Social Choice and Welfare

, Volume 35, Issue 3, pp 415–433 | Cite as

Consistency in one-sided assignment problems

  • Bettina KlausEmail author
  • Alexandru Nichifor
Open Access
Original Paper


One-sided assignment problems combine important features of two well-known matching models. First, as in roommate problems, any two agents can be matched and second, as in two-sided assignment problems, the division of payoffs to agents is flexible as part of the solution. We take a similar approach to one-sided assignment problems as Sasaki (Int J Game Theory 24:373–397, 1995) for two-sided assignment problems, and we analyze various desirable properties of solutions including consistency and weak pairwise-monotonicity. We show that for the class of solvable one-sided assignment problems (i.e., the subset of one-sided assignment problems with a non-empty core), if a subsolution of the core satisfies [Pareto indifference and consistency] or [invariance with respect to unmatching dummy pairs, continuity, and consistency], then it coincides with the core (Theorems 1 and 2). However, we also prove that on the class of all one-sided assignment problems (solvable or not), no solution satisfies consistency and coincides with the core whenever the core is non-empty (Theorem 4). Finally, we comment on the difficulty in obtaining further positive results for the class of solvable one-sided assignment problems in line with Sasaki’s (1995) characterizations of the core for two-sided assignment problems.


Assignment Problem Pareto Optimality Solvable Problem Payoff Vector Assignment Game 
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We thank Çağatay Kayı, William Thomson, an anonymous referee, and an associate editor for helpful comments. We thank the Netherlands Organisation for Scientific Research (NWO) for its support under grant VIDI-452-06-013.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


  1. Eriksson K, Karlander J (2001) Stable outcomes of the roommate game with transferable utility. Int J Game Theory 29: 555–569CrossRefGoogle Scholar
  2. Gale D, Shapley LS (1962) College admissions and the stability of marriage. Am Math Mon 69: 9–15CrossRefGoogle Scholar
  3. Roth AE, Sotomayor MAO (1990) Two-sided matching: a study in game-theoretic modeling and analysis. Cambridge University Press, CambridgeGoogle Scholar
  4. Sasaki H (1995) Consistency and monotonicity in assignment problems. Int J Game Theory 24: 373–397CrossRefGoogle Scholar
  5. Shapley L, Shubik M (1972) The assignment game I: the core. Int J Game Theory 1: 111–130CrossRefGoogle Scholar
  6. Sotomayor M (2003) Some further remark on the core structure of the assignment game. Math Soc Sci 46(3): 261–265CrossRefGoogle Scholar
  7. Sotomayor M (2005) On the core of the one-sided assignment game. MimeoGoogle Scholar
  8. Talman DJJ, Yang Z (2008) A model of partnership formation. Center Discussion Paper Series No. 2008-103Google Scholar
  9. Thomson W (2009) Consistent allocation rules. Monograph (forthcoming)Google Scholar
  10. Toda M (2005) Axiomatization of the core of assignment games. Games Econ Behav 53: 248–261CrossRefGoogle Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Faculty of Business and EconomicsUniversity of LausanneLausanneSwitzerland
  2. 2.Department of EconomicsMaastricht UniversityMaastrichtThe Netherlands

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