Homogeneous Algorithms Can Reduce Competition in Personalized Pricing

TL;DR

This paper addresses how homogeneous algorithm development can lead to correlated pricing in a duopoly with personalized pricing. It develops a game-theoretic model where prediction quality and correlation affect downstream pricing and proves that consumer welfare deteriorates with higher correlation, especially as price sensitivity rises. It further shows that firms may strategically pursue correlation by sharing data or selecting similar models, even at the cost of predictive accuracy, and supports these findings with an empirical study using demographic-based income prediction. The work highlights significant antitrust implications, suggesting that correlation-enabled tacit collusion can arise without explicit communication and calling for reevaluation of regulatory frameworks in digital markets.

Abstract

Firms' algorithm development practices are often homogeneous. Whether firms train algorithms on similar data, aim at similar benchmarks, or rely on similar pre-trained models, the result is correlated predictions. We model the impact of correlated algorithms on competition in the context of personalized pricing. Our analysis reveals 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 underscore the ease with which algorithms facilitate price correlation without overt communication, which raises concerns about a new frontier of anti-competitive behavior. We analyze the implications of our results on the application and interpretation of US antitrust law.

Figures (9)

• Figure 1: Regions where firms following the algorithm's recommendation is a Bayes Nash Equilibrium (BNE) for independent models only ($\rho = 0$, light gray), identical models only ($\rho = 1$, dark gray), and both (gradient). The gradient represents the difference in firm utility when $\rho = 1$ relative to $\rho=0$; blue (red) signifies positive (negative) difference. Columns represent two values of $\theta \in \{0.5, 0.75\}$, while rows represent two values of $H/L \in \{2, 4\}$. The x-axis in each subfigure is $\sigma$ and the y-axis is $a = a_1 = a_2$.
• Figure 2: Regions where firms using both correlated models and independent models are Pure Nash Equilibra (first-stage game). An additional condition is that firms following the algorithm's recommendation must be a Bayes Nash Equilibrium (second-stage game). The x-axis is the performance of the correlated algorithm $a_c$, and the y-axis is the performance of the independent algorithm $a_i$. The gradient represents the difference in firm utility when $\rho = \rho_c$ (correlated) at performance $a_c$ relative to the utility at $\rho=0$ (independent) at performance $a_i$; blue (red) signifies positive (negative) difference. All subfigures show parameters for which firms have a preference for correlation at $a_c = a_i$ as per Theorem \ref{['thm:cons_welfare_corr']}, with $H/L=3, \theta=0.75$.
• Figure 3: (a) [Left] Accuracy, precision, recall, true negative rate (TNR), and area under ROC curve (AUC) for a given firm deploying a logistic regression (LR) or random forest (RF) model. Since both firms 1 and 2 face the same model options, their results are identically distributed. [Right] Correlation between both firms' models when they both use logistic regression (LR-LR) or both use random forests (RF-RF). Error bars indicate 95% confidence intervals over 15 seeds. (b) Utility when both firms use logistic regression models (LR-LR) subtracted by utility when both firms use random forests (RF-RF). Greater than 0 indicates a preference for correlation at the expense of predictive performance. $x$-axis varies the proportion of $H$ (high price) to $L$ (low price), and line colors indicate different values of $\sigma$, where a lighter color means higher consumer sensitivity to price. Shaded region indicates 95% confidence intervals over 15 seeds.
• Figure 4: Best response matrices for the two firms where the action space is to deploy a logistic regression (LR) or random forest (RF) model, over five model parameters. Best response for Firms 1 and 2 are highlighted in blue and red. Nash equilibria exist when both blue and red are highlighted in the same box (e.g., (LR, LR) in the middle subfigure). When both (LR, LR) and (RF, RF) are equilibria, a yellow square indicates higher firm utility between the two. Grey boxes are "invalid" regions because $(s^*, s^*)$ would not have been a BNE in the downstream game where firms compete on prices. These results use the average firm utility over 15 seeds.
• Figure 5: (a) [Left] Correlation between both firms' models in the empirical study across various values of $\gamma$. $\gamma = 0$ (1) corresponds to no overlap (full overlap) in training data. [Middle and Right] Accuracy of Firm 1 and 2's models over various values of $\gamma$. Error bars are 95% confidence intervals over 15 seeds. (b) Difference in utility between $\gamma$ at the x-axis and $\gamma = 0$ (no overlap in training data) for the empirical study, over various values of $H/L$ and $\sigma$. Top and bottom rows correspond to Firm 1 and 2's utilities, respectively. Shaded regions indicate 95% confidence intervals over 15 seeds.

Theorems & Definitions (10)

• Theorem 4.1
• Theorem 4.2
• Lemma 5.1
• Corollary 5.2: Corollary to Thm \ref{['thm:price_sens']}
• Theorem 5.3
• proof
• proof
• proof
• proof
• proof