New Characterizations of Core Imputations of Matching and b-Matching Games

Author Vijay V. Vazirani

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Vijay V. Vazirani
  • University of California, Irvine, CA, USA


I wish to thank Hervé Moulin for asking the interesting question of extending results obtained for the assignment game to general graph matching games having a non-empty core. I also wish to thank Federico Echenique, Hervé Moulin and Thorben Trobst for several valuable discussions.

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Vijay V. Vazirani. New Characterizations of Core Imputations of Matching and b-Matching Games. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 28:1-28:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We give new characterizations of core imputations for the following games: 1) The assignment game. 2) Concurrent games, i.e., general graph matching games having non-empty core. 3) The unconstrained bipartite b-matching game (edges can be matched multiple times). 4) The constrained bipartite b-matching game (edges can be matched at most once). The classic paper of Shapley and Shubik [Shapley and Shubik, 1971] showed that core imputations of the assignment game are precisely optimal solutions to the dual of the LP-relaxation of the game. Building on this, Deng et al. [Deng et al., 1999] gave a general framework which yields analogous characterizations for several fundamental combinatorial games. Interestingly enough, their framework does not apply to the last two games stated above. In turn, we show that some of the core imputations of these games correspond to optimal dual solutions and others do not. This leads to the tantalizing question of understanding the origins of the latter. We also present new characterizations of the profits accrued by agents and teams in core imputations of the first two games. Our characterization for the first game is stronger than that for the second; the underlying reason is that the characterization of vertices of the Birkhoff polytope is stronger than that of the Balinski polytope.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algorithmic game theory
  • LP-duality theory
  • cooperative game theory
  • core of a game
  • assignment game
  • general graph matching game
  • bipartite b-matching game


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  1. Garrett Birkhoff. Three observations on linear algebra. Univ. Nac. Tacuman, Rev. Ser. A, 5:147-151, 1946. Google Scholar
  2. Péter Biró, Walter Kern, and Daniël Paulusma. Computing solutions for matching games. International journal of game theory, 41(1):75-90, 2012. Google Scholar
  3. Christopher P Chambers and Federico Echenique. The core matchings of markets with transfers. American Economic Journal: Microeconomics, 7(1):144-64, 2015. Google Scholar
  4. Gabrielle Demange, David Gale, and Marilda Sotomayor. Multi-item auctions. Journal of political economy, 94(4):863-872, 1986. Google Scholar
  5. Xiaotie Deng, Toshihide Ibaraki, and Hiroshi Nagamochi. Algorithmic aspects of the core of combinatorial optimization games. Mathematics of Operations Research, 24(3):751-766, 1999. Google Scholar
  6. Kimmo Eriksson and Johan Karlander. Stable outcomes of the roommate game with transferable utility. International Journal of Game Theory, 29(4):555-569, 2001. Google Scholar
  7. David Gale and Lloyd S Shapley. College admissions and the stability of marriage. The American Mathematical Monthly, 69(1):9-15, 1962. Google Scholar
  8. L. Lovász and M.D. Plummer. Matching Theory. North-Holland, Amsterdam-New York, 1986. Google Scholar
  9. Hervé Moulin. Cooperative microeconomics: a game-theoretic introduction, volume 313. Princeton University Press, 2014. Google Scholar
  10. Marina Núñez and Carles Rafels. On the dimension of the core of the assignment game. Games and Economic Behavior, 64(1):290-302, 2008. Google Scholar
  11. Lloyd S Shapley and Martin Shubik. The assignment game I: The core. International Journal of Game Theory, 1(1):111-130, 1971. Google Scholar
  12. Vijay V Vazirani. New characterizations of core imputations of matching and b-matching games. arXiv preprint, 2022. URL: