Private-Response Coordination: Information Limits and Feasible Decisions

A coordinator can query each party's allocation at a price without observing its objective. We study when these responses suffice to produce and certify a feasible joint decision. We construct coupled strongly convex objectives with different optima that admit the same error-bounded vector response at every price. Their two-world minimax regret equals the conditional-error budget, even with arbitrary adaptive queries and unlimited memory. A comparison family separately gives sharp query, state, and ambiguity bounds. We then certify feasible candidates from response positions, public smoothness, and local error bounds without fitting their objectives. On identical transcripts and recovered averages, the position certificate verifies target accuracy after a mean of $8.6$ and $20.3$ high-fidelity calls for quadratic and log-cosh objectives. Neither recovery-based certificate does so within $512$ calls. Coupled nonlinear studies show how response fitting and validation complement each other. The results distinguish irreducible response ambiguity from avoidable certificate conservatism.