Many-to-One Stable Matching: Geometry and Fairness

Abstract

characterized the stable admissions polytope using a system of linear inequalities. The structure of feasible solutions to this system of inequalities -fractional stable matchings-is the focus of this paper. The main result associates a geometric structure with each fractional stable matching. This insight appears to be interesting in its own right, and can be viewed as a generalization of the lattice structure (for integral stable matchings) to fractional stable matchings. In addition to obtaining simple proofs of many known results, the geometric structure is used to prove the following two results: first, it is shown that assigning each agent their "median" choice among all stable partners results in a stable matching, which can be viewed as a "fair" compromise; second, sufficient conditions are identified under which stable matchings exist in a problem with externalities, in particular, in the stable matching problem with couples.

Median Stable Matching for College Admissions

“…The main result of this section is a very simple proof that for any college admissions market, generalized median stable matchings are well-defined and stable (Theorem 3.2). In contrast to Sethuraman et al (2004), who used a linear programming approach, our proof is based on the lattice structure of the set of stable matchings. Proceeding from the existence of generalized median stable matchings, we define the subset of median stable matchings and discuss their fairness properties (Definition 3.5, Remark 3.6, and Example 3.7).…”

“…Similarly, the l-th college optimal generalized median stable matching is defined by function α C l that assigns all colleges to their l-th (weakly) best match among all k stable matchings. 6 Sethuraman et al (2004) used linear programming tools to prove the following theorem. We give a simple proof of this result by exploiting the lattice structure of the set of stable matchings.…”

“…When the number of stable matching is odd, Teo and Sethuraman (1998) and Sethuraman et al (2004) proved the existence of the so-called median stable matching for the marriage model and the college admissions model, respectively, i.e., giving each student and each college the 'median' match in the set of stable matches yields again a stable matching. Next, independently of the number of stable matching being odd or even, we define 'median stable matchings.…”

“…Using another approach, Teo and Sethuraman (1998) and Sethuraman et al (2004) established the existence of natural deterministic 'compromising mechanisms' for marriage and college admissions models respectively. Specifically, they showed that if all agents order their (possibly non-distinct) matches at the, say, k stable matchings from best to worst, then the map that assigns to each agent of one side of the market its l-th best match and to each agent of the other side its (k − l + 1)-st best match constitutes a stable matching.…”

“…Specifically, they showed that if all agents order their (possibly non-distinct) matches at the, say, k stable matchings from best to worst, then the map that assigns to each agent of one side of the market its l-th best match and to each agent of the other side its (k − l + 1)-st best match constitutes a stable matching. Teo and Sethuraman (1998) and Sethuraman et al (2004) used linear programming tools to prove that these '(generalized) median stable matchings' are indeed well-defined and stable. We use the term '(generalized) median' to emphasize not only the formal equivalence of this solution concept to (generalized) medians in voting theory, but also to its similar spirit of compromise (Moulin, 1980, Barberà et al, 1993.…”