. Econometrica 58(6), 1255–1277 (1990)
Milgrom, P., Roberts, J.: Adaptive and sophisticated learning in repeated normal form games. In: Games and Economic Behavior, pp. 82–100 (February 1991)
Rosen, J.: Existense and uniqueness of equilibrium points for concave n-person games
Sheth, J.N., Sisodia, R.S.: The Rule of Three: Surviving and Thriving in Competitive Markets
Theocharis, R.D.: On the stability of the cournot solution on the oligopoly problem. The Review of Economic Studies 27(2), 133–134 (1960)
Vives, X.: Oligopoly Pricing: Old Ideas and New Tools. The MIT Press, Cambridge (2001)
Wu, J.: Correlated equilibrium of bertrand competition. In: Papadimitriou, C., Zhang, S. (eds.) WINE 2008. LNCS, vol. 5385, pp. 166–177. Springer, Heidelberg (2008)
Yi, S.-S.: On the existence of a unique correlated equilibrium in cournot oligopoly. Economic Letters 54, 235–239 (1997)
Young, H.: Strategic Learning and Its Limits. Oxford University Press, Oxford (2004)
Zinkevich, M.: Online convex programming and generalized infitesimal gradient ascent. In: ICML (2003)
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Nadav, U., Piliouras, G. (2010). No Regret Learning in Oligopolies: Cournot vs. Bertrand. In: Kontogiannis, S., Koutsoupias, E., Spirakis, P.G. (eds) Algorithmic Game Theory. SAGT 2010. Lecture Notes in Computer Science, vol 6386. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16170-4_26
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DOI: https://doi.org/10.1007/978-3-642-16170-4_26
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