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Algorithms 2013, 6(4), 782-804; doi:10.3390/a6040782

Stability, Optimality and Manipulation in Matching Problems with Weighted Preferences

1,* , 2
1 Department of Information Engineering, University of Padova, Padova 35131, Italy 2 Department of Mathematics, University of Padova, Padova 35121, Italy 3 Department of Computer Science, Tulane University and IHMC, New Orleans, LA 70118, USA 4 NICTA and UNSW, Sydney, NSW 1466, Australia
* Author to whom correspondence should be addressed.
Received: 9 July 2013 / Revised: 7 November 2013 / Accepted: 8 November 2013 / Published: 18 November 2013
(This article belongs to the Special Issue Special Issue on Matching under Preferences)
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The stable matching problem (also known as the stable marriage problem) is a well-known problem of matching men to women, so that no man and woman, who are not married to each other, both prefer each other. Such a problem has a wide variety of practical applications, ranging from matching resident doctors to hospitals, to matching students to schools or, more generally, to any two-sided market. In the classical stable marriage problem, both men and women express a strict preference order over the members of the other sex, in a qualitative way. Here, we consider stable marriage problems with weighted preferences: each man (resp., woman) provides a score for each woman (resp., man). Such problems are more expressive than the classical stable marriage problems. Moreover, in some real-life situations, it is more natural to express scores (to model, for example, profits or costs) rather than a qualitative preference ordering. In this context, we define new notions of stability and optimality, and we provide algorithms to find marriages that are stable and/or optimal according to these notions. While expressivity greatly increases by adopting weighted preferences, we show that, in most cases, the desired solutions can be found by adapting existing algorithms for the classical stable marriage problem. We also consider the manipulability properties of the procedures that return such stable marriages. While we know that all procedures are manipulable by modifying the preference lists or by truncating them, here, we consider if manipulation can occur also by just modifying the weights while preserving the ordering and avoiding truncation. It turns out that, by adding weights, in some cases, we may increase the possibility of manipulating, and this cannot be avoided by any reasonable restriction on the weights.
Keywords: stable matching; weighted preferences; manipulation stable matching; weighted preferences; manipulation
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Pini, M.S.; Rossi, F.; Venable, K.B.; Walsh, T. Stability, Optimality and Manipulation in Matching Problems with Weighted Preferences. Algorithms 2013, 6, 782-804.

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