Data-Driven Collusion and Competition in a Pricing Duopoly With Multinomial Logit Demand

Abstract

We consider dynamic pricing and demand learning in a duopoly, both from the perspective where the firms compete against each other and from the perspective where the firms aim to collude to increase their revenues. We adopt the widely studied multinomial logit demand model and construct a sustainable notion of collusion, called fair Pareto optimal pricing, that ensures equal relative revenue improvements for both firms compared to the Nash equilibrium. In contrast to other notions of collusion such as joint-revenue maximization, we show that fair Pareto optimal pricing is always detrimental for consumers and profitable for both firms in the duopoly, regardless of the model parameters. Next, we construct a price algorithm that learns the fair Pareto optimal price from accumulating data if deployed by both firms in the duopoly, and prove theoretical performance bounds. In addition, we propose a mechanism to infer demand observations from the competitor's price path, so that our algorithm can operate in a setting where prices are public but demand is private information. We also construct a price algorithm for the case that the firms compete against each other, and show that it learns to respond optimally against a class of algorithms that includes best-response and fixed-price policies. Our work contributes to the understanding of well-performing price policies in a competitive multi-agent setting, and also shows that collusion by algorithms is in theory possible and deserves the attention of lawmakers and competition policy regulators.