https://www.nature.com/articles/s42256-023-00646-0 Skip to main content Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. Advertisement Advertisement Nature Machine Intelligence * View all journals * Search * Log in * Explore content * About the journal * Publish with us * Subscribe * Sign up for alerts * RSS feed 1. nature 2. nature machine intelligence 3. articles 4. article * Article * Published: 04 May 2023 Adversarial competition and collusion in algorithmic markets * Luc Rocher ORCID: orcid.org/0000-0002-9956-1187^1,2^ na1, * Arnaud J. Tournier^2,3^ na1 & * Yves-Alexandre de Montjoye ORCID: orcid.org/0000-0002-2559-5616^ 2,3 Nature Machine Intelligence volume 5, pages 497-504 (2023)Cite this article * 1099 Accesses * 66 Altmetric * Metrics details Subjects * Computational science * Economics Abstract Algorithms are now playing a central role in digital marketplaces, setting prices and automatically responding in real time to competitors' behaviour. The deployment of automated pricing algorithms is scrutinized by economists and regulatory agencies, concerned about its impact on prices and competition. Existing research has so far been limited to cases where all firms use the same algorithm, suggesting that anti-competitive behaviour might spontaneously arise in that setting. Here we introduce and study a general anti-competitive mechanism, adversarial collusion, where one firm manipulates other sellers that use their own pricing algorithm. We propose a network-based framework to model the strategies of pricing algorithms on iterated two-firm and three-firm markets. In this framework, an attacker learns to endogenize competitors' algorithms and then derive a strategy to artificially increase its profit at the expense of competitors. Facing a drastic loss of profits, competitors will eventually intervene and revise or turn off their pricing algorithm. To disincentivize this intervention, we show that the attacker can instead unilaterally increase both its profits and the profits of competitors. This leads to a collusive outcome with symmetric and supra-competitive profits, sustainable in the long run. Together, our findings highlight the need for policymakers and regulatory agencies to consider adversarial manipulations of algorithmic pricing, which might currently fall outside of the scope of current competition laws. Access through your institution Buy or subscribe This is a preview of subscription content, access via your institution Access options Access through your institution Access through your institution Change institution Buy or subscribe Access Nature and 54 other Nature Portfolio journals Get Nature+, our best-value online-access subscription $29.99 / 30 days cancel any time Learn more Subscribe to this journal Receive 12 digital issues and online access to articles $119.00 per year only $9.92 per issue Learn more Rent or buy this article Prices vary by article type from$1.95 to$39.95 Learn more Prices may be subject to local taxes which are calculated during checkout Additional access options: * Log in * Learn about institutional subscriptions * Read our FAQs * Contact customer support Fig. 1: Representation of the two-firm market game and experimental design. [42256_2023_646_Fig1_HTML] Fig. 2: Representation of the exploration and adversarial pricing strategies. [42256_2023_646_Fig2_HTML] Fig. 3: Adversarial pricing enables one firm to increase its profits after learning over time how its competitor reacts to price changes. [42256_2023_646_Fig3_HTML] Fig. 4: Harms caused to consumers by adversarial competition and collusion. [42256_2023_646_Fig4_HTML] Data availability Data files to reproduce figures are on the Open Science Framework repository at https://doi.org/10.17605/osf.io/2yuvm (ref. ^44). Code availability The source code to reproduce the results of this article is also available on the Open Science Framework repository at https://doi.org /10.17605/osf.io/2yuvm (ref. ^44). References 1. Dalgleish, R. Retail sales, Great Britain: December 2020. Office for National Statistics https://www.ons.gov.uk/ businessindustryandtrade/retailindustry/bulletins/retailsales/ december2020 (2021). 2. Buck, R. et al. 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Authors and Affiliations 1. Oxford Internet Institute, University of Oxford, Oxford, UK Luc Rocher 2. Data Science Institute, Imperial College London, London, UK Luc Rocher, Arnaud J. Tournier & Yves-Alexandre de Montjoye 3. Department of Computing, Imperial College London, London, UK Arnaud J. Tournier & Yves-Alexandre de Montjoye Authors 1. Luc Rocher View author publications You can also search for this author in PubMed Google Scholar 2. Arnaud J. Tournier View author publications You can also search for this author in PubMed Google Scholar 3. Yves-Alexandre de Montjoye View author publications You can also search for this author in PubMed Google Scholar Contributions L.R. and A.J.T. contributed to conceptualization, methodology development, software development, experimental validation and writing. Y.-A.d.M. contributed to conceptualization, methodology development and writing. Corresponding authors Correspondence to Luc Rocher or Yves-Alexandre de Montjoye. Ethics declarations Competing interests L.R. acknowledges support from EPSRC (EP/W016419/1). The other authors declare no competing interests. Peer review Peer review information Nature Machine Intelligence thanks Stefan Thomas, Aniko Hannak and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Additional information Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Extended data Extended Data Fig. 1 The exploration phase efficiently discovers the optimal price sequence to maximize the profits of both the attacker and the competitor. We report the likelihood L[G] of having found the optimal sequence, averaged over all 25 x 25 initial market configuration, as well as its 95% confidence interval. Each panel displays the likelihood L[G] for each of the three studied scenarios (a. TES, b. KLN, c. CAL), increasing from L[G] = 0 initially to L[G] = 100% across all scenarios and for all 25 x 25 initial market conditions. Extended Data Fig. 2 Profits obtained with the best price sequence currently found at time t, showing that the attacker can find a good price sequence even with limited exploration. We report the profits r[1] (competitor \(\mathcal{A}_1\)) and r[2] (attacker \(\mathcal{A}_2\)) for the best sequence found within the explored vertices (market price configuration) during the exploration phase for each of the three studied scenarios (a. TES, b. KLN, c. CAL). We display the median profits along with the 25% and 75% quartiles, averaged over each of the \(25\times25\) initial market configuration. For instance, stopping after only half of the complete exploration phase duration would yield the optimal profits against TES and KLN, and 39% of the optimal profits against CAL (median profits, filled triangles). The curves are not necessarily monotonous and profits symmetric: the adversary estimates the competitor's profits by assuming symmetry of the demand function and marginal costs (see Methods) and therefore visits configurations where the competitor's profits cannot be estimated yet. Once all vertices have been explored, all profits can be estimated, and the best sequence corresponds to symmetric profits. Supplementary information Supplementary Information Supplementary Figs. 1-6, Tables 1-4, Notes 1-3 and Methods. Rights and permissions Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Reprints and Permissions About this article Check for updates. Verify currency and authenticity via CrossMark Cite this article Rocher, L., Tournier, A.J. & de Montjoye, YA. Adversarial competition and collusion in algorithmic markets. Nat Mach Intell 5, 497-504 (2023). https://doi.org/10.1038/s42256-023-00646-0 Download citation * Received: 10 December 2021 * Accepted: 16 March 2023 * Published: 04 May 2023 * Issue Date: May 2023 * DOI: https://doi.org/10.1038/s42256-023-00646-0 Share this article Anyone you share the following link with will be able to read this content: Get shareable link Sorry, a shareable link is not currently available for this article. 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