[Paper Review] Coalition manipulations of the Gale-Shapley algorithm

Yuan Deng, Weiran Shen+1 more|arXiv (Cornell University)|Feb 27, 2015

Game Theory and Voting Systems3 citations

TL;DR

This paper studies coalition manipulations in the Gale-Shapley algorithm, focusing on women forming a group (liars) who strategically alter their preference lists to achieve better matchings. It proves that in the incomplete preference list setting, a strong Nash equilibrium always exists and yields a unique, strongly Pareto-dominant outcome for all liars—matching each to the same partner as in her best single-agent manipulation.

ABSTRACT

It is well-known that the Gale-Shapley algorithm is not truthful for all agents. Previous studies on this front mostly focus on blacklist manipulations by a single woman and by the set of all women. Little is known about manipulations by a coalition of women or other types of manipulations, such as manipulation by permuting preference lists. In this paper, we consider the problem of finding an equilibrium for a coalition of women (aka. liars) in the Gale-Shapley algorithm. We restrict attentions on manipulations that induce stable matchings. For the incomplete preference list setting, where liars can truncate their preference lists, we show that a strong Nash equilibrium always exists and the matching from such equilibria is unique. The equilibrium outcome is strongly Pareto dominant for all liars among the set of matchings achievable by manipulation: every woman is matched with the same man as the one she matches in her best single-agent manipulation. For the complete preference list setting where liars can permute their preference list, we first show that a coalition of women can get worse off by manipulating jointly than each performing a single-agent manipulation, thus a strongly Pareto-dominant outcome may not exist by manipulation. We then put forward an efficient algorithm to compute a strong Nash equilibrium that is strongly Pareto-optimal for all liars. We derive connections between the stable marriage problem and stable roommate problem, and use tools there to prove our results for this part. This approach is highly nontrivial and of independent interest.

Motivation & Objective

• To investigate the strategic behavior of a coalition of women in manipulating the Gale-Shapley algorithm to achieve better matchings.
• To determine whether joint manipulation by a coalition of women can lead to outcomes that are Pareto-dominant compared to individual manipulations.
• To develop efficient algorithms for computing strong Nash equilibria in both incomplete and complete preference list settings.
• To establish connections between the stable marriage problem and the stable roommate problem to prove results in the complete preference list setting.

Proposed method

• Analyzes manipulation via truncation of incomplete preference lists, proving existence of strong Nash equilibria.
• Uses tools from the stable roommate problem to analyze the complete preference list setting, where liars can permute their lists.
• Introduces an efficient algorithm to compute a strong Nash equilibrium that is strongly Pareto-optimal for all liars in the complete list case.
• Employs structural analysis of stable matchings and equilibrium conditions to prove uniqueness and optimality of outcomes.
• Leverages known results from the stable roommate problem to derive new theoretical results in the stable marriage context.
• Proves that in the incomplete list model, the equilibrium outcome is uniquely determined and Pareto-dominant for all liars.

Experimental results

Research questions

• RQ1 Does a strong Nash equilibrium exist for a coalition of women manipulating the Gale-Shapley algorithm via preference list truncation?
• RQ2 Can joint manipulation by a coalition of women lead to a matching that is strongly Pareto-dominant compared to individual manipulations?
• RQ3 Is there an efficient algorithm to compute a strong Nash equilibrium that is Pareto-optimal for all liars in the complete preference list model?
• RQ4 What is the relationship between the stable marriage problem and the stable roommate problem in the context of coalition manipulation?
• RQ5 Can manipulation by a coalition lead to worse outcomes than individual manipulation, and if so, under what conditions?

Key findings

• In the incomplete preference list setting, a strong Nash equilibrium always exists and produces a unique matching outcome.
• The equilibrium matching in the incomplete list model is strongly Pareto-dominant for all liars, with each woman matched to the same partner as in her best single-agent manipulation.
• In the complete preference list setting, joint manipulation can lead to worse outcomes than individual manipulation, so a strongly Pareto-dominant outcome may not exist.
• An efficient algorithm is provided to compute a strong Nash equilibrium that is strongly Pareto-optimal for all liars in the complete list model.
• The paper establishes nontrivial connections between the stable marriage and stable roommate problems to prove results in the complete list setting.
• The results demonstrate that coalition manipulation in the Gale-Shapley algorithm is strategically complex, with outcomes depending critically on the type of manipulation allowed.