Introduction

Algorithmic pricing has become ubiquitous: ride-share platforms reprice every few seconds, e-commerce bots adjust millions of listings daily, and airlines update fares in real time. Each seller's algorithm is trained independently and without explicit communication. Yet economists and regulators have raised a troubling question: can independent learning algorithms spontaneously sustain supra-competitive prices—tacit collusion—without any human coordination?

The Federal Trade Commission, the European Commission, and the UK Competition and Markets Authority have each opened investigations into algorithmic pricing practices[ezrachi2016virtual, harrington2018developing]. A key obstacle to enforcement is the absence of a shared, reproducible detection methodology. Traditional collusion-detection tools assume human decision-makers who communicate; algorithmic agents leave no such paper trail.

Calvano et al.[calvano2020artificial] provided the foundational empirical result: tabular Q-learning agents in a symmetric Bertrand duopoly consistently converge to supra-competitive prices, sustaining collusion through learned punishment strategies. Their finding sparked a literature examining conditions under which algorithmic collusion emerges[klein2021autonomous, banchio2022artificial] and how it might be detected[brown2023competition].

We make three contributions:

• A parametric simulation engine covering three realistic market presets (e-commerce, ride-share, commodity), five agent algorithms, variable memory lengths ($M \in {1, 3, 5}$), and two shock types.
• A multi-agent auditor panel with four complementary detection methods and three aggregation strategies, calibrated against known-competitive and known-collusive baselines.
• An agent-executable skill (SKILL.md) that encodes the full experimental pipeline so that any AI coding agent can reproduce all 324 simulations and analyses from scratch.

Market Model

Logit Bertrand Competition

We model $N$ sellers competing in a repeated Bertrand game with logit demand. At each round $t$, seller $i$ sets a price $p_i^t$ on a discrete grid of $K = 15$ prices. Market shares follow the multinomial logit: $$s_i(\mathbf{p}) = \frac{\exp(-\alpha p_i)}{\sum_{j=1}^{N} \exp(-\alpha p_j)},$$ where $\alpha > 0$ is the price sensitivity parameter. Seller $i$'s per-round profit is: $$\pi_i = (p_i - c_i) \cdot s_i(\mathbf{p}),$$ with marginal cost $c_i$ (normalized so $c_i = 1.0$ in symmetric markets). Prices are expressed as unitless markup ratios: $p = 1.0$ is break-even, $p = 1.5$ is a 50% markup.

The one-shot Nash equilibrium price $p^$p∗ and the joint monopoly price $p^m$ serve as lower and upper benchmarks, respectively, and are computed analytically for each configuration. We define the collusion index: $$\Delta = \frac{\bar{p} - p^$$}{p^m - p^*} \in [0, 1], where $\bar{p}$ is the mean price in the final 20% of training rounds. $\Delta = 0$ indicates competitive pricing; $\Delta = 1$ indicates full monopoly pricing.

Market Presets

Three domain presets capture qualitatively different competitive environments:

Market presets. α controls price sensitivity; higher α means more commodity-like competition.

Pricing Agents

Five agent types span the range from adaptive learners to game-theoretic benchmarks:

• Q-learner: Tabular Q-learning with $\varepsilon$-greedy exploration and discount factor $\delta = 0.95$. The baseline from Calvano et al., known to collude under standard parameters.
• SARSA: On-policy TD learning. More conservative update rule; expected to collude less aggressively than Q-learning.
• Policy Gradient (PG): REINFORCE with softmax action selection and a running-average baseline. Gradient-based exploration dynamics differ qualitatively from tabular methods.
• Tit-for-Tat (TFT): Matches the competitor's price from the previous round. A classical game-theoretic benchmark that can sustain cooperation without learning.
• Competitive: Always prices at the analytically computed Nash equilibrium. The non-collusive control group.

Agents observe only the public price history of the last $M$ rounds (state space $K^{N \times M}$). For $M \geq 3$, tile coding reduces the effective state space to a tractable number of tiles (8 tilings, 16 tiles per dimension). Exploration decays linearly from $\varepsilon = 1.0$ to $\varepsilon = 0.01$ over the first 40% of training rounds ($T = 200,000$ for Q-learning and SARSA matchups; $T = 100,000$ for others).

Auditor Panel

Detection Methods

Four auditors independently analyze the same price history and each produce a collusion score $\sigma_a \in [0, 1]$:

Auditor 1: Price-Cost Margin (PCM). Computes the average markup $(\bar{p} - c)/c$ in the final 20% of rounds and maps it onto the Nash--Monopoly spectrum to obtain $\sigma_\text{PCM}$. Strength: simple and interpretable. Weakness: cannot distinguish collusion from slow convergence.

Auditor 2: Deviation-Punishment (DP). Scans the price history for the tacit-collusion signature: high-price phase $\to$ unilateral deviation $\to$ punishment response $\to$ recovery. Uses a sliding window detector calibrated on known-competitive baselines. Strength: detects the mechanism, not just the outcome. Weakness: requires enough deviation events for statistical power.

Auditor 3: Counterfactual Simulator (CF). Loads agent policies at round $0.9T$, then re-simulates the final $0.1T$ rounds with one agent replaced by the Competitive (Nash-pricing) bot. Measures the resulting price destabilization. Strength: provides causal evidence. Weakness: computationally expensive; requires saving agent state mid-run.

Auditor 4: Welfare Analyst (WA). Computes consumer surplus (CS), producer surplus (PS), and deadweight loss relative to Nash and monopoly benchmarks: $$\sigma_\text{WA} = \frac{\text{CS}^* - \overline{\text{CS}}}{\text{CS}^* - \text{CS}^m},$$ where $\text{CS}^*$ is Nash consumer surplus and $\text{CS}^m$ is monopoly consumer surplus. Strength: directly measures consumer harm. Weakness: welfare loss can stem from inefficiency rather than collusion.

Panel Aggregation

Three aggregation strategies are reported:

• Majority vote: Collusion flagged if $\geq 3$ of 4 auditors score $> 0.5$.
• Weighted average: Scores weighted by empirical reliability estimated on calibration runs.
• Unanimous: All 4 auditors must agree (most conservative; minimizes false positives).

False positive rates are calibrated on runs with the Competitive control agent; false negative rates are calibrated on known-collusive Q-learner $\times$ Q-learner runs at $M = 5$.

Experiments and Results

Experimental Design

Table summarizes the full factor matrix. The 324 simulations span all combinations; each (agent matchup $\times$ memory $\times$ preset $\times$ shock) cell is replicated with 3 random seeds, and we report mean $\pm$ standard deviation for all metrics. Statistical significance of supra-competitive pricing is assessed with one-sample $t$-tests against the Nash price, with Bonferroni correction over all 108 conditions ($\alpha_{\text{corrected}} = 0.05 / 108$); the collusion index $\Delta$ (Eq.) is the primary effect size measure.

Experimental factor matrix. 6 × 3 × 3 × 2 × 3 = 324 total simulations.

Collusion Heatmap

Table shows the mean collusion index $\Delta$ for the e-commerce preset. Values in bold exceed $\Delta > 0.1$; negative values indicate prices below Nash equilibrium.

Mean collusion index \Delta (e-commerce preset, no shock, mean over 3 seeds). Bold: \Delta > 0.1.

Auditor Agreement

Table reports pairwise auditor agreement rates (fraction of simulations where both auditors agree on collusion/no-collusion). All three panel aggregation methods are evaluated; F1 is computed against $\Delta > 0.1$ as ground truth.

Pairwise auditor agreement (no-shock runs, n = 162) and panel F1 vs. \Delta > 0.1 ground truth.

Memory Effect

We assess how collusion intensity scales with memory length $M$ for learning-agent matchups (QQ, SS, QS). The $M = 1 \to M = 3$ transition produces a clear jump in collusion index: Q vs. Q goes from $\Delta = 0.071$ to $\Delta = 0.109$; SARSA vs. SARSA from $0.012$ to $0.111$. Averaged across all presets, Q/SARSA matchups reach mean $\Delta = 0.121$ at $M = 1$ and $\Delta = 0.151$ at $M = 3$, a 25% increase. This transition is also where statistical significance emerges under raw $t$-tests ($p < 0.005$ at $M \geq 3$ for QQ and SS, vs. $p > 0.05$ at $M = 1$ for e-commerce and commodity presets). The policy implication is direct: if longer memory windows reliably produce more collusion, regulators should scrutinize the lookback window of deployed pricing algorithms.

Shock Robustness

A cost shock at round $0.6T$ (one seller's cost increases by 30%) and a demand shock at round $0.8T$ ($\alpha$ shifts 20%) are injected into half the simulations. We measure whether supra-competitive pricing persists after shocks. For Q vs. Q at $M = 3$ (e-commerce), mean prices are 1.703 without shocks and 1.702 with shocks, a difference of $< 0.1$. Across all learning-agent matchups, the shock/no-shock average price difference is less than 0.5%, indicating that once collusive pricing patterns are established they are resilient to moderate cost and demand perturbations. Auditor detection rates are unchanged between shock and no-shock conditions.

Discussion

Policy implications. Q-learner $\times$ Q-learner results replicate Calvano et al.[calvano2020artificial] across all three market presets, confirming that tacit collusion is a robust property of independent Q-learning in repeated pricing games, not an artifact of a specific parameterization. The auditor panel provides regulators with a principled detection toolkit: the Counterfactual Simulator offers causal evidence closest to a "but-for" price analysis, while the Welfare Analyst directly quantifies consumer harm. The memory effect result implies that auditing deployed algorithms for their lookback window length could serve as a practical regulatory screen: the $M = 1 \to M = 3$ transition is where statistically significant supra-Nash pricing emerges.

Limitations. This framework uses tabular and linear function approximation agents; deep RL agents (e.g., DQN, PPO) may exhibit qualitatively different collusion dynamics that warrant separate study. The logit demand model is stylized: it assumes homogeneous consumers, no product differentiation beyond price, and no entry or exit. Market presets are calibrated to plausible parameter ranges but are not fitted to real transaction data. The counterfactual auditor requires saving agent state mid-run, which may not be feasible for proprietary black-box pricing systems in practice.

Future work. Natural extensions include: (i) deep RL agents with shared-encoder architectures, (ii) heterogeneous consumer populations, (iii) fitting parameters to real-world pricing datasets, (iv) multi-market collusion where the same agent operates across products, and (v) adversarial auditing—agents that are aware of the detection panel and actively try to evade it.

\section*{Reproducibility}

All code, data, and results are packaged as an executable SKILL.md. Any AI coding agent can reproduce the full 324-simulation experiment by following the skill steps: set up the virtual environment, run unit tests, execute run.py, and validate with validate.py. No API keys, GPU, or external data downloads are required; the framework is a pure Python simulation with pinned dependencies.

References

• [calvano2020artificial] E. Calvano, G. Calzolari, V. Denicolò, and S. Pastorello, "Artificial Intelligence, Algorithmic Pricing, and Collusion," American Economic Review, vol. 110, no. 10, pp. 3267--3297, 2020.

• [ezrachi2016virtual] A. Ezrachi and M. E. Stucke, Virtual Competition: The Promise and Perils of the Algorithm-Driven Economy. Harvard University Press, 2016.

• [harrington2018developing] J. E. Harrington, "Developing Competition Law for Collusion by Autonomous Artificial Agents," Journal of Competition Law & Economics, vol. 14, no. 3, pp. 331--363, 2018.

• [klein2021autonomous] T. Klein, "Autonomous Algorithmic Collusion: Q-Learning Under Sequential Pricing," The RAND Journal of Economics, vol. 52, no. 3, pp. 538--558, 2021.

• [banchio2022artificial] M. Banchio and G. Mantegazza, "Artificial Intelligence and Spontaneous Collusion," Working Paper, 2022.

• [brown2023competition] Z. Y. Brown and A. MacKay, "Competition in Algorithmic Pricing," The Review of Economic Studies, 2023.

• [mnih2015human] V. Mnih et al., "Human-level Control through Deep Reinforcement Learning," Nature, vol. 518, pp. 529--533, 2015.