Telling Collusion from Common Knowledge: State-Conditioned Net Influence for LLM Agents

Collusion between AI agents may run through channels an auditor cannot read, so detectors that work from actions alone are valuable. But agents that act alike are notnecessarily colluding: they may be reacting to the same public information, the "parallel conduct" that antitrust law refuses to treat as evidence of agreement. We give the first evaluation of pairwise net influence (Bonjour et al., 2022), an informationtheoretic collusion statistic, on language-model agents, and test whether it can make this distinction. Four Qwen2.5-7B-Instruct agents play 60-round, four-player RockPaper-Scissors under five conditions (100 games). Net influence finds a covertly colluding pair in 19 of 20 games, while an LLM judge reading the same logs finds none. When two innocent agents follow the same public rule (83% of rounds), a calibrated correlation detector flags them in 19 of 20 games; net influence conditioned on a coarse public state flags 2 of 20 and never the innocent pair (McNemar p =1.5e-5). We also show that the original fixed threshold is unsafe at LLM-scale sample sizes (12-32% false alarms at 60 rounds), and introduce a per-game permutation test; combined with state conditioning it keeps false alarms at 2.5-6.5% in every scripted setting we tried, including fully predictable players. State conditioning costs power (14 vs 20 of 20 covert pairs), so we recommend it as a second-stage test that separates collusion from common knowledge.