Algorithms for new types of fair stable matchings
Abstract
We study the problem of finding "fair" stable matchings in the Stable Marriage problem with Incomplete lists (SMI). For an instance $I$ of SMI there may be many stable matchings, providing significantly different outcomes for the sets of men and women. We introduce two new notions of fairness in SMI. Firstly, a regretequal stable matching minimises the difference in ranks of a worstoff man and a worstoff woman, among all stable matchings. Secondly, a minregret sum stable matching minimises the sum of ranks of a worstoff man and a worstoff woman, among all stable matchings. We present two new efficient algorithms to find stable matchings of these types: the RegretEqual Degree Iteration Algorithm finds a regretequal stable matching in $O(d_0 n^3)$ time, where $d_0$ is the absolute difference in ranks between a worstoff man and a worstoff woman in the manoptimal stable matching, and $n$ is the number of men or women; and the MinRegret Sum Algorithm finds a minregret sum stable matching in $O(d_s n^2)$ time, where $d_s$ is the difference in the ranks between a worstoff man in each of the womanoptimal and manoptimal stable matchings. Experiments to compare several types of fair optimal stable matchings were conducted and show that the RegretEqual Degree Iteration Algorithm produces matchings that are competitive with respect to other fairness objectives. On the other hand, existing types of "fair" stable matchings did not provide as close an approximation to regretequal stable matchings.
 Publication:

arXiv eprints
 Pub Date:
 January 2020
 arXiv:
 arXiv:2001.10875
 Bibcode:
 2020arXiv200110875C
 Keywords:

 Computer Science  Data Structures and Algorithms
 EPrint:
 12 page paper, 1 page of references, 17 pages of appendices, 7 figures, 11 tables