The Recoverability Gap in Coordination with Private Objectives

Accurate local optimization responses need not reveal the objective differences that matter for a shared decision. We define the recoverability gap as the excess expected loss of the best transcript policy relative to the best instance-aware planner under the same protocol. For strongly convex objectives, we identify changes that preserve every allowed local response yet alter the shared optimum. When response maps coincide, the gap is the cost of choosing one common allocation rather than instance-specific optima. It vanishes precisely when the instances share an optimum almost surely. The result covers continuous priors and boundary optima. Constrained sensitivity quantifies the loss from small hidden changes. Local messages can select an optimum within a known family or certify a proposed allocation without a candidate list. Controlled eight-party experiments test the predicted loss and recovery using the same posterior-optimal decision rule. In generated resource-allocation tasks without inserted hidden perturbations, a record search finds decision-relevant ambiguity in short transcripts. Greater tested depths remain unresolved. These results link the differences an interface conceals to the shared decisions it can support.