Strategy-Proof Approximation Algorithms for the Stable Marriage Problem with Ties and Incomplete Lists

In the stable marriage problem (SM), a mechanism that always outputs a stable matching is called a stable mechanism. One of the well-known stable mechanisms is the man-oriented Gale-Shapley algorithm (MGS). MGS has a good property that it is strategy-proof to the men's side, i.e., no man can obtain a better outcome by falsifying a preference list. We call such a mechanism a man-strategy-proof mechanism. Unfortunately, MGS is not a woman-strategy-proof mechanism. Roth has shown that there is no stable mechanism that is simultaneously man-strategy-proof and woman-strategy-proof, which is known as Roth's impossibility theorem. In this paper, we extend these results to the stable marriage problem with ties and incomplete lists (SMTI). Since SMTI is an extension of SM, Roth's impossibility theorem takes over to SMTI. Therefore, we focus on the one-sided-strategy-proofness. In SMTI, one instance can have stable matchings of different sizes, and it is natural to consider the problem of finding a largest stable matching, known as MAX SMTI. Thus we incorporate the notion of approximation ratio used in the theory of approximation algorithms. We say that a stable-mechanism is c-approximate-stable mechanism if it always returns a stable matching of size at least 1/c of a largest one. We also consider a restricted variant of MAX SMTI, which we call MAX SMTI-1TM, where only men's lists can contain ties. Our results are summarized as follows: (i) MAX SMTI admits both a man-strategy-proof 2-approximate-stable mechanism and a woman-strategy-proof 2-approximate-stable mechanism. (ii) MAX SMTI-1TM admits a woman-strategy-proof 2-approximate-stable mechanism. (iii) MAX SMTI-1TM admits a man-strategy-proof 1.5-approximate-stable mechanism. All these results are tight in terms of approximation ratios. Also, all these strategy-proofness results apply for strategy-proofness against coalitions.

• 7 publications
• 9 publications
• 4 publications
research
05/11/2020

On the Approximability of the Stable Matching Problem with Ties of Constant Size up to the Integrality Gap

Finding a stable matching is one of the central problems in algorithmic ...
research
07/07/2021

Deep Learning for Two-Sided Matching

We initiate the use of a multi-layer neural network to model two-sided m...
research
05/11/2020

Approximating Stable Matchings with Ties of Bounded Size

Finding a stable matching is one of the central problems in algorithmic ...
research
05/07/2021

Maximally Satisfying Lower Quotas in the Hospitals/Residents Problem with Ties

Motivated by a serious issue that hospitals in rural areas suffer from s...
research
11/16/2018

Understanding popular matchings via stable matchings

Let G = (A ∪ B, E) be an instance of the stable marriage problem with st...
research
04/09/2019

Stability-Preserving, Incentive-Compatible, Time-Efficient Mechanisms for Increasing School Capacity

We address the following dynamic version of the school choice question: ...
research
08/28/2023

Complementarities in childcare allocation under priorities

We investigate the allocation of children to childcare facilities and pr...