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Valuation Monotonicity, Fairness and Stability in Assignment Problems

Author

Listed:
  • Rene (J.R.) van den Brink

    (VU Amsterdam)

  • Marina Nunez

    (Universitat de Barcelona)

  • Francisco Robles

    (Universidad Carlos III de Madrid)

Abstract

In this paper, we investigate the possibility of having stable rules for two-sided markets with transferable utility, that satisfy some valuation monotonicity and fairness axioms. Valuation fairness requires that changing the valuation of a buyer for the object of a seller leads to equal changes in the payoffs of this buyer and seller. This is satisfied by the Shapley value, but is incompatible with stability. A main goal in this paper is to weaken valuation fairness in such a way that it is compatible with stability. It turns out that requiring equal changes only for buyers and sellers that are matched to each other before as well as after the change, is compatible with stability. In fact, we show that the only stable rule that satisfies weak valuation fairness is the well-known fair division rule which is obtained as the average of the buyers-optimal and the sellers-optimal payoff vectors. Our second goal is to characterize these two extreme rules by valuation monotonicity axioms. We show that the buyers-optimal (respectively sellers-optimal) stable rule is characterized as the only stable rule that satisfies buyer-valuation monotonicity which requires that a buyer cannot be better off by weakly decreasing his/her valuations for all objects, as long as he is assigned the same object as before (respectively object-valuation antimonotonicity which requires that a buyer cannot be worse off when all buyers weakly decrease their valuations for the object that is assigned to this specific buyer, as long as this buyer is assigned the same object as before). Finally, adding a consistency axiom, the two optimal rules are characterized in the general domain of allocation rules for two-sided assignment markets with a variable population.

Suggested Citation

  • Rene (J.R.) van den Brink & Marina Nunez & Francisco Robles, 2018. "Valuation Monotonicity, Fairness and Stability in Assignment Problems," Tinbergen Institute Discussion Papers 18-071/II, Tinbergen Institute.
    • Handle: RePEc:tin:wpaper:20180071
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    References listed on IDEAS

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    1. René van den Brink, 2002. "An axiomatization of the Shapley value using a fairness property," International Journal of Game Theory, Springer;Game Theory Society, vol. 30(3), pages 309-319.
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    Cited by:

    1. Domènech, Gerard & Núñez, Marina, 2022. "Axioms for the optimal stable rules and fair-division rules in a multiple-partners job market," Games and Economic Behavior, Elsevier, vol. 136(C), pages 469-484.
    2. Solymosi, Tamás, 2023. "Sensitivity of fair prices in assignment markets," Mathematical Social Sciences, Elsevier, vol. 126(C), pages 1-12.
    3. Gerard Domènech Gironell & Marina Núñez Oliva, 2022. "Axioms for the optimal stable rules and fair-division rules in a multiple-partners job market," UB School of Economics Working Papers 2022/419, University of Barcelona School of Economics.

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    More about this item

    Keywords

    assignment problems; stability; valuation monotonicity; valuation fairness; fair division rule; optimal rules;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • C78 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Bargaining Theory; Matching Theory

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