Learning to Collude in a Pricing Duopoly

Manufacturing and Service Operations Management

49 Pages Posted: 22 Jan 2021 Last revised: 28 Dec 2021

See all articles by Janusz M Meylahn

Janusz M Meylahn

University of Amsterdam - Korteweg-de Vries Institute for Mathematics; University of Amsterdam - Dutch Institute for Emergent Phenomena, Korteweg de Vries Institute, Informatics Institute, Amsterdam Business School; University of Amsterdam - Faculty of Economics and Business (FEB)

Arnoud V. den Boer

University of Amsterdam - Korteweg-de Vries Institute for Mathematics; University of Amsterdam Business School

Date Written: December 1, 2020

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 Kiefer--Wolfowitz 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 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.

Keywords: Dynamic Pricing, Demand Learning, Competition, Algorithmic Collusion, Kiefer-Wolfowitz Algorithm

Suggested Citation

Meylahn, Janusz and Meylahn, Janusz and Meylahn, Janusz and den Boer, Arnoud V., Learning to Collude in a Pricing Duopoly (December 1, 2020). Manufacturing and Service Operations Management, Available at SSRN: https://ssrn.com/abstract=3741385 or http://dx.doi.org/10.2139/ssrn.3741385

Janusz Meylahn (Contact Author)

University of Amsterdam - Korteweg-de Vries Institute for Mathematics ( email )

Netherlands

University of Amsterdam - Dutch Institute for Emergent Phenomena, Korteweg de Vries Institute, Informatics Institute, Amsterdam Business School ( email )

Plantage Muidergracht 12
Amsterdam, Noord Holland 1018TV
Netherlands

University of Amsterdam - Faculty of Economics and Business (FEB) ( email )

Roetersstraat 11
Amsterdam, 1018 WB
Netherlands

Arnoud V. Den Boer

University of Amsterdam - Korteweg-de Vries Institute for Mathematics ( email )

Netherlands

University of Amsterdam Business School ( email )

Roetersstraat 18
Amsterdam, 1018WB
Netherlands

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