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Adversarial competition and collusion in algorithmic markets


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.

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Fig. 1: Representation of the two-firm market game and experimental design.
Fig. 2: Representation of the exploration and adversarial pricing strategies.
Fig. 3: Adversarial pricing enables one firm to increase its profits after learning over time how its competitor reacts to price changes.
Fig. 4: Harms caused to consumers by adversarial competition and collusion.

Data availability

Data files to reproduce figures are on the Open Science Framework repository at (ref. 44).

Code availability

The source code to reproduce the results of this article is also available on the Open Science Framework repository at (ref. 44).


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We thank all members of the Computational Privacy Group for discussions and suggestions. We also thank J. Cremer and H. Piffaut for comments on earlier versions of the paper.

Author information

Authors and Affiliations



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.

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Competing interests

L.R. acknowledges support from EPSRC (EP/W016419/1). The other authors declare no competing interests.

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Nature Machine Intelligence thanks Stefan Thomas, Aniko Hannak and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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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 LG of having found the optimal sequence, averaged over all 25 × 25 initial market configuration, as well as its 95% confidence interval. Each panel displays the likelihood LG for each of the three studied scenarios (a. TES, b. KLN, c. CAL), increasing from LG = 0 initially to LG = 100% across all scenarios and for all 25 × 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 r1 (competitor \(\mathcal{A}_1\)) and r2 (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.

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Rocher, L., Tournier, A.J. & de Montjoye, YA. Adversarial competition and collusion in algorithmic markets. Nat Mach Intell 5, 497–504 (2023).

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