Encyclopedia of Algorithms

2016 Edition
| Editors: Ming-Yang Kao

Matching Market Equilibrium Algorithms

  • Ning ChenEmail author
  • Mengling Li
Reference work entry
DOI: https://doi.org/10.1007/978-1-4939-2864-4_788

Years and Authors of Summarized Original Work

  • 1971; Shapley, Shubik

  • 1982; Kelso, Crawford

  • 1986; Demange, Gale and Sotomayor

Problem Definition

The study of matching market equilibrium was initiated by Shapley and Shubik [13] in an assignment model. A classical instance of the matching market involves a set B of n unit-demand buyers and a set Q of m indivisible items, where each buyer wants to buy at most one item and each item can be sold to at most one buyer. Each buyer i has a valuation vij ≥ 0 for each item j, representing the maximum amount that i is willing to pay for item j. Each item j has a reserve price rj ≥ 0, below which it won’t be sold. Without loss of generality, one can assume there is a null item whose value is zero to all buyers and whose price is always zero.

An output of the matching market is a tuple (p, x), where p = (p1, , pm) ≥ 0 is a price vector with pj denoting the price charged for item j and x = (x1, , xn) ≥ 0 is an allocation vector with xidenoting the...

Keywords

Competitive equilibrium Matching market Maximum competitive equilibrium Minimum competitive equilibrium 
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Recommended Reading

  1. 1.
    Aggarwal G, Muthukrishnan S, Pal D, Pal M (2009) General auction mechanism for search advertising. In: WWW, Madrid, pp 241–250Google Scholar
  2. 2.
    Becker GS (1981) A treatise on the family. Harvard University Press, CambridgeGoogle Scholar
  3. 3.
    Chen N, Deng X (2011) On Nash dynamics of matching market equilibria. CoRR abs/1103.4196Google Scholar
  4. 4.
    Chen N, Deng X, Ghosh A (2010) Competitive equilibria in matching markets with budgets. ACM SIGecom Exch 9(1):1–5Google Scholar
  5. 5.
    Crawford VP, Knoer EM (1981) Job matching with heterogeneous firms and workers. Econometrica 49(2):437–450zbMATHCrossRefGoogle Scholar
  6. 6.
    Demange G, Gale D (1985) The strategy structure of two-sided matching markets. Econometrica 53(4):873–883MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Demange G, Gale D, Sotomayor M (1986) Multi-item auctions. J Pol Econ 94(4):863–872CrossRefGoogle Scholar
  8. 8.
    Gale D (1984) Equilibrium in a discrete exchange economy with money. Int J Game Theory 13:61–64MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Gale D, Shapley LS (1962) College admissions and the stability of marriage. Am Math Mon 69(1):9–15MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Kelso AS, Crawford VP (1982) Job matching, coalition formation, and gross substitutes. Econometrica 50(6):1483–1504MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Kuhn HW (1955) The Hungarian method for the assignment problem. Nav Reserv Logist Q 2(1):83–97MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Quinnzi M (1984) Core and competitive equilibrium with indivisibilities. Int J Game Theory 13:41–60CrossRefGoogle Scholar
  13. 13.
    Shapley LS, Shubik M (1971) The assignment game I: the core. Int J Game Theory 1(1):110–130MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Division of Mathematical SciencesSchool of Physical and Mathematical Sciences, Nanyang Technological UniversitySingaporeSingapore
  2. 2.Division of Mathematical SciencesNanyang Technological UniversitySingaporeSingapore