Online Appendix to When are Local Incentive Constraints Sufficient?

This online appendix gives a more detailed study of conditions under which the basic method of proof used for the sufficiency results in the main paper can be applied, with an eye to understanding how much the method might potentially be further generalized, and whether the results still hold when the method does not apply. We restrict ourselves to cardinal type spaces and no transfers, as in Subsection 3.1. All of the proofs of sufficiency results in the main paper follow the general method of showing that the linear inequality corresponding to any desired incentive constraint can be obtained by adding up inequalities corresponding to local incentive constraints. We show here that for finite type spaces, whenever a set S of incentive constraints is sufficient, there exists a proof of sufficiency by adding up (Lemma OA-1 below). Moreover, with minor exceptions, whenever an incentive constraint (u, v) is provable by adding up, there exists such a proof that uses only types along the line segment [u, v], or types cardinally equivalent to them (Proposition OA-1). The conclusion, then, is that for finite type spaces, there exist essentially no sufficiency results beyond those that can be proven using the method of Proposition 1. On the other hand, for infinite type spaces, the conclusions are not as tight. We give an example (Proposition OA-2) of a type space where local incentive constraints are sufficient, but sufficiency cannot be proven by adding up. In that example, we prove sufficiency by a combination of adding-up arguments and limiting arguments exploiting the compactness of the space ∆(X). To begin the investigation, we must first be precise about what it means for an incentive constraint to be provable by adding up other constraints. Let T be a cardinal