No Regret Learning in Oligopolies: Cournot vs. Bertrand

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. Department of Computer Science, Stanford University, Stanford CA 94305. E-mail: [email protected]. Department of Computer Science, Cornell University, Ithaca NY 14853. Supported by NSF grant CCF-0325453. E-mail: [email protected].