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On Transversals and Systems of Distinct Representatives

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  • L. Hurwicz
  • S. Reiter

Abstract

A transversal generated by a system of distinct representatives (SDR) for a colection 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, and does not use transfinite induction. 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 econbomic 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 partiton. This is of interest because partitions have special properties of informational efficiency.

Suggested Citation

  • L. Hurwicz & S. Reiter, 1997. "On Transversals and Systems of Distinct Representatives," Discussion Papers 1176R, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    • Handle: RePEc:nwu:cmsems:1176r
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    References listed on IDEAS

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    1. Hurwicz, Leonid & Radner, Roy & Reiter, Stanley, 1975. "A Stochastic Decentralized Resource Allocation Process: Part I," Econometrica, Econometric Society, vol. 43(2), pages 187-221, March.
    2. Reichelstein, Stefan & Reiter, Stanley, 1988. "Game Forms with Minimal Message Spaces," Econometrica, Econometric Society, vol. 56(3), pages 661-692, May.
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