Advertisement

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

Chapter
  • 113 Downloads
Part of the Studies in Economic Design book series (DESI)

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aharoni, R., Nash-Williams, C. St. J.A., Shelah S. (1983) A general criterion for the existence of transversals. Proceedings London Mathematical Society, 47: 43–68CrossRefGoogle Scholar
  2. 2.
    Berge, C. (1963) Topological Spaces. Macmillan, New YorkGoogle Scholar
  3. 3.
    Brualdi, R.A., Scrimgen, E.B. (1968) Exchange systems, matchings and transversals. Journal of Combinatorial Theory 5: 242–257CrossRefGoogle Scholar
  4. 4.
    Damerell, R.M., Milner, E.C. (1974) Necessary and sufficient conditions for transversals of countable set systems. Journal of Combinatorial Theory Series A 17: 350–379CrossRefGoogle Scholar
  5. 5.
    Dugundji, J. (1966) Topology. Allyn and Bacon, Inc., BostonGoogle Scholar
  6. 6.
    Everett, C.J., Whaples, G. (1949) Representations of sequences of sets. American Journal of Mathematics 71: 287–293CrossRefGoogle Scholar
  7. 7.
    Folkman, K.J. (1968) Transversals of Infinite Families With Finitely Many Infinite Members. RAND Corp. memorandum, RM — 5676 — PRGoogle Scholar
  8. 8.
    Gale, D., Shapley, L. (1962) College Admissions and the Stability of Marriage. American Mathematical Monthly 69: 9–15CrossRefGoogle Scholar
  9. 9.
    Hall, M., Jr. (1948) Distinct representatives of subsets. Bulletin American Mathematics Society 54: 922–926CrossRefGoogle Scholar
  10. 10.
    Hall, P. (1935) On representatives of subsets. Journal of London Mathematic Society 10: 26–30CrossRefGoogle Scholar
  11. 11.
    Hurwicz, L. (1976) Mathematical Models in Economics. Papers and Proceedings of a U.S. — U.S.S.R. Seminar, MoscowGoogle Scholar
  12. 12.
    Hurwicz, L., Radner, R., Reiter S. (1975) A stochastic decentralized allocation process: Part I. Econometrica 43 (2): 187–221CrossRefGoogle Scholar
  13. 13.
    Hurwicz, L., Radner, R., Reiter S. (1975) A stochastic decentralized resource allocation process: Part II. Econometrica 43 (3): 363–393CrossRefGoogle Scholar
  14. 14.
    Hurwicz, L., Reiter S. (1990) Constructing Decentralized Mechanisms by the Method of Rectangles. Decentralization Conference, Northwestern UniversityGoogle Scholar
  15. 15.
    Hurwicz, L., Reiter S. (1993) Designing Mechanisms by the `Method of Rectangles’. Decentralization Conference, University of California, BerkeleyGoogle Scholar
  16. 16.
    Kelso, A.S., Jr., Crawford, V.P. (1982) Job matching, coalition formation, and gross substitutes. Econometrica 50: 1483–1504CrossRefGoogle Scholar
  17. 17.
    Mirsky, L. (1971) Transversal Theory. Science and Engineering 75, Academic Press, New York and LondonGoogle Scholar
  18. 18.
    Nash-Williams, C. St. J.A. (1978) Another criterion for marriage in denumerable societies. Annals Discrete Mathematics 3: 165–179CrossRefGoogle Scholar
  19. 19.
    Podewski, K.P., Steffens, K. (1976) Injective choice functions for countable families. Journals of Combination Theory Series B 21: 40–46CrossRefGoogle Scholar
  20. 20.
    Radner, R. (1972a) Normative theories of organization: an introduction. In: McGuire, C.C., Radner, R. (eds.) Decision and Organization. North Holland / American Elsevier, pp. 177–188Google Scholar
  21. 21.
    Radner, R. (1972b) Teams. In: McGuire, C.C., Radner, R. (eds.) Decision and Organization. North Holland / American Elsevier, pp. 189–216Google Scholar
  22. 22.
    Radner, R. (1972c) Allocation of a scarce resource under uncertainty: an example of a team. In: McGuire, C.C., Radner, R. (eds.) Decision and Organization. North Holland / American Elsevier, pp. 217–236Google Scholar
  23. 23.
    Reichelstein, S. (1984) Incentive compatibility and informational requirements. Journal of Economic Theory 32: 384–390CrossRefGoogle Scholar
  24. 24.
    Reichelstein, S., Reiter, S. (1988) Game forms with minimal message spaces. Econometrica 56 (3): 661–692CrossRefGoogle Scholar
  25. 25.
    Rado, R. (1967) Note on the transfinite case of Hall’s theorem on representatives. Journal of London Mathematic Society 42: 321–324CrossRefGoogle Scholar
  26. 26.
    Roth, A.E. (1984) The evolution of the labor market for medical interns and residents: A case study in game theory. Journal of Political Economy 92: 991–1016Google Scholar
  27. 27.
    Roth, A.E., Sotomayor, M. (1990) Two-sided matching: A study in game-theoretic modeling and analysis. Cambridge University Press, CambridgeGoogle Scholar
  28. 28.
    Shelah, S. (1973) Notes on partition calculus. In: Hajinal, A., Rado, R., Sos, V.T. (eds.) Infinite and Finite Sets. (Colloq. Math. Soc. Janos Bolyai ) 10: 1257–1276Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Department of EconomicsUniversity of MinnesotaMinneapolisUSA
  2. 2.Center for Mathematical Studies in Economics and Management SciencesNorthwestern UniversityEvanstonUSA

Personalised recommendations