Entangled vs. Separable Choice

A judge observes the joint probabilistic choice rule of two decision makers: the frequency of action pairs across pairs of local covariates. The rule is separable if behavior can be generated as if the decision makers were in separate rooms, unable to communicate at the time of choice. Separability allows arbitrary correlation in tastes, beliefs, information, and randomization devices; it rules out only covariate-dependent coordination. It is therefore the revealed-preference null of social independence, not statistical independence. We construct an exact judge for separability: a complete nonparametric characterization requiring no rationality, utility maximization, equilibrium, or parametric structure. In the binary-covariate, binary-action case, separability is equivalent to no-signaling plus Bell-type CHSH inequalities. In general finite domains, it is equivalent to a no-signaling extension to finitely many virtual replicas of one decision maker, yielding a finite system of linear restrictions. The judge matters because entangled choice rules can pass no-signaling tests while violating separability. We illustrate this in match rigging, plea bargaining, classroom cheating, and an LLM-based agentic system.