Online Optimization Algorithms in Repeated Price Competition: Equilibrium Learning and Algorithmic Collusion

Problem definition: This paper examines whether widely used online learning algorithms used in pricing can independently reach competitive outcomes or whether they may instead foster tacit collusion. This issue has drawn considerable attention from competition regulators, because algorithmic pricing is increasingly common in digital markets. Understanding when such algorithms lead to equilibrium prices or to supra-competitive prices is critical for buyers, sellers, and policymakers. Methodology/results: We study the behavior of multiarmed bandit algorithms in repeated price competition. These algorithms only observe profits from the prices actually chosen, making them realistic models of automated pricing. Using formal analysis, we show that an important class of online learning algorithms, called mean-based algorithms, reliably converges to the Nash equilibrium in Bertrand competition. This finding is notable because, in general, online learning algorithms do not guarantee convergence to equilibrium. In addition, we run extensive numerical experiments with different widely used bandit algorithms. The experiments confirm that most of them, including those that are not mean based, also converge to equilibrium. We observe supra-competitive prices only in special cases where all sellers implement the same symmetric version of certain algorithms, such as upper confidence bound. Even then, supra-competitive pricing vanishes as the number of competing sellers increases. Managerial implications: Our results highlight that the risk of algorithmic collusion in competitive pricing markets is often overstated. For most practical implementations of bandit algorithms, sellers’ prices converge to competitive levels. Only under very specific and symmetric setups do prices remain above competitive benchmarks, and this effect diminishes with more competitors. These insights provide reassurance to regulators concerned with consumer welfare, as well as to managers considering algorithmic pricing tools. They suggest that, although vigilance is warranted, fears of widespread algorithm-driven collusion may be exaggerated. Funding: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon Europe research and innovation programme [Grant agreement 101198689]. This project was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - GRK 2201/2 - Project Number 277991500. Supplemental Material: The online appendices are available at https://doi.org/10.1287/msom.2024.1389 .

• 저널Manufacturing & Service Operations Management
• 토픽Auction Theory and Applications · Management Science and Operations Research
• DOI10.1287/msom.2024.1389
• 저자Julius Durmann, Matthias Oberlechner, Martin Bichler