Recent work shows that pricing with symmetric LLM agents leads to algorithmic collusion. We show that collusion is fragile under the heterogeneity typical of real deployments. In a stylized repeated-pricing model, heterogeneity in patience or data access reduces the set of collusive equilibria. Experiments with open-source LLM agents (totaling over 2,000 compute hours) align with these predictions: patience heterogeneity reduces price lift from 22% to 10% above competitive levels; asymmetric data access, to 7%. Increasing the number of competing LLMs breaks up collusion; so does cross-algorithm heterogeneity, that is, setting LLMs against Q-learning agents. But model-size differences (e.g., 32B vs. 14B weights) do not; they generate leader-follower dynamics that stabilize collusion. We discuss antitrust implications, such as enforcement actions restricting data-sharing and policies promoting algorithmic diversity.
Figures & tables
Experiments
Collusion strength
Price elevation
(Low/Med./High)
(% vs. competitive)
Homogeneous Benchmarks
Two myopic LLMs (no collusion emerges)
Low
≈0%
Two patient LLMs (tacit collusion emerges)
High
+22%
Main Heterogeneity dimensions (predicted by model)
1 patient LLM agent vs. 1 myopic LLM agent
Medium
+10%
Table 1: Results at a Glance: Collusion Outcomes under Agent Heterogeneity.
Variable
Definition
N
Number of sellers
a
Vertical differentiation parameter
b
Own-price sensitivity parameter
d
Cross-price sensitivity parameter
μ
Price elasticity parameter
a0
Outside option parameter
Table 2: Summary of Key Notation
Theory
Experiment
Interpretation in practice
Discount factor δ
Prompted optimization horizon
Governance decisions
Monitoring precision ρ
Access to rival price history
Data infrastructure differences
Grim-trigger strategies
Not imposed
Test if punishment emerges endogenously
Linear demand
MNL demand
More realistic demand model
Table 3: Mapping from Theory to Experiment
Figure 1: Experimental workflow of two LLM agents (Section 3.3.1 ). In each period, each agent re-prompts its LLM with (i) its previous pricing strategy and (ii) recorded market history to generate a price. The market receives the prices and calculates quantity demanded and profit for each agent. Each agent’s history records both sellers’ prices and its own quantities demanded and profits for the last 100 periods.
Heterogeneity
# of
# of
Max. Periods
Approximate
Dimension
Conditions
Runs
Per Run
Runtimes
Patience levels (Section 4.1 – 4.2 )
3
30
1,000
≈ 238 hrs
Data access (Section 4.3 )
1
10
1,000
≈ 73 hrs
Algorithm type (Section 5.1 )
2
20
1,000
≈ 520 hrs
Number of sellers (Section 5.2 )
3
30
1,000
≈ 1,184 hrs
Model size (Section 5.3 )
1
10
1,000
≈ 220 hrs
Table 4: Experimental Coverage Across All Heterogeneity Dimensions
Condition
Rounds to
Avg. Price
% Price Elevation
Convergence (SD)
(SD)
vs. Competitive
Homogeneous Benchmarks (Section 4.1 )
Two Myopic LLMs
159.0 (26.0)
1.472 (0.012)
≈0%
Two Patient LLMs
195.1 (28.8)
1.801 (0.027)
+22%
Main Heterogeneity Dimensions (predicted by model)
Patient vs. Myopic LLMs (§ 4.2 )
140.8 (29.0)
1.619 (0.023)
+10%
Table 5: Summary of Experimental Results
Figure 2: Representative price dynamics under homogeneous LLM competition. Panel (a): high patience agents ( δ=0.95 ); prices satisfy convergence criterion by period 176, at ≈22% above the competitive benchmark ( pC ). Panel (b): low patience agents ( δ=0 ); prices converge near competitive benchmark ( pC ) by period 150. Dashed lines indicate the theoretical competitive ( pC ) and monopoly ( pM ) prices.
Figure 3: Representative price dynamics under heterogeneous LLM competition. Panel (a): one patient agent ( δ=0.95 ) versus one myopic agent ( δ=0 ) (Section 4.2 ); prices satisfy the convergence criterion by period 103. Panel (b): one high-data agent versus one low-data agent (Section 4.3 ); prices satisfy the convergence criterion by period 162. In both cases, price elevation is reduced (but remains positive) relative to the homogeneous benchmark in Figure 2 (a).
Figure 4: Representative price dynamics under heterogeneous cross-algorithm competition. Panel (a): one patient LLM agent ( δ=0.95 ) versus one frozen Q-learning agent ( δ=0.95 ) (Section 5.1 ); prices do not satisfy the convergence criterion within 1,000 periods. Panel (b): one patient LLM agent ( δ=0.95 ) versus one adaptive Q-learning agent ( δ=0.95 , α=0.15 , β=0.004 ) (Section 5.1 ); prices do not satisfy the convergence criterion within 1,000 periods. Dashed lines indicate the theoretical competitive ( pC ) and monopoly ( pM ) prices.
Figure 5: Representative price dynamics as the number of homogeneous LLM sellers increases (Section 5.2 ). Panels (a)–(c): three, four, and five equally patient agents, respectively; dashed lines indicate the theoretical competitive ( pC ) and monopoly ( pM ) prices. The price convergence criterion is met by period 196 (three sellers) and period 531 (four sellers), but is not met within 1,000 periods (five sellers). Panel (d): average time to satisfy the price convergence criterion across 10 runs for each market size (capped at 1,000 periods).
Figure 6: Representative price dynamics under model-size heterogeneous LLM competition (Section 5.3 ). One 14B agent ( δ=0.95 ) competes against one 32B agent ( δ=0.95 ); prices satisfy the convergence criterion by period 445. Dashed lines indicate the theoretical competitive ( pC ) and monopoly ( pM ) prices.
Condition
Rounds to
Avg. Price
% Price Elevation
Convergence (SD)
(SD)
vs. Competitive
Homogeneous Benchmarks (Section 4.1 )
Two Myopic LLMs
419.0 (216.0)
1.046 (0.045)
−29%
Two Patient LLMs
227.8 (134.5)
2.312 (0.029)
+57%
Main Heterogeneity Dimensions (predicted by model)
Patient vs. Myopic LLMs (§ 4.2 )
194.2 (133.9)
1.999 (0.032)
+36%
Table 6: Summary of Robustness Results (GPT-5-mini)
Appendix figures & tables3 assets
Supplementary material from the paper’s appendix.
Appendix
Experiment
Valid Output
Correct Pricing
Bertrand Duopoly
50/50, 50/50
46/50, 42/50
Monopoly
50/50, 50/50
48/50, 48/50
Appendix
Table 7: One-Shot Game Performance
Figure 7: Prices chosen by a monopolist LLM agent (Section B.2 ) compared to the theoretical monopoly price pM , over 300 periods. The parameters are a=2 , μ=0.25 , c=1 , a0=0 .
Condition
Rounds to
Avg. Price
% Price Elevation
Convergence (SD)
(SD)
vs. Competitive
Homogeneous Benchmarks (Section 4.1 )
Two Myopic LLMs
419.0 (216.0)
1.046 (0.045)
−29%
Two Patient LLMs
227.8 (134.5)
2.312 (0.029)
+57%
Main Heterogeneity Dimensions (predicted by model)
Patient vs. Myopic LLMs (§ 4.2 )
194.2 (133.9)
1.999 (0.032)
+36%
Appendix
Table 8: Summary of Robustness Results (GPT-5-mini)
We conduct experiments with algorithmic pricing agents based on Large Language Models (LLMs). In oligopoly settings, LLM-based pricing agents quickly and autonomously reach supracompetitive prices and profits. Variation in seemingly innocuous phrases in LLM instructions ("prompts") substantially influence the degree of supracompetitive pricing. We develop novel techniques for behavioral analysis of LLMs and use them to uncover price-war concerns as a contributing factor. Our results extend to auction settings. Our findings uncover unique challenges to any future regulation of LLM-based pricing agents, and AI-based pricing agents more broadly.
Sara Fish, Yannai A. Gonczarowski, Ran I. Shorrer
†School of Engineering and Applied Sciences, Harvard University · ‡Department of Economics and Department of Computer Science, Harvard University · §Department of Economics, Penn State University
Large language models (LLM) deployed as autonomous pricing agents may sustain supracompetitive prices through tacit coordination. We develop a causal graph divergence framework that separately measures structural faithfulness and intent faithfulness of LLM pricing agents in Bertrand competition. Across nine LLMs under duopoly and triopoly conditions, collusive behavior and chain-of-thought (CoT) faithfulness dissociate along both dimensions: the most collusive model accurately reports cooperative intent yet reasons structurally unfaithfully, while the most structurally faithful model sustains supra-Nash pricing under both market structures. These findings establish that CoT monitoring alone cannot serve as a standalone safeguard against algorithmic collusion.
Dohun Lee, Hyunwoo Park
Graduate School of Data Science, Seoul National University
Evaluations of large language model (LLM) agents report that pricing agents collude at others' expense. Yet collusion is defined by enforcement: a cartel, in which competitors share monopoly profits, holds only if a member who cheats is punished. We introduce an audit that tests enforcement by forcing a member to cheat in counterfactual copies of a market, and use it to study what reinforcement learning (RL) teaches agents in pricing and trading. Small LLMs trained by RL learn what their training rivals reward, but only what they readily try. In self-play, they learn fixed rules ignoring their rivals; against a punishing rival, to be deterred; against a cheating rival, to match its cuts, never the costly price war that would deter it, which they rarely try. A war forced into training is learned, but mostly only where it was forced. Against their own copies, deterred and matching agents can earn the same cartel profits, but only matching ones can enforce them, and then thinly: as a by-product and only against large cuts. Q-learners enforce collusion only in pricing. Prompted LLMs reach cartel-like outcomes mostly through rules that ignore the market. Outcome measures cannot tell these cases apart; the audit can.
Hieu Tran, Hong Yu
University of Massachusetts Amherst · University of Massachusetts Lowell