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Regulation of Algorithmic Collusion

Jason D. Hartline, Sheng Long, Chenhao Zhang

TL;DR

The paper tackles the risk that pricing algorithms in competitive markets may yield supra-competitive outcomes and proposes a regulator-friendly, data-driven framework to audit algorithmic non-collusion. It defines plausible non-collusion through unilateral, information-compatible, and optimization-based properties grounded in calibrated vanishing regret, connecting dynamic learning to static correlated equilibria. A concrete instantiation uses an empirical propensity-score test to estimate calibrated regret from logged price and demand data, derives a finite-sample complexity bound, and discusses audit-compatibility and robustness. The work enables regulators to assess algorithmic pricing without access to proprietary code, providing a practical path toward enforcing non-collusion through data-driven verification and highlighting avenues for reducing statistical complexity in future work.

Abstract

Consider sellers in a competitive market that use algorithms to adapt their prices from data that they collect. In such a context it is plausible that algorithms could arrive at prices that are higher than the competitive prices and this may benefit sellers at the expense of consumers (i.e., the buyers in the market). This paper gives a definition of plausible algorithmic non-collusion for pricing algorithms. The definition allows a regulator to empirically audit algorithms by applying a statistical test to the data that they collect. Algorithms that are good, i.e., approximately optimize prices to market conditions, can be augmented to contain the data sufficient to pass the audit. Algorithms that have colluded on, e.g., supra-competitive prices cannot pass the audit. The definition allows sellers to possess useful side information that may be correlated with supply and demand and could affect the prices used by good algorithms. The paper provides an analysis of the statistical complexity of such an audit, i.e., how much data is sufficient for the test of non-collusion to be accurate.

Regulation of Algorithmic Collusion

TL;DR

The paper tackles the risk that pricing algorithms in competitive markets may yield supra-competitive outcomes and proposes a regulator-friendly, data-driven framework to audit algorithmic non-collusion. It defines plausible non-collusion through unilateral, information-compatible, and optimization-based properties grounded in calibrated vanishing regret, connecting dynamic learning to static correlated equilibria. A concrete instantiation uses an empirical propensity-score test to estimate calibrated regret from logged price and demand data, derives a finite-sample complexity bound, and discusses audit-compatibility and robustness. The work enables regulators to assess algorithmic pricing without access to proprietary code, providing a practical path toward enforcing non-collusion through data-driven verification and highlighting avenues for reducing statistical complexity in future work.

Abstract

Consider sellers in a competitive market that use algorithms to adapt their prices from data that they collect. In such a context it is plausible that algorithms could arrive at prices that are higher than the competitive prices and this may benefit sellers at the expense of consumers (i.e., the buyers in the market). This paper gives a definition of plausible algorithmic non-collusion for pricing algorithms. The definition allows a regulator to empirically audit algorithms by applying a statistical test to the data that they collect. Algorithms that are good, i.e., approximately optimize prices to market conditions, can be augmented to contain the data sufficient to pass the audit. Algorithms that have colluded on, e.g., supra-competitive prices cannot pass the audit. The definition allows sellers to possess useful side information that may be correlated with supply and demand and could affect the prices used by good algorithms. The paper provides an analysis of the statistical complexity of such an audit, i.e., how much data is sufficient for the test of non-collusion to be accurate.
Paper Structure (22 sections, 11 theorems, 74 equations, 1 figure)

This paper contains 22 sections, 11 theorems, 74 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related literature
  3. Preliminaries
  4. Dynamic Imperfect Price Competition
  5. Learning Problem of Sellers
  6. Collusive Equilibria in Repeated Games
  7. Framework for Regulating Collusion
  8. Empirical Propensity Score Test
  9. Conclusion
  10. Proofs
  11. Theorem \ref{['thm:pscore-sample-complexity']}
  12. Upper Tail
  13. Lower Tail
  14. Main Result of Theorem \ref{['thm:pscore-sample-complexity']}
  15. Skipped Algebraic Steps
  16. ...and 7 more sections

Key Result

Theorem 2.1

For a seller $i$ with cost $c_i$, there exists an online algorithm such that:

Figures (1)

  • Figure 1: Geometric illustration of the example at a particular round. Two sellers post prices $p_1$ and $p_2$ respectively. The buyer buys from Seller 1 if his valuation $\text{v} \equiv (v_1, v_2)$ (represented by the black dot), lies in the region $A_1$. He buys from Seller 2 if $\text{v}$ lies in the $A_2$. The buyer buys from neither seller if $\text{v}$ lies in the region $\emptyset$.

Theorems & Definitions (35)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.1: blum2007externalstoltz2005incomplete
  • Definition 2.4
  • Definition 2.5
  • Corollary 2.2
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • ...and 25 more