Transversals, systems of distinct representatives, mechanism design, and matching

Abstract.

A transversal generated by a system of distinct representatives (SDR) for a collection of sets consists of an element from each set (its representative) such that the representative uniquely identifies the set it belongs to. Theorem 1 gives a necessary and sufficient condition that an arbitrary collection, finite or infinite, of sets, finite or infinite, have an SDR. The proof is direct, short. A Corollary to Theorem 1 shows explicitly the application to matching problems. In the context of designing decentralized economic mechanisms, it turned out to be important to know when one can construct an SDR for a collection of sets that cover the parameter space characterizing a finite number of economic agents. The condition of Theorem 1 is readily verifiable in that economic context. Theorems 2–5 give different characterizations of situations in which the collection of sets is a partition. This is of interest because partitions have special properties of informational efficiency.