510K: When-to-Cooperate under Latent Team Relationships and the Other-Regarding Value of Information

Agents acting with others must first decide whether and with whom to cooperate, often by inferring relationships from observed behaviour. We study this when-to-cooperate problem in a setting where the relationship is learnable in principle yet unused in practice. We introduce 510K, a four-player card game in which team membership is hidden at deal time and the reward itself depends on the unknown relationship. We separate four questions that benchmarks conflate — availability, decodability, utilization, and value — and report a consistent pattern. (1) The relationship is decodable but unused. 67% of decisions are logically determined by public play and a history probe reaches AUC 0.72/1.00, yet 1M-step PPO never conditions on the team: actions change in only 0.5–3.6% of decisions when the team bits are zeroed, and adding more evidence, explicit revelation, stronger win incentives, or an undiscounted objective changes nothing. (2) Information value is other-regarding. Informing the agent is inert; informing partners moves outcomes — team-aware partners raise the agent's first-place rate by +0.066 (p=0.007, replicated on a fresh deal set), adding team-awareness to both teams lowers the agent's reward relative to the ignorant condition, and training against aware bots yields behavioural adaptation without reciprocity. Small language models reproduce the same gap, and a full-information search (first-place 0.43–0.48 vs. the best policy's 0.21–0.36) shows large headroom. We release the environment, the evaluation protocol, and belief-labelled trajectories as an instrument for when-to-cooperate decisions.