Ex-post Implementation and Preference Aggregation via Potentials

We introduce several notions of potentials for mechanism design problems with interdependent values, and relate them to implementation in ex-post equilibrium. Whereas ex-post implementation is closely linked to the ordinal concept of best-alternative potentials, the celebrated Vickrey-Clarke-Groves mechanism corresponds to the stronger notion of a cardinal potential: agents agree not only on the best alternative, but also on the quantitative di¤erences between all alternatives. We characterize all valuations that allow for cardinal potentials, and use this characterization for: 1) Identifying valuations for which ex-post implementation is possible; 2) Identifying classes of valuations for which all ex-post implementable choice rules correspond to cardinal potentials. The latter allows us to extend to interdependent valuations a result for dominant strategy implementation in private values settings, due to Roberts (1979).