Valuation Monotonicity, Fairness and Stability in Assignment Problems

Tinbergen Institute Discussion Paper 2018-071/II

22 Pages Posted: 28 Sep 2018

See all articles by Rene van den Brink

Rene van den Brink

Vrije Universiteit Amsterdam, School of Business and Economics

Marina Núñez

University of Barcelona

Francisco Robles

University of Barcelona

Multiple version iconThere are 2 versions of this paper

Date Written: July 19, 2018

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.

Keywords: assignment problems, stability, valuation monotonicity, valuation fairness, fair division rule, optimal rules

JEL Classification: C71, C78

Suggested Citation

van den Brink, Rene and Núñez, Marina and Robles, Francisco, Valuation Monotonicity, Fairness and Stability in Assignment Problems (July 19, 2018). Tinbergen Institute Discussion Paper 2018-071/II, Available at SSRN: https://ssrn.com/abstract=3246539 or http://dx.doi.org/10.2139/ssrn.3246539

Rene Van den Brink (Contact Author)

Vrije Universiteit Amsterdam, School of Business and Economics ( email )

De Boelelaan 1105
Amsterdam, 1081HV
Netherlands

Marina Núñez

University of Barcelona ( email )

Gran Via de les Corts Catalanes, 585
Barcelona, 08007
Spain

Francisco Robles

University of Barcelona ( email )

Gran Via de les Corts Catalanes, 585
Barcelona, 08007
Spain

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