Regulation of Algorithmic Collusion, Refined: Testing Pessimistic Calibrated Regret

TL;DR

This work revisits algorithmic collusion regulation by auditing pricing data through the lens of calibrated regret, extending Hartline et al.'s framework with a refined method that tests pessimistic calibrated regret and relaxes the fully-supported price-distribution requirement via aggregated distributions. It argues that vanishing calibrated regret is essential to rule out supra-competitive pricing even without side information, and demonstrates that best-in-hindsight regret can be manipulated, creating false negatives under weaker non-collusion notions. The paper provides a regret-estimation framework with one-sided consistency and shows how to compute a pessimistic regret bound $\overline{R}^T(c,\bm{x}^T)$ by aggregating price data, minimizing over plausible costs, and comparing to a threshold with finite-sample guarantees. It also highlights practical regulatory implications by illustrating unknown-cost scenarios where a per se audit may fail, advocating a rule-of-reason extension to bound reasonable costs and guide market-context considerations for effective antitrust enforcement.

Abstract

We study the regulation of algorithmic (non-)collusion amongst sellers in dynamic imperfect price competition by auditing their data as introduced by Hartline et al. [2024]. We develop an auditing method that tests whether a seller's pessimistic calibrated regret is low. The pessimistic calibrated regret is the highest calibrated regret of outcomes compatible with the observed data. This method relaxes the previous requirement that a pricing algorithm must use fully-supported price distributions to be auditable. This method is at least as permissive as any auditing method that has a high probability of failing algorithmic outcomes with non-vanishing calibrated regret. Additionally, we strengthen the justification for using vanishing calibrated regret, versus vanishing best-in-hindsight regret, as the non-collusion definition, by showing that even without any side information, the pricing algorithms that only satisfy weaker vanishing best-in-hindsight regret allow an opponent to manipulate them into posting supra-competitive prices. This manipulation cannot be excluded with a non-collusion definition of vanishing best-in-hindsight regret. We motivate and interpret the approach of auditing algorithms from their data as suggesting a per se rule. However, we demonstrate that it is possible for algorithms to pass the audit by pretending to have higher costs than they actually do. For such scenarios, the rule of reason can be applied to bound the range of costs to those that are reasonable for the domain.

Figures (3)

• Figure 1: The frequencies of each pair of strategies in the last 10 rounds of the competition are shown in the heatmap. The highest payoff equilibrium is marked in red.
• Figure 2: The true regret and estimated regret plotted against different assumed costs of seller 1, with a zoomed-view on the right. The red dashed line shows the example threshold $6\times10^{-2}$ for estimated regret. The seller maintains exploration. The true regret is the regret estimated with the knowledge of the ground truth demand, while the estimated regret is the regret estimated using our auditing method.
• Figure 3: The true regret at the true cost $c=0.1$ and at the plausible cost $c'=0.25$ plotted with different time horizons.

Theorems & Definitions (47)

• Remark 1.1
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Remark 2.4
• Definition 3.1
• Lemma 3.2
• Definition 3.3: Consistency, one-sided
• Definition 3.4: Consistency, two-sided
• Proposition 3.5