Learning to Collude in a Pricing Duopoly

Janusz Meylahn*, Arnoud den Boer

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

4 Citations (Scopus)


Problem definition: This paper addresses the question whether or not self-learning algorithms can learn to collude instead of compete against each other, without violating existing competition law. Academic/practical relevance: This question is practically relevant (and hotly debated) for competition regulators, and academically relevant in the area of analysis of multi-agent data-driven algorithms. Methodology: We construct a price algorithm based on simultaneous-perturbation Kiefer–Wolfowitz recursions. We derive theoretical bounds on its limiting behavior of prices and revenues, in the case that both sellers in a duopoly independently use the algorithm, and in the case that one seller uses the algorithm and the other seller sets prices competitively. Results: We 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 competitive equilibrium otherwise. We prove this latter convergence result under the assumption that the firms use a misspecified monopolist demand model, thereby providing evidence for the so-called market-response hypothesis that both firms’ pricing as a monopolist may result in convergence to a competitive equilibrium. If the competitor is not willing to collaborate but prices according to a strategy from a certain class of strategies, we prove that the prices generated by our algorithm converge to a best-response to the competitor’s limit 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.
Original languageEnglish
Pages (from-to)2577–2594
Number of pages18
JournalManufacturing & service operations management
Issue number5
Early online date10 Feb 2022
Publication statusPublished - Sept 2022
Externally publishedYes


  • Dynamic pricing
  • Demand learning
  • Competition
  • Algorithmic collusion
  • Kiefer–Wolfowitz algorithm
  • n/a OA procedure

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