Characterizing Incentive Compatibility for Convex Valuations

We study implementability in dominant strategies of social choice functions when types are multi-dimensional, sets of outcomes are arbitrary, valuations for outcomes are convex functions in the type, and utilities over outcomes and payments are quasi-linear. Archer and Kleinberg [1] have proven that in case of convex sets of types and linear valuation functions monotonicity in combination with locally disappearing line integrals on triangles are equivalent with implementability. We generalize this characterization from linear to convex valuation functions. We also get rid of a technical assumption which they had to make in the case of linear valuations. Saks and Yu [7] have shown that for the same setting but finite set of outcomes, monotonicity alone is sufficient for implementability. Later Archer and Kleinberg [1], Monderer [4] and Vohra [8] have given alternative proofs for the same theorem. Using our characterization, we show that the Saks and Yu theorem generalizes to convex valuations. Thereby we provide yet another, but very short proof for the special case of linear valuations. All our results are stated and proven in terms of single agent models. The characterization result for arbitrary set of outcomes immediately generalizes to a characterization of dominant strategy implementable rules as well as Bayes-Nash implementable rules in the case of multiple agents. The generalization of the Saks and Yu theorem for finite set of outcomes carries over to dominant strategy implementation in the case of multiple agents.