Homogeneous Algorithms Can Reduce Competition in Personalized Pricing

Thank you for the helpful comments.

Weakness 1 and Question 1. Since we are proposing a new mechanism of algorithmic collusion, we aim to demonstrate this phenomenon in a minimal working example. We did extend the base model by considering asymmetric market power (Appendix D) and n competing firms (Appendix E), which showed that the main findings generally hold with more complexity. In general, we expect various other extensions to still provide the same intuition as well.

We would like to clarify that our problem is that of personalized discounting rather than setting base prices (which is a continuous problem and depends on first principles of supply and demand). In the context of discounting in personalized pricing, firms in reality typically consider discrete price options rather than continuous prices (see, e.g., Dubé & Misra, 2023 or Uber and Lyft discounts). If we were to model the continuous pricing regime, we would expect to see a race to the bottom where firms constantly undercut each other; hence we would need to make additional subjective modeling choices that we avoid in the first place with the discrete model.

Weakness 2. We would like to clarify what our solution concept (BNE) captures. BNE (and equilibria in general) characterize the set of limiting behaviors in repeated games given all players are rational. The existence of a BNE solution (say, s^) does not necessarily mean that firms will play that strategy in the one-shot setting, but rather that after sufficient rounds of interacting, s^ is one possible solution that firms will “settle” on and will not be incentivized to deviate. Therefore, BNE bakes in dynamic interactions through time. Collusive equilibria in the one-shot game we study are in a sense stronger than collusion in repeated games. Consider, for example, the (repeated) prisoner’s dilemma game. In the one-shot setting, the only equilibrium is non-collusive. But collusion is possible in repeated games because agents fear the threat of future retaliation. Indeed folk theorems in game theory show that the set of equilibria in repeated games can be much broader than those in one-shot games. Roughly speaking, collusive equilibria in the one-shot setting imply collusive equilibria in repeated games, but the converse isn’t true. We took great care to elucidate this point in lines 194-200 as this is what sets us apart from existing works on algorithmic collusion.

Weakness 3. We agree with the reviewer that our basic conclusion – algorithmic collusion is bad for consumers – is relatively understood in economics. However, our paper is novel in that it reveals a novel mechanism for collusion: that firms can tacitly coordinate through algorithmic homogenization. Investigating specific mechanisms is particularly important when analyzing the legal implications of algorithms, since the law is concerned with both process and outcomes. Our model allows us to apply concepts from antitrust law to this new mechanism (Section 7.1), where we show for the first time that collusion due to algorithmic homogenization lies in a legal gray area despite potential for consumer harm. Therefore, we believe our insights are quite important and not obvious from previous literature.

Question 2. In modeling $\sigma$, we are implicitly assuming that firms do not know which consumers shop around for a lower price. Under this assumption, many natural behavioral models reduce to ours. For example, if consumers have heterogeneous search costs (unknown to the firms), then only a fraction of them will be willing to shop around, yielding our model. We do not consider network structures on the consumers, though this could be an interesting future extension.

Question 3. Our results are agnostic to precisely how the firms approach equilibria. In Bayesian games, no-regret learning dynamics can converge to equilibria (Hartline et al., 2015), so one can think of our model as describing the limiting behavior of learning agents interacting with one another. Our goal is to characterize these limiting behaviors as opposed to analyzing the dynamics of particular learning strategies, but we believe that our equilibrium concept captures the outcomes that reasonable learning dynamics would converge to. See also our response to Weakness 2 for a related discussion.

Question 4. Great question! Our conclusion implies that proving intent in this situation is indeed very difficult, partly because there exists almost no legal precedent for these types of cases, and partly because antitrust law captures many complex scenarios with varying burdens of proof. While we discuss certain ways the law or regulation could evolve in our paper, we view our contribution as highlighting this novel mechanism of collusion in service of future work (both legal scholars and the courts/government agencies) to continue this important line of work.

Thank you for the rebuttal. My questions are partly solved. I understand that many pricing problems need to be studied in a simplified setting, otherwise it is not easy to derive some theoretical insights. However, I also agree with Reviewer zrwy's concern that the model studied is indeed too restricted to be applied in any practical senses.

My overall assessment for the paper is that it lies just on the borderline of acceptance (or rejection), like between 3 and 4. Given the current rating distribution, I'd like to increase my rating to 4, otherwise I think there is no chance for the paper to be considered for acceptance.

Thank you for the questions!

When true positive rate and true negative rate are not the same (often due to class imbalance as the reviewer points out), we might expect that the metric corresponding to the dominant class to be the relevant metric precisely because it applies to more consumers. Generally, separating TPR and TNR is a valid extension and can be modeled by adding one extra parameter; our framework can easily capture that.

Regarding the question about the definition of correlation – throughout the paper we made sure to clarify that the quantity of interest is excess correlation, i.e., correlation that is unexplained by model performance. Rho is formally defined in Table 3; indeed, it does not correspond to the typical notions of correlation (e.g., Pearson) precisely because we are measuring prediction similarity in excess of performance. We note that in our case of two Bernoulli’s, all models of correlation are equivalent up to reparametrization.

The reviewer also mentioned that we treat a1, a2, and rho as exogenous in Theorem 4.1, suggesting to treat them as endogenous instead. This is precisely what we model in Section 5, and we find similar results.

In general, our model is indeed quite simple; since we are proposing a new mechanism, we aim to demonstrate this phenomenon in a minimal working example. We did extend the base model by considering asymmetric market power (Appendix D) and n competing firms (Appendix E). However, we did our best to justify any assumptions we made – for example, we argued that assuming shared knowledge of relevant parameters is standard and somewhat expected in oligopolies with frequent interactions, as is the case with e-commerce markets. Note also that our experiments relax many of these assumptions, and our theoretical findings still hold.

Finally, the white regions in Figure 1 signify parameters where firms would not have used the algorithm in the first place (i.e., always price high or low, since the cost of mistakes does not matter when H/L is too high or too low). The caption in Table 1 is a typo and we will fix it – thank you for pointing it out!

Thank you to the reviewer for the helpful comments. To address your concern, we added the extensions of asymmetric market power and $N$ firms as a robustness check that our main results and mechanisms hold even when we relax certain conditions. Our main text focuses on the simple case to shed light on a novel mechanism of collusion, which allows us to analyze its legal implications.

To answer your question – if there were more than two discounting levels, we expect our main results to hold simply because correlation will always be a mechanism for tacit collusion. In the context of personalized discounting, firms in reality typically consider discrete price options rather than continuous ones (see, e.g., Dubé & Misra, 2023). If we were to model the continuous pricing regime, we would expect to see a race to the bottom where firms constantly undercut each other; hence we would need to make additional subjective modeling choices that we avoid in the first place with the discrete model. We would also need to rule out perfect price competition. While we currently do this with $\sigma$, we would need a much stronger set of assumptions for continuous-price models.

Thank you to the reviewer for the comments! We do indeed assume that firms have information about competing models’ accuracies and correlations. This assumption is realistic in oligopolies that frequently interact with each other (i.e., small but frequent purchases from many consumers, as is the case with our motivating scenarios of Uber and Lyft). Firms that frequently observe the market outcomes from their predictions allow them to better estimate the relevant parameters in equilibrium. Another perspective is that when competing firms play a no-regret strategy in this simple game, they will eventually converge to an equilibrium without explicit knowledge of parameters, see Hartline et al. (2015).

Question (1). Interesting question. We suspect that the regulator would typically need enough information about firm behavior and the market in order to offer good incentives. However, regulators typically do not have this much information unless they conduct a formal investigation on possible antitrust violations.

Question (2). In general, the cited references investigate different mechanisms/scenarios related to competition. For example, Einav & Rosenfeld (2025) show that instead of maximizing accuracy, firms might try to differentiate and segment the market – this outcome is of course possible, but many goods/services (e.g., Uber/Lyft, Amazon goods) are largely similar save for brand loyalty (which we can model via market asymmetry). We see a similar market segmentation effect in Jagadeesan et al. (2023). There are many scenarios that competitive firms face in real life, and we are merely analyzing a novel mechanism – one of many – that has important antitrust implications.

Question (3). We believe our model can capture agents with different dynamics as well. The reviewer’s question about LLM providers adjusting prices would fit in our model; instead of making predictions about a user, they could be making decisions based on time, demand, etc. But of course, our model is limited to two possible prices (which can be easily extended to multiple and the same results will hold).

Thank you for the detailed response! After reading the other reviews, I concur with the other reviewers' concerns about the model possibly being too simplistic. However, I also appreciate the paper's focus on legal implications, and acknowledge the benefit of conciseness in such settings.

To help reconcile these perspectives, I wonder if the authors can further elaborate on the limitations of their simplifying assumptions. In particular:

• Is it possible to describe concrete scenarios in which the proposed collusion mechanism is expected to occur, or is already believed to occur in some sense? If so, which model parameters (i.e., values of $p, \\rho, L, H$, etc.) best capture these scenarios? Is it possible to use the model to quantify the negative societal effects of collusion in these settings?
• Conversely, would it be possible to describe scenarios in which the proposed collusion mechanism is theoretically possible, but is in practice likely to be dominated by other economic effects, or counteracted by other forces? (for instance, which additional economic effects could limit the applicability of the proposed collusion model in the "Uber-Lyft" scenario?)

In addition, I also have a couple of additional (very minor) formatting remarks:

• In Figure 3, the graph legends seem to hide some of the lines and error bars. Given the available whitespace around the plots, maybe it would be beneficial to relocate the legend for improved readability.
• Some tables and equations appear to overflow outside the margins (e.g., Table 1, L510, L517, Table 6).