Learning to Collude in a Pricing Duopoly

Abstract

Problem definition: This paper addresses the question -- hotly debated in competition regulation circles -- whether or not self-learning algorithms can learn to collude instead of compete against each other, without violating existing competition law. Methodology/results: We construct a price algorithm based on simultaneous-perturbation stochastic-approximation and mathematically prove that, if implemented independently by two price-setting firms in a duopoly, prices will converge to those that maximize the firms' joint revenue in case this is profitable for both firms, and to a Nash equilibrium otherwise. In addition, if the competitor is not willing to collaborate but behaves according to a reaction function, we prove that the prices generated by our algorithm converge to a best-response to the competitor's price. Managerial implications: Our algorithm can learn to collude under self-play while simultaneously learn to price competitively against a `regular' competitor, in a setting where the price-demand relation is unknown and within the boundaries of competition law. This demonstrates that algorithmic collusion is a genuine threat in realistic market scenarios. Moreover, our work exemplifies how algorithms can be explicitly designed to learn to collude, and demonstrates that algorithmic collusion is facilitated (a) by the empirically observed practice of (explicitly or implicitly) sharing demand information, and (b) by allowing different firms in a market to use the same price algorithm. These are important and concrete insights for lawmakers and competition policy professionals struggling with how to respond to algorithmic collusion.