Paper: arXiv 2610.21867

Authors: Kaue Lopes de Moraes, Rodrigo De-Losso

Abstract

Independent tabular Q-learning agents repricing in a simulated Bertrand duopoly converge to supracompetitive prices, and the seminal evidence attributes this to a learned reward-punishment scheme: undercut your rival and it retaliates, so deviation never pays. This mechanistic reading is what turns high prices into a cartel, and into a regulatory target. We replicate the environment of Calvano et al. (2020) and confirm their impulse-response pattern within co-trained pairs. We then audit the pattern on frozen policies, with learning off, and find that on its own it does not identify a mechanism: the return is over-determined rather than selective. In nine of ten converged pairs (914 of 1,024) a single attractor’s basin is the entire state space, so every perturbation returns whatever provoked it, and a forced price increase triggers the same retaliatory pattern as an undercut. What co-training produces is a narrow, dyad-specific basin of attraction: re-match frozen agents from independent runs, as Calvano et al. did with learning on, and coordination collapses by 80% with nothing carried across. The deterrence claim is incomplete on its own terms: the one-period undercut of Calvano et al.’s impulse-response test is unprofitable, which we confirm in 92% of pairs, but the optimal deviation, computed by dynamic programming against each partner’s frozen strategy, profits by 17.5% on average. The scheme deters only the deviation no optimizer would choose; collusion survives because deviation is never tried, not because it would not pay. Supracompetitive prices are robust; the punishment mechanism read into them does not survive these tests, which changes what the standard test of algorithmic collusion establishes, and what pairing-disruption remedies can be expected to do.

Complexity vs Empirical Score

  • Math Complexity: 6.0/10
  • Empirical Rigor: 8.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper demonstrates strong empirical rigor through replication and novel auditing techniques. The mathematical underpinnings of Q-learning and dynamic programming are present, contributing to a high overall score due to its significant contribution to the algorithmic collusion debate.

Research Flowchart

  flowchart TD
    A[Research Goal: Audit Q-Learning Punishment Mechanism in Algorithmic Collusion] --> B{Key Methodology Steps};
    B -- Replicated Environment --> C[Calvano et al. (2020) Bertrand Duopoly Simulation];
    B -- Audited Frozen Policies --> D[Impulse-Response Test on Frozen Policies];
    B -- Rematched Frozen Agents --> E[Cross-Pairing of Independently Trained Agents];
    B -- Optimal Deviation Calculation --> F[Dynamic Programming for Optimal Deviation];
    D -- Identifies: No Selective Mechanism --> G{Key Findings/Outcomes};
    E -- Identifies: Collusion Collapse --> G;
    F -- Identifies: Optimal Deviation is Profitable --> G;
    G --> H[Punishment Mechanism is Incomplete: Supracompetitive Prices Robust, But Deterrence Claim is Flawed];