Title: Homogeneous Algorithms Can Reduce Competition in Personalized Pricing
Venue: NeurIPS 2025
Accepted: yes
Keywords: Algorithmic Pricing

Personalized Pricing

Collusion

Competition

Outcome Homogenization

Algorithmic Monoculture

Antitrust

Tacit Collusion
Abstract: Firms' algorithm development practices are often homogeneous Whether firms train algorithms on similar data or rely on similar pre-trained models, the result is correlated predictions. In the context of personalized pricing, correlated algorithms can be viewed as a means to collude among competing firms, but whether or not this conduct is legal depends on the mechanisms of achieving collusion. We investigate the precise mechanisms through a formal game-theoretic model. Indeed, we find that (1) higher correlation diminishes consumer welfare and (2) as consumers become more price sensitive, firms are increasingly incentivized to compromise on the accuracy of their predictions in exchange for coordination. We demonstrate our theoretical results in a stylized empirical study where two firms compete using personalized pricing algorithms. Our results demonstrate a new mechanism for achieving collusion through correlation, which allows us to analyze its legal implications. Correlation through algorithms is a new frontier of anti-competitive behavior that is largely unconsidered by US antitrust law.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates a mechanism for collusion between competing firms using personalized pricing algorithms. The authors propose that firms can reduce price competition by using \"homogeneous\" algorithms—that is, algorithms that produce correlated predictions. The paper develops a formal game-theoretic model of a duopoly where two firms use algorithms to segment consumers and decide whether to offer a high or a low price.\n\nThe main theoretical findings are: (1) higher correlation between firms' algorithms harms consumer welfare, and (2) firms are more incentivized to use correlated algorithms when consumers are highly price-sensitive. The authors show that firms may strategically prefer to use a less accurate but more correlated algorithm over a more performant but independent one. These theoretical results are demonstrated in an experiment using public census data.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n* The paper studies an important problem in algorithmic fairness and economics—how shared components in ML pipelines can lead to anti-competitive outcomes. The focus on homogeneity as a specific mechanism for collusion is a good contribution.\n* The work is well-motivated by current events and its connection to antitrust law is a strength. The discussion of legal implications is good.\n* The paper effectively combines a formal game-theoretic model with an empirical study. The experiments, while simple, provide a demonstration that the theoretically-derived incentives can manifest with standard machine learning models on real-world data.\n\n### Weaknesses\n* The model's primary weakness is its simplicity. The assumption of a two-price world (High/Low) is a major abstraction from real-world pricing. The consumer choice model, governed by a single price-sensitivity parameter sigma, is also highly simplified. These abstractions make it difficult to be confident that the core results would hold in settings with continuous pricing or more complex consumer search behaviors.\n* The paper relies on a static Bayes Nash Equilibrium analysis. This approach deliberately abstracts away from the learning dynamics through which firms might arrive at a collusive equilibrium. While this isolates the correlation mechanism, it also makes the model less descriptive of the \"algorithmic\" part of algorithmic collusion, which often implies adaptation and learning over time.\n* The underlying principle (reduce strategic uncertainty through correlation can soften competition and lead to higher prices) is a known concept in the economics of oligopolies. The paper's novelty lies in applying this principle to the specific context of ML model development practices, which is a valuable but incremental step.\n\nQUESTIONS:\n* How robust are the findings to a more realistic pricing model? If firms could choose prices from a continuous range, would the incentive to sacrifice accuracy for correlation still exist, or would competitive pressures dominate?\n* The sigma parameter for consumer choice is essential to the results. How would the equilibria change under alternative models of consumer behavior, such as those involving search costs, network effects, or reference effects?\n* The paper argues that its static model is a feature that isolates the correlation mechanism. However, couldn't this mechanism be amplified in a dynamic setting? For instance, could RL agents learn to choose simpler, more correlated models as a way to facilitate convergence to a collusive pricing policy?\n* From a practical, regulatory standpoint, how can one distinguish between firms intentionally correlating via a less accurate model (the proposed \"plus factor\") versus firms that are simply less sophisticated and independently choose a simpler (e.g., linear) model that happens to be more correlated? The paper touches on this, but the problem of inferring intent seems very difficult.\n\nLIMITATIONS:\nThe authors correctly note some limitations. My main concerns are stated in the Weaknesses and Questions sections.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-20"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the possibility of algorithmic collusion through using highly correlated predicting algorithms for price discrimination. The authors consider a duopoly model and model their competition with a few parameters including consumer’s price sensitivity ($\\sigma$), extent of algorithm correlation $\\rho$, price discrimination ratio $H/L$, the competitors’ prediction accuracy ($a_1$) and his own prediction accuracy $a_2$. Under this stylized model with two-firm, binary action, symmetric prediction error, the paper shows the following main results: \n(1) Given these known parameters, when Bayesian Nash equilibrium exists, the authors prove that consumer welfare decreases when $\\rho$ (correlation between two firm’s action) increases. \n(2) When consumers are more price sensitive, firms prefer highly correlated algorithms than independent ones.\n(3) An extended study about the situation where the firm can strategically choose an independent algorithm (by training on their own data) or a correlated algorithm (by buying data from vendors), and identifies conditions under which correlated algorithm is always preferred. \n(4) They verify their sights through simulation on ACSIncome data.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\n-\tThe problem studied in this paper is very interesting. Algorithmic collusion is indeed a critical, intriguing and difficult problem that touches AI, laws, policy and regulation\n-\t The paper illustrate interesting insights into the issue of price discrimination from a tractable economic model. It highlights how algorithmic correlations, a growing concern in the AI domain, can negatively impact consumer welfare.\n-\tThis paper leverages publicly available data and machine learning algorithms in a very smart way to efficiently demonstrate the insights behind its proposed method.\n\nWeakness:\nThe reviewer has a very mixed feeling. While the reviewer understands that it is difficult to strike a balance between insights and model complexity, the reviewer feels the studied model is a bit too simplistic and several assumptions are too strong – a learning algorithm is summarized into a single number $a$, pricing actions is simplified into two numbers, and correlation is captured by a single number $rho$ (PS: I seem to be missing how explicitly $rho$ affects the correlation of $p_1, p_2$), and all the knowledge is publicly shared. While these simplifications make the models very tractable (I think most quantities can be characterized in closed form under this simple model), I am not sure how generalizable the obtained insights are. \n\nOn the other hand, the reviewer also found the conclusions derived from the model is mostly expected, which reduces the novelty and insightfulness of the paper. \n\nTo concretize my concerns above, the reviewer has the following questions: \n\n-\tTrue positive rate most likely is very different from true negative rate, especially when the class label (customer type) is unbalanced. Will the insights still hold if this assumption has been relaxed?\n-\tThe definition of correlation is not rigorous. What is the mathematical formula of the correlation studied in this paper? Does this author consider the Pearson correlation, which is a ratio between condition covariance over conditional variance? The unrigorous definition leads to unrigorous proof of Theorem 4.1, which treats correlation as an exogenous variable, and independent of a1, a2, the performance of algorithms employed by the companies. Usually, a1, a2, and $rho$ moves at the same time, which the reviewer suggest to add simulations to study this problem\n\nQUESTIONS:\nSee weakness for concrete questions.\n\nSome additional questions\n-\tWhat does the white region in Figure 1 mean?\n-\tIn Table 1 caption, should “top” “bottom” be left, right?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-25"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the problem of algorithmic homogeneity in personalized pricing. Specifically, it proposes a game-theoretic model to demonstrate how correlated algorithms can facilitate tacit collusion among competing firms, ultimately harming consumer welfare. Empirical studies on a stylized game between two firms are presented to demonstrate the theoretical results. Finally, the paper connects its technical findings to current antitrust law, suggesting new plus factors for detecting collusion.\n\nSTRENGTHS AND WEAKNESSES:\nOn the positive side, this paper presents a timely and important investigation into the anti-competitive implications of algorithmic homogeneity in personalized pricing. The game-theoretic model is well-defined, and the theorems clearly present the core findings: consumer welfare decreases with higher correlation, and firms are incentivized to choose correlated algorithms, especially when consumers are price-sensitive. I found the problem presented by the paper interesting and relevant. In general, this paper proposed an innovative research topic.\n\nMy main concern with this paper is its focus on the oversimplified model with two firms selling identical goods with a binary pricing scheme. While the extensions of asymmetric market power and N-firms are discussed in the appendix, it suggests that they are secondary. Given the importance of real-world market complexities, a more integrated discussion of how firm asymmetry and multiple firms impact the core findings (e.g., Theorems 4.1, 4.2, 5.3) could improve the paper's scope.\n\nQUESTIONS:\nThe model simplifies pricing to a binary choice. How would the results change if firms could set prices on a continuous spectrum, or if there were more than two discrete price points? Would the incentives for correlation still hold?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\n* This paper develops a game-theoretic model of price discrimination and collusion through the choice of predictive models.\n* In the setting, there are two firms selling identical goods. The goods may either sell at a high or low price leading to per-unit profits $H$ or $L$, respectively. A fraction $\\\\theta$ of consumers have valuations higher than the high price, and their sensitivity to price is characterized by a parameter $\\\\sigma$. Firms predict the type of each consumer (high/low valuation) using a classifier $p_i(x)$, with group accuracy characterized by a parameter $a_i$. Predications have correlation parameterized by $\\\\rho$. The strategy space of each firm is the price they offer given a prediction, and the goal is to characterize the equilibria of the resulting game.\n* The main results show that correlation hurts user welfare (Theorem 4.1), and characterize the firm’s preference towards correlation as a function of price sensitivity (Theorem 4.2). The second set of results analyzes the resulting equilibria when firms can decide whether to train prediction models independently or using common data - Showing the firms may benefit from correlated but less accurate prediction models (Theorem 5.3). Empirical analysis is presented based on an income-level prediction task, showing that similar phenomena are also evident in real-world datasets.\n* Finally, broader implications are discussed with an emphasis on legal implications.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n* The paper is well-written and very easy to follow. Formal statements of results are clear, and intuition for the results is provided.\n* The proposed model is intuitive and seems to capture key aspects of collusion dynamics in a concise way.\n* Results offer actionable insights that have potential to influence policy decisions.\n\nWeaknesses:\n* Results seem to rely on the assumption that all the relevant information about the setting is known to all parties, while in practice such information may be hard to obtain. In particular it seems that estimation of the correlation factor $\\\\rho$ in practical settings would require sharing fine-grained model performance metrics between competitors. It’s unclear whether results hold in settings in which the core assumptions are not met.\n\nQUESTIONS:\n* In cases where user welfare is negatively affected by the strategic decisions of the firms, is it possible to incentivize the firms to act in a way which benefits the users?\n* How does the work relate to the existing literature on AI and competition? (e.g., Ben-Porat & Tennenholtz, EC 2019; Jagadeesan et al., NeurIPS 2023; Einav & Rosenfeld, ICML 2025; Tsoy & Konstantinov, NeurIPS 2023; Laufer et al., WWW 2024, etc.). Does the interaction model generalize existing models in this literature? Is it possible to interpret some of these results from the perspective of collusion?\n* What happens if other types of agents are introduced to the dynamics? For example, what are the implications of LLM providers being allowed to adjust their price dynamically based on demand?\n\nLIMITATIONS:\nLimitations and broader impact are discussed.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-02"}