Stable marriage problems with quantitative preferences

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 quantitative 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 which are stable and/or optimal according to these notions. While expressivity greatly increases by adopting quantitative preferences, we show that in most cases the desired solutions can be found by adapting existing algorithms for the classical stable marriage problem.

💡 Research Summary

The paper extends the classic Stable Marriage Problem (SMP) by allowing participants to express quantitative preferences instead of merely ranking the opposite sex. In the traditional SMP each man and each woman provides a strict total order over the members of the opposite set, and a matching is called stable if there is no blocking pair—a man and a woman who both strictly prefer each other to their current partners. While this qualitative model is elegant, many real‑world applications (e.g., resident‑hospital assignments, school‑student placements, or any two‑sided market) involve scores that represent profit, cost, distance, or other numeric criteria.

Model definition
The authors introduce two n × n score matrices:

• (S = (s_{ij})) where (s_{ij}) is the score that man (i) assigns to woman (j).
• (T = (t_{ji})) where (t_{ji}) is the score that woman (j) assigns to man (i).
  Higher scores indicate stronger preference. Scores may be real numbers and ties are allowed, which is a departure from the strict ordering assumption of the classic model.

New stability notions
Because a simple “both prefer each other” condition becomes ambiguous when scores are involved, the authors propose two mathematically precise stability concepts:

1. Sum‑stability – A pair ((i,j)) blocks a matching (M) if
   \