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

## TL;DR

This paper extends the study of strategy-proof stable mechanisms to the stable marriage problem with ties and incomplete lists, proposing approximation algorithms that are strategy-proof and optimal within certain bounds.

## Contribution

It introduces the first known strategy-proof approximation algorithms for the stable marriage problem with ties and incomplete lists, including tight bounds for various cases.

## Key findings

- Existence of 2-approximate man- and woman-strategy-proof mechanisms for MAX SMTI.
- A 2-approximate woman-strategy-proof mechanism for MAX SMTI-1TM.
- A 1.5-approximate man-strategy-proof mechanism for MAX SMTI-1TM.

## Abstract

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.