Net Influence Meets Language-Model Agents: Detecting a Colluding Pair from Actions Alone

Pairwise net influence (Bonjour et al., 2022) flags two agents as colluding when each influences the other's actions more than any third agent does, measured by conditional mutual information. It assumes exactly two colluders and had not been tested on language-model agents. We implement it with unit tests, replicate the original Rock–Paper–Scissors results (matching the published accuracy table to 2.6 points on average once two unstated details are inferred), and apply it to a four-seller repeated market game played by Qwen2.5-1.5B-Instruct agents, in which one seeded pair runs a market-allocation cartel through a covert per-round message. Across 150 LLM games (15 conditions × 10 seeds) with negative and scripted positive controls, net influence ranked the true pair first in 0.91 [0.86, 0.96] of cartel runs, against 0.89 for Cramér's V and 0.11 for a frozen-prompt LLM judge; a variant that uses the agents' exact action probabilities reached 0.95. The "net" step adds calibration rather than ranking power: with a rich state it cuts fixed-threshold false alarms in RPS from 100% to 0–23%, and conditional MI without it matched its ranking accuracy in every condition. Conditioning on public state is the key ingredient: it defeats an observed common cause that fools plain correlation (RPS top-1 1.00 vs 0.07), but it makes a cartel keyed on public information invisible (0/10 vs 10/10). Net influence also fails under hidden common causes, near-deterministic policies and more than one cartel. Action-level dependence is a cheap, useful screen for LLM collusion, not proof of agreement.