[Digital ISBN: e-2184-898X |]

ERBE 01 1 05

Collusion between Algorithms: A Literature Review and Limits to Enforcement


a REM/Univ. Lisboa and GOVCOPP/Univ. Aveiro, Portugal.

To cite this article:

Gata, J. E. 2021. Collusion between Algorithms: A Literature Review and Limits to Enforcement. European Review of Business Economics I(1): 73-94.


Received: 1 December 2020. Accepted: 5 April 2021. Published: 31 December 2021

Language: English

View full text (PDF)


Algorithms play an increasingly important role in economic activity, as they become faster and smarter. Together with the increasing use of ever-larger data sets, they may lead to significant changes in the way markets work. These developments have raised concerns not only over the right to privacy and consumers’ autonomy, but also concerning competition. Infringements of antitrust laws involving the use of algorithms have occurred in the past. However, current concerns are of a different nature as they relate to the role algorithms can play as facilitators of collusive behavior in repeated games, and the role increasingly sophisticated algorithms can play as autonomous implementers of firms’ strategies, as they learn to collude without any explicit instructions provided by human agents. In particular, it is recognized that the use of ‘learning algorithms’ can facilitate tacit collusion and lead to an increased blurring of borders between tacit and explicit collusion. Several authors, who have addressed the possibilities for achieving tacit collusion equilibrium outcomes by algorithms interacting autonomously, have also considered some form of ex- ante assessment and regulation over the type of algorithms used by firms. By using well- known results in the theory of computation, this paper shows that such an option faces serious challenges to its effectiveness due to undecidability results. Ex-post assessment may be constrained as well. Notwithstanding several challenges faced by current software testing methodologies, competition law enforcement and policy have much to gain from interdisciplinary collaboration with computer science and mathematics.


Anitha, P., G. Krithka and M. D. Choudhry (2014): “Machine Learning Techniques for learning features of any kind of data: A Case Study”, International Journal of Advanced Research in Computer Engineering & Technology, Vol. 3(12), pp. 4324-4331.

Aumann, Robert (1976): “Agreeing to Disagree”, The Annals of Statistics, Vol. 4(6), pp. 1236-1239. Ben-Porath, Elchanan (1990): “The Complexity of Computing a Best Response Automaton in

Repeated Games with Mixed Strategies,” Games and Economic Behavior, Vol. 2(1), pp. 1-12. Borenstein, Severin (2004): “Rapid Price Communication and Coordination: The Airline Tariff Publishing Case (1994)”, in John E. Kwoka Jr. & Lawrence J. White (Eds.), The Antitrust

Revolution: Economics, Competition, and Policy, Oxford University Press.
Buyers, John C. (2018): Artificial Intelligence: The Practical Legal Issues. Law Brief Publishing Calvano, Emilio, et al. (2018a): “Algorithmic Pricing: What Implications for Competition Policy?”

WP, University of Bologna, CEPR & Toulouse School of Economics, June.
Calvano, Emilio, et al. (2020): “Artificial Intelligence, Algorithmic Pricing and Collusion”, American Economic Review, Vol. 110(10) pp. 3267-3297. Previously published as a WP, University of Bologna, Toulouse School of Economics, EUI & CEPR, Dec. 2018.
Chopra, Samir & White, Laurence F. (2011): A Legal Theory for Autonomous Artificial Agents. University of Michigan Press.
Competition & Markets Authority (2018): “Pricing algorithms: Economic working paper on the use of algorithms to facilitate collusion and personalized pricing”, CMA94, 8th October.
Conlisk, John (1996): “Why Bounded Rationality?” Journal of Economic Literature, Vol. 34, pp. 669-700.
Davis, Martin & Weyuker, Elaine (1983): Computability, Complexity, and Languages: Fundamentals of Theoretical Computer Science. Academic Press, Inc.
Desai, Deven & Kroll, Joshua (2018): “Trust but Verify: A Guide to Algorithms and the Law”, Harvard Journal of Law and Technology, Vol. 31(1), pp. 1-64.
Deutsch, David (1985): “Quantum theory, the Church-Turing principle and the universal quantum computer”, Proceedings of the Royal Society of London, A 400, pp. 97-117.
Ezrachi, Ariel & Stucke, Maurice (2016): Virtual Competition: The Promise and Perils of the Algorithm-Driven Economy. Harvard University Press.
Fudenberg, Drew & Levine, David K. (1998): The Theory of Learning in Games. MIT Press. Fudenberg, Drew & Tirole, Jean (1991): Game Theory. MIT Press.
Gal, Michal S. (2017): “Algorithmic-Facilitated Coordination: Market and Legal Solutions”, Competition Policy International (CPI), May.
Gal, Michal S. & Elkin-Koren, Niva (2017): “Algorithmic Consumers,” Harvard Journal of Law and Technology, Vol. 30(2), pp. 309-353.
Gal, Michal S. (2019): “Algorithms as Illegal Agreements”, Berkeley Technology Law Journal, Vol. 34, pp. 67-118.
Gal, Michal S. (2020): “Algorithms & Competition Law”, Interview of M. Gal by Thibault Schrepel, e-Competitions Special Issue Algorithms, 14 May 2020.
Gata, Joao E. (1995): “Infinite Regression in Strategic Decision Making: An Application of Rice’s Theorem”, Discussion Paper No. 95/38, DERS/University of York, UK.
Gilboa, Itzhak (1988): “The Complexity of Computing Best-Response Automata in Repeated Games,” Journal of Economic Theory, Vol. 45, pp. 342-352.
Harrington, J. E. (2018): “Developing Competition Law for Collusion by Autonomous Artificial Agents”, WP, Wharton School/University of Pennsylvania, April.
Ivaldi, Mark et al. (2003): “The Economics of Tacit Collusion”, Final Report for the DG COMP/EC, March.
Kalai, Ehud (1990): “Bounded Rationality and Strategic Complexity in Repeated Games,” in Tatsuro Ichiishi, Abraham Neyman & Yari Tauman (Eds.), Game Theory and Applications, pp. 131-157, Academic Press.
Kaplow, Louis (2011): “On the Meaning of Horizontal Agreements in Competition Law”, California Law Review, Vol. 99, pp. 683-808.
Klein, Joel I. (1999): “Competition in the Airline Industry”, Testimony as Assistant Attorney General, Antitrust Division, US Department of Justice, before the Committee on Commerce, Science and Transportation, US Senate, March 12.
Klein, Timo (2018): “Assessing Autonomous Algorithmic Collusion: Q-Learning under Sequential Pricing”, Amsterdam Law School Legal Studies Research Paper No. 2018-15, June.
Kovacic, W. E., Marshall, R. C., Marx, L. M. & White, H. L. (2011): “Plus Factors and Agreement in Antitrust Law”, Michigan Law Review, Vol. 110, pp. 393-436.

Kroll, Joshua et al. (2017): “Accountable Algorithms”, University of Pennsylvania Law Review, Vol. 165, pp. 633-705.

Lewis, David (1969): Convention: a philosophical study. Reprint 2002. Blackwell Publishers. Lewis, Harry & Papadimitriou, Christos (1981): Elements of the Theory of Computation. Prentice-Hall.
McSweeny, Terrell (2017): “Algorithms and Coordinated Effects”. Presentation given at University of Oxford Center for Competition Law and Policy, Oxford, UK, May 22nd.
Mehra, Salil K. (2016): “Antitrust and the Robo-Seller Competition in the Time of Algorithms”, Minnesota Law Review, Vol. 100, pp. 1323-1375.
OECD (2017): “Algorithms and Collusion: Competition Policy in the Digital Age”. Paris. Ohlhausen, Maureen (2017): “Should We Fear the Things That Go Beep In the Night? Some Initial Thoughts on the Intersection of Antitrust Law and Algorithmic Pricing”, Remarks from the Concurrences Antitrust in the Financial Sector Conference, New York, May 23rd.
Oliveira, Arlindo (2017): The Digital Mind: How Science Is Redefining Humanity. The MIT Press. OXERA (2017): “When algorithms set prices: winners and losers”, DP, 19 June, Papadimitriou, Christos (1992): “On Players with a Bounded Number of States,” Games and Economic Behavior, Vol. 4(1), pp. 122-131.
Petit, Nicolas (2017): “Antitrust and Artificial Intelligence: A Research Agenda,” Journal of European Competition Law and Practice, Vol. 8(6), pp. 361-362.
Rogers, Hartley (1987): Theory of Recursive Functions and Effective Computability. MIT Press. Rubinstein, Ariel (1986): “Finite Automata Play the Repeated Prisoner’s Dilemma”, Journal of Economic Theory, Vol. 39, pp. 83-96.
Rubinstein, Ariel (1998): Modeling Bounded Rationality. MIT Press.
Russell, Stuart & Norvig, Peter (2016): Artificial Intelligence: a Modern Approach (3rd edition). Pearson Education Ltd.
Salcedo, Bruno (2015): “Pricing Algorithms and Tacit Collusion”, WP, Pennsylvania State University. Schwalbe, Ulrich (2019): “Algorithms, Machine Learning, and Collusion”, Journal of Competition

Law and Economics, Vol. 14(4), pp. 568-607.
Vestager, Margrethe (2017): “Algorithms and Competition”. Presentation at the Bundeskartellamt 18th Conference on Competition, March 16th.

© Copyright CICEE 2023. All rights reserved

Back To Top