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Mitigating Emergent Collusion in LLM Pricing Agents

Abdullah Garra

arXiv:2609.13037v1cs.GT

TL;DR

The study asks whether regulation can reduce prompt-induced supracompetitive pricing by LLM pricing agents and which intervention channels perform best. It reproduces the P1/P2 effect with a lower-cost model and tests instruction, payoff, and market-structure interventions; active entry has the strongest reported outcome-level effect.

  • Problem

    The study asks whether regulatory interventions can reduce prompt-induced supracompetitive pricing by LLM pricing agents and which intervention channels perform best.

    Source: §2 · p. 3

  • Method

    The authors reproduce Fish et al.’s qualitative P1/P2 effect with a lower-cost model, then test instruction, expected-damages payoff, and active-entry interventions.

    Source: §2 · p. 4

  • Results

    Active entry has the strongest outcome-level effect, pushing both LLM firms below the random-entrant Nash benchmark; the Harrington regulator moves P1 close to Nash and reduces prompt sensitivity.

    Source: §7 · p. 10

  • Takeaways & Limitations

    In these experiments, interventions changing payoffs or market participation have stronger reported effects than prompt warnings alone.

    Source: §7 · p. 10

    The reproduction uses a lower-cost model because of computational-budget constraints, with the API-based setup serving as a practical fidelity–resource compromise.

    Source: §2 · p. 4, §4 · p. 5

Abstract

from arXiv · show

Recent work shows that LLM-based pricing agents can produce supracompetitive outcomes in repeated oligopoly environments without being explicitly instructed to collude. We reproduce the qualitative prompt-sensitivity effect of Fish et al. using DeepSeek-V3.1: the P1 prompt produces significantly higher prices and profits than P2, although our outcomes are less monopoly-like than the original GPT-4 results. We then evaluate three regulatory interventions: a prompt-only warning, a Harrington-inspired expected-damages payoff regulator, and an active random entrant. The prompt-only regulator reduces but does not eliminate above-Nash pricing. The Harrington regulator brings P1 outcomes close to the duopoly Nash benchmark and removes the statistically significant P1--P2 gap. The active entrant produces the strongest effect, pushing both prompts below the appropriate random-entrant Nash benchmark. Overall, our experiments provide preliminary evidence that interventions that alter incentives or market participation can reduce supracompetitive pricing more effectively than prompt warnings alone.

1 Algorithmic Pricing Collusion in LLMs

Fish et al. test whether pretrained LLM pricing agents can produce supracompetitive outcomes without explicit instructions to collude, and what mechanisms may sustain them. Their findings link prompt wording to pricing differences and identify reward-punishment-consistent behavior.

1 Algorithmic Pricing Collusion in LLMs

  • Fish et al. situate LLM pricing agents as a new test case because pretrained systems can be deployed without costly environment-specific training, yet are opaque and stochastic.They ask whether these agents produce collusive-like outcomes without being instructed to collude and what mechanisms drive them.Source: §1 · p. 1

1.1 Experimental Design

  • The experiments use independent LLM agents in repeated Bertrand markets, comparing two profit-maximizing prompt prefixes that differ in their emphasis on profitability versus exploration and undercutting.Agents receive market history and their prior plans to provide continuity across periods.Source: §1.1 · pp. 1–2

1.2 Findings

  • Both prompts produce supracompetitive prices and profits, but P1 prices are substantially higher than P2 prices and can exceed the monopoly price.Neither prompt instructs agents to collude; the authors argue wording changes the strategic frame rather than supplying new economic knowledge.Source: §1.2 · p. 2

1.3 Reward-Punishment and Textual Reasoning

  • Textual price-war concern is more common under P1, and implantation experiments find significantly higher prices after plans are replaced with price-war-avoidance reasoning.Regression evidence also shows positive responses to competitors’ previous prices and persistence in agents’ own prices, consistent with reward-punishment behavior; the authors caution that textual correlation alone does not establish causality.Source: §1.3 · p. 3

2 Study Overview

The study asks whether regulation can reduce prompt-induced supracompetitive pricing by LLM agents and which intervention channels perform best. It first reproduces the P1/P2 effect using a lower-cost model, then evaluates changes to instructions, payoffs, and market structure.

2 Study Overview

  • The study asks whether regulatory interventions can reduce prompt-induced supracompetitive pricing by LLM pricing agents and which intervention channels perform best.This follows from the concern that instructions and market histories can shape pricing behavior.Source: §2 · p. 3
  • The authors first seek to reproduce Fish et al.’s qualitative P1/P2 effect using a lower-cost model because of computational-budget constraints.Source: §2 · p. 4
  • After establishing a baseline, the study tests interventions that modify agent instructions, realized payoffs through expected damages, or market structure through an economically active additional participant.Source: §2 · p. 4

3 Related Work

The study draws on work on algorithmic pricing, reward-punishment mechanisms, and antitrust enforcement. These literatures motivate its focus on LLM pricing behavior and expected-damages regulation.

3 Related Work

  • Prior work shows that Q-learning agents can sustain supracompetitive prices, while Fish et al. extend the concern to pretrained, prompt-driven LLM agents.Other cited work examines sequential pricing and platform-level interventions for algorithmic sellers.Source: §3 · p. 4
  • Reward-punishment theory explains how firms may avoid profitable price cuts when deviations risk future price wars, motivating analogous analyses of LLM agents.The cited literature includes Stigler (1964), Friedman (1971), Green and Porter (1984), Harrington (2018), Calvano et al. (2020), and Fish et al. (2026).Source: §3 · p. 4
  • Harrington’s enforcement model distinguishes price-sensitive damages from fixed fines: damages accumulate with overcharge harm, and cartel prices decrease with the damage multiple and detection probability.The discussion also notes complementary work on audit-based regulation of pricing algorithms.Source: §3 · p. 4

4 Reproducing Fish et al.’s Results

The DeepSeek-V3.1 reproduction recovers the direction of Fish et al.’s P1/P2 effect at a weaker level, then examines textual reasoning and realized price responses consistent with reward-punishment behavior.

4.1 Experimental Design

  • The reproduction preserves Fish et al.’s repeated Bertrand-duopoly design and agent protocol, using DeepSeek-V3.1 after model screening and under API-cost constraints.The authors selected DeepSeek-V3.1 after it produced usable single-agent behavior; the API-based setup was a practical compromise given limited local compute.Source: §4 · pp. 4–5, §4.1 · p. 5

4.2 No-Regulator Reproduction Results

  • P1 prices and profits significantly exceed P2’s, but DeepSeek-V3.1 is less monopoly-like: 76% of P1 runs, versus 29% of P2 runs, exceed Nash.The normalized Nash and monopoly benchmarks are pNash/α = 1.473 and pM/α = 1.925.Source: Figure 1 · p. 5, §4.2 · pp. 5–6

4.3 Reward-Punishment Analyses

  • Of 1,610 price-war sentences, 73.5% of AvoidPriceWar sentences are from P1 and 69.7% of StartPriceWar sentences are from P2; clusters reflect P1 stability versus P2 exploration.The off-path analysis is descriptive; the authors did not perform implantation because it would require additional counterfactual simulations.Source: §4.3 · p. 6, Figure 2 · p. 7
  • With N = 2100 observations per prompt, competitor-price coefficients are positive and significant; P2 responds more, while P1 shows greater own-price stickiness.The regression uses periods 101–300 and follows Fish et al.’s specification.Source: §4.3 · pp. 6–7, Table 2 · p. 7
  • Because DeepSeek-V3.1 agents appear to plateau faster, the periods 101–300 regression may be dominated by post-convergence price persistence; Appendix C checks earlier windows.Source: §4.3 · p. 7

5 Regulatory Interventions

The section evaluates three regulatory channels—prompt warnings, expected-damages payoffs, and added market participation—under both P1 and P2.

5 Regulatory Interventions

  • The interventions target language, realized payoffs, or market structure, with the goal of moving outcomes toward the Bertrand–Nash benchmark and reducing the prompt-prefix gap.All interventions are evaluated under both P1 and P2.Source: §5 · pp. 7–8
  • The prompt-only regulator adds a monitoring warning while leaving the economic environment and profit calculation unchanged.Source: §5 · p. 8
  • The Harrington-inspired regulator penalizes persistent above-Nash pricing through expected damages based on overpricing, detection probability, and accumulated overcharge exposure.Agents observe realized profit but are not told the regulator’s formula.Source: §5 · p. 8
  • The active entrant adds a random price centered around the original duopoly Nash price; outcomes use the random-entrant Nash benchmark, not the duopoly or symmetric triopoly benchmark.The entrant is random rather than strategic and is not an aggressive undercutter.Source: §5 · p. 8

6 Results

At α = 1, prompt warnings reduce but do not eliminate above-Nash pricing, while the Harrington payoff regulator brings P1 close to duopoly Nash. Active entry produces the largest reduction, placing both prompts below the random-entrant Nash benchmark.

6 Results

  • Prompt-only regulation lowers P1’s collusion index from ∆ = 0.569 to ∆ = 0.242, but most P1 runs remain above Nash.The P1–P2 gap is no longer statistically significant under this treatment.Source: §6 · p. 8, Table 3 · p. 9
  • Without regulation, all seven P1 runs exceed duopoly Nash and one exceeds monopoly; P2 has only two of seven above Nash.Figures 3 and 4 show P1 mostly above Nash and nearer monopoly, while P2 is lower; their profit outcomes are similarly separated.Source: §6 · pp. 8–9, Figure 3 · p. 9, Figure 4 · p. 9
  • ∆ = −0.367 for P1 and −0.571 for P2 under active entry; no runs exceed random-entrant Nash, the largest reduction among treatments.The entrant samples prices around the previous duopoly Nash benchmark rather than aggressively undercutting.Source: §6 · pp. 8–9, Table 3 · p. 9
  • ∆ = 0.080 for Harrington-regulated P1, close to duopoly Nash; P1 remains +2.45% above Nash on average and faces a 4.74% average penalty share.Penalty pressure is concentrated during sustained above-Nash pricing and decays as prices return toward Nash.Source: §6 · pp. 8, 10
  • Across treatments, prompt warnings partially reduce prompt sensitivity, Harrington regulation disciplines the duopoly, and active entry exerts the strongest downward pressure.The figures show active-entry prices below random-entrant Nash and joint LLM profits substantially below those in duopoly treatments.Source: §6 · pp. 9–10, Figure 3 · p. 9, Figure 4 · p. 9

7 Conclusion

The conclusion finds that DeepSeek-V3.1 reproduces prompt-sensitive pricing, though less strongly than the original GPT-4 results. It contrasts prompt warnings with payoff incentives and market entry, while noting exploratory scope and implementation limits.

7 Conclusion

  • DeepSeek-V3.1 reproduces P1>P2 prompt sensitivity more weakly than GPT-4; Harrington regulation moves P1 near Nash, while active entry pushes both prompts below benchmark.The seven-run-per-condition, single-model experiments are exploratory; active entry changes demand allocation, and below-Nash pricing may overshoot an ideal regulatory target.Source: §7 · p. 10

A Prompt Prefixes Used in the Main Experiment

  • P1 and P2 share a long-run user-profit objective, but differ in their final instruction about pricing strategy.P1 emphasizes exploring strategies without undermining profitability; P2 explicitly allows risky or aggressive exploration and notes that lower prices can win sales.Source: A Prompt Prefixes Used in the Main… · p. 12

B Computation of Nash and Monopoly Benchmarks

  • In the symmetric two-firm setup, α scales profits but leaves the normalized best-response problem unchanged, so Nash and monopoly benchmarks are reported as normalized prices.This makes the normalized benchmarks applicable across currency-scale settings in the described environment.Source: B Computation of Nash and Monopoly… · p. 12

B.1 Bertrand–Nash benchmark

The section defines the one-shot Bertrand–Nash and joint-monopoly price benchmarks, then uses them to interpret a normalized collusion index.

B.1 Bertrand–Nash benchmark

  • The Bertrand–Nash benchmark is the symmetric Nash equilibrium of the one-shot pricing game, with the symmetric price satisfying xN = c + µ(1 − s(xN)).The first-order condition for a firm's best response is used to solve for the symmetric equilibrium price.Source: §B.1 · p. 13

B.2 Joint-monopoly benchmark

  • The joint-monopoly benchmark is the common price chosen to maximize total profit when one monopolist controls both firms, and it satisfies xM = c + µ(1 − 2s(xM)).The section solves for this benchmark numerically using the experimental parameters.Source: §B.2 · pp. 13–14

B.3 Collusion index

  • The collusion index locates the average normalized price between the Bertrand–Nash and joint-monopoly benchmarks; values above zero indicate supracompetitive pricing, while values near one indicate prices near monopoly.The average price is taken over the evaluation window.Source: §B.3 · p. 14

C Early-Window Reward-Punishment Regressions

Early-window regressions indicate stronger competitor responsiveness before prices stabilize, while the later window is more dominated by price persistence; the accompanying analysis distinguishes P1 and P2 reasoning patterns.

C Early-Window Reward-Punishment Regressions

  • In periods 2–150, competitor-lag coefficients are δ = 0.329 for P1 and δ = 0.526 for P2, larger than in periods 101–300, while own-price persistence dominates the later window.The early window captures price discovery; the later period-101–150 window preserves the full-history logic while isolating the adjustment phase.Source: C Early-Window Reward-Punishment… · pp. 14–15, Table 4 · p. 14
  • Across windows, P2 has the larger competitor-response coefficient, whereas P1 shows stronger price persistence and higher average prices in the no-regulator experiment.The authors interpret this as P1 being more supracompetitive in price levels while P2 is more reactive to competitor prices.Source: C Early-Window Reward-Punishment… · p. 15

D Fish et al.’s Textual Cluster Analysis

  • Fish et al.’s textual cluster analysis associates P1 with maintaining profitable prices and avoiding price wars, and P2 with undercutting, exploration, and quantity considerations.The analysis compares the relative prevalence of clusters in generated plan sentences under each prompt.Source: D Fish et al.’s Textual Cluster Analysis · p. 15

E Robustness to the Unit Parameter α

The appendix describes robustness and text-classification procedures, then details the payoff warning and market-entry interventions used to alter pricing incentives or participation.

E Robustness to the Unit Parameter α

  • Normalized no-regulator outcomes vary mainly between P1 and P2; within-prefix differences across α values are not statistically significant.The figure evaluates robustness across unit-parameter values.Source: Figure 6 · p. 15

E.1 Semantic Classifier for Price-War Sentences

  • Price-war sentences are classified by comparing sentence embeddings with avoidance and initiation/escalation reference directions; StartPriceWar also covers aggressive or reactive framing.Filtered sentences contain “price war” or “pricing war,” and cosine similarity determines the assigned label.Source: §E.1 · p. 16

F Details of the Harrington-Inspired Regulator

  • The Harrington-inspired regulator subtracts expected accumulated damages from realized profit, with detection responding to persistent overpricing and upward price movements.Its calibration uses a 25-period window and a 0.2 threshold; the authors describe the parameter values as a reasonable finite-horizon calibration, not theoretically optimal constants.Source: F Details of the Harrington-Inspired… · pp. 16–17

G Prompt-Only Regulator

  • The prompt-only intervention adds a warning that sustained supracompetitive pricing may trigger regulatory intervention or penalties.The warning is added to the agents’ prompt.Source: G Prompt-Only Regulator · p. 17

H Third-Party Market-Participant Regulator

  • The active entrant is a non-strategic third firm that draws a random price centered around the original duopoly Nash price and affects the LLM firms’ demand.Because entry changes demand allocation, the treatment requires a random-entrant Nash benchmark rather than the original duopoly benchmark.Source: H Third-Party Market-Participant… · p. 18

H.1 Nash and Monopoly Benchmarks for the Third-Party Market-Participant Regulator

The active-entrant treatment requires benchmarks that account for an exogenous third seller, and P1 trajectories show increasingly strong movement toward and below the relevant Nash benchmark across interventions. Textual analyses compare reasoning patterns across prompts and treatments.

H.1 Nash and Monopoly Benchmarks for the Third-Party Market-Participant Regulator

  • With an economically active random entrant, the relevant Nash benchmark is the symmetric one-shot equilibrium of the two strategic LLM firms facing that entrant, not the fully symmetric triopoly equilibrium.The section also defines a random-entrant monopoly benchmark using the expected joint profit of the two LLM firms while the entrant continues drawing prices randomly.Source: §H.1 · pp. 18–20

I P1 Price Trajectories Across Treatments

  • The active entrant produces the strongest P1 trajectory effect: prices remain above Nash without regulation, generally above Nash with the prompt warning, hover around Nash under Harrington, and fall below the random-entrant Nash benchmark with entry.The figure uses the duopoly Nash benchmark for NoReg, Prompt, and Harrington, and the random-entrant Nash benchmark for ActiveEntrant.Source: I P1 Price Trajectories Across… · p. 20, Figure 7 · p. 20

J Textual Analysis of Regulatory Treatments

  • The text analysis extracts plan sentences, classifies price-war references, and clusters all plan sentences; its reported patterns are descriptive rather than causal.Across regulators, explicit price-war language declines, with the prompt warning retaining P1 stability reasoning, Harrington shifting P1 reasoning toward Nash-like stability, and entry making price-war reasoning nearly balanced across prompts.Source: J Textual Analysis of Regulatory… · p. 21

J.1 No-Regulator Baseline

  • In the no-regulator baseline, P1 is more associated with avoiding price wars and stable price maintenance, while P2 is more associated with aggressive response, escalation, or price testing.Among price-war sentences, 73.5% of AvoidPriceWar sentences come from P1, while 69.7% of StartPriceWar sentences come from P2.Source: §J.1 · p. 21

J.2 Prompt-Only Regulator

  • The warning reduces explicit price-war language but does not remove P1’s stability-oriented reasoning; P2 remains more associated with contingency planning and aggressive response framing.P1-dominant clusters emphasize proven profitable equilibria and stability, while P2-dominant clusters emphasize testing prices and monitoring volume sensitivity.Source: §J.2 · pp. 21–22, Figure 8 · p. 22

J.3 Harrington Regulator

  • Under Harrington, price-war references decline and P1 clusters emphasize coordination near the duopoly Nash price, while P2 clusters emphasize testing and attempts to regain price advantage.The most P1-dominant cluster refers to stable coordination around $1.47, approximately the duopoly Nash benchmark.Source: §J.3 · pp. 22–23, Figure 9 · p. 23

J.4 Active Random Entrant

  • The active entrant yields the sharpest reduction in price-war language and nearly balanced prompt shares, with P1 clusters focused on strategy assessment and P2 clusters on testing and fallback rules.Only 160 sentences contain price-war references, compared with 1,610 in the no-regulator baseline and 670 under Harrington.Source: §J.4 · p. 23, Figure 10 · p. 24

K Harrington Regulator Diagnostic Metrics

The diagnostic metrics track above-Nash pricing and expected-penalty pressure; Figure 5 shows pressure concentrated in above-Nash regions and declining as prices return toward the competitive benchmark. The accompanying keyword check found little evidence of explicit regulator recognition.

K Harrington Regulator Diagnostic Metrics

  • The top panels measure average-price deviation from duopoly Nash; zero marks Nash, positive values above Nash, and negative values below Nash.The dotted monopoly reference is approximately 30.7% above Nash.Source: K Harrington Regulator Diagnostic… · p. 25
  • Duopoly Nash price is approximately 1.4729 and monopoly price 1.925, placing monopoly about 30.7% above Nash.Source: K Harrington Regulator Diagnostic… · p. 25
  • Bottom-panel penalty share normalizes total expected penalties by combined gross profit; a value of 5 means a 5% share, shown as a five-period rolling average.Source: K Harrington Regulator Diagnostic… · p. 25
  • Penalty pressure is concentrated in above-Nash regions and declines as prices return toward the competitive benchmark.The authors describe these diagnostics as a consistency check, not a causal estimate, because penalty pressure is endogenous.Source: K Harrington Regulator Diagnostic… · p. 25

L Explicit Regulator-Awareness Check

  • A keyword sanity check found little evidence agents explicitly recognized the hidden regulator or penalty formula; hits often concerned quantity losses, self-regulation, or false positives.The authors note that the search is not a full semantic analysis.Source: L Explicit Regulator-Awareness Check · p. 25, Table 6 · p. 26

M Robustness of On-Path Regressions

Across alternate windows, lagged-price regressions retain positive competitor responsiveness in baseline and prompt-only conditions, while active entry most clearly changes dynamics, especially for P1.

M Robustness of On-Path Regressions

  • Under active entry, P1's competitor-lag coefficient is near zero and insignificant in all windows; P2 remains positive and significant in periods 2–150 and 101–300, but not 101–150.Source: M Robustness of On-Path Regressions · p. 27
  • Baseline competitor responsiveness is positive and significant for both prompts in all windows, and the prompt-only regulator preserves sticky prices and positive responsiveness.Source: M Robustness of On-Path Regressions · p. 27
  • The Harrington regulator retains positive competitor responsiveness, but P1 is only marginally significant in periods 101–150 and significant in periods 101–300.The authors caution that positive responsiveness under Harrington need not indicate collusive punishment; prices may adjust jointly around a lower regulated region.Source: M Robustness of On-Path Regressions · p. 27
  • Table 7 distinguishes own-price stickiness γ from responsiveness to the other LLM firm's lagged price δ; under ActiveEntrant, δ refers to the other LLM firm, not the entrant.The regressions use periods 2–150, 101–150, and 101–300, with N = 525, 175, and 700 per treatment-prompt cell, respectively.Source: M Robustness of On-Path Regressions · pp. 27–28, Table 7 · p. 28
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