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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aharoni, R., Nash-Williams, C. St. J.A., Shelah S. (1983) A general criterion for the existence of transversals. Proceedings London Mathematical Society, 47: 43–68
Berge, C. (1963) Topological Spaces. Macmillan, New York
Brualdi, R.A., Scrimgen, E.B. (1968) Exchange systems, matchings and transversals. Journal of Combinatorial Theory 5: 242–257
Damerell, R.M., Milner, E.C. (1974) Necessary and sufficient conditions for transversals of countable set systems. Journal of Combinatorial Theory Series A 17: 350–379
Dugundji, J. (1966) Topology. Allyn and Bacon, Inc., Boston
Everett, C.J., Whaples, G. (1949) Representations of sequences of sets. American Journal of Mathematics 71: 287–293
Folkman, K.J. (1968) Transversals of Infinite Families With Finitely Many Infinite Members. RAND Corp. memorandum, RM — 5676 — PR
Gale, D., Shapley, L. (1962) College Admissions and the Stability of Marriage. American Mathematical Monthly 69: 9–15
Hall, M., Jr. (1948) Distinct representatives of subsets. Bulletin American Mathematics Society 54: 922–926
Hall, P. (1935) On representatives of subsets. Journal of London Mathematic Society 10: 26–30
Hurwicz, L. (1976) Mathematical Models in Economics. Papers and Proceedings of a U.S. — U.S.S.R. Seminar, Moscow
Hurwicz, L., Radner, R., Reiter S. (1975) A stochastic decentralized allocation process: Part I. Econometrica 43 (2): 187–221
Hurwicz, L., Radner, R., Reiter S. (1975) A stochastic decentralized resource allocation process: Part II. Econometrica 43 (3): 363–393
Hurwicz, L., Reiter S. (1990) Constructing Decentralized Mechanisms by the Method of Rectangles. Decentralization Conference, Northwestern University
Hurwicz, L., Reiter S. (1993) Designing Mechanisms by the `Method of Rectangles’. Decentralization Conference, University of California, Berkeley
Kelso, A.S., Jr., Crawford, V.P. (1982) Job matching, coalition formation, and gross substitutes. Econometrica 50: 1483–1504
Mirsky, L. (1971) Transversal Theory. Science and Engineering 75, Academic Press, New York and London
Nash-Williams, C. St. J.A. (1978) Another criterion for marriage in denumerable societies. Annals Discrete Mathematics 3: 165–179
Podewski, K.P., Steffens, K. (1976) Injective choice functions for countable families. Journals of Combination Theory Series B 21: 40–46
Radner, R. (1972a) Normative theories of organization: an introduction. In: McGuire, C.C., Radner, R. (eds.) Decision and Organization. North Holland / American Elsevier, pp. 177–188
Radner, R. (1972b) Teams. In: McGuire, C.C., Radner, R. (eds.) Decision and Organization. North Holland / American Elsevier, pp. 189–216
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. 217–236
Reichelstein, S. (1984) Incentive compatibility and informational requirements. Journal of Economic Theory 32: 384–390
Reichelstein, S., Reiter, S. (1988) Game forms with minimal message spaces. Econometrica 56 (3): 661–692
Rado, R. (1967) Note on the transfinite case of Hall’s theorem on representatives. Journal of London Mathematic Society 42: 321–324
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: 991–1016
Roth, A.E., Sotomayor, M. (1990) Two-sided matching: A study in game-theoretic modeling and analysis. Cambridge University Press, Cambridge
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: 1257–1276
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Hurwicz, L., Reiter, S. (2003). Transversals, systems of distinct representatives, mechanism design, and matching. In: Ichiishi, T., Marschak, T. (eds) Markets, Games, and Organizations. Studies in Economic Design. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24784-5_10
Download citation
DOI: https://doi.org/10.1007/978-3-540-24784-5_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-53465-2
Online ISBN: 978-3-540-24784-5
eBook Packages: Springer Book Archive