Manipulation of Stable Matchings using Minimal Blacklists

Gale and Sotomayor (1985) have shown that in the Gale-Shapley matching algorithm (1962), the proposed-to side W (referred to as women there) can strategically force the W-optimal stable matching as the M-optimal one by truncating their preference lists, each woman possibly blacklisting all but one man. As Gusfield and Irving have already noted in 1989, no results are known regarding achieving this feat by means other than such preference-list truncation, i.e. by also permuting preference lists. We answer Gusfield and Irving's open question by providing tight upper bounds on the amount of blacklists and their combined size, that are required by the women to force a given matching as the M-optimal stable matching, or, more generally, as the unique stable matching. Our results show that the coalition of all women can strategically force any matching as the unique stable matching, using preference lists in which at most half of the women have nonempty blacklists, and in which the average blacklist size is less than 1. This allows the women to manipulate the market in a manner that is far more inconspicuous, in a sense, than previously realized. When there are less women than men, we show that in the absence of blacklists for men, the women can force any matching as the unique stable matching without blacklisting anyone, while when there are more women than men, each to-be-unmatched woman may have to blacklist as many as all men. Together, these results shed light on the question of how much, if at all, do given preferences for one side a priori impose limitations on the set of stable matchings under various conditions. All of the results in this paper are constructive, providing efficient algorithms for calculating the desired strategies.Comment: Hebrew University of Jerusalem Center for the Study of Rationality discussion paper 64

Tiered Random Matching Markets: Rank Is Proportional to Popularity

We study the stable marriage problem in two-sided markets with randomly generated preferences. Agents on each side of the market are divided into a constant number of "soft" tiers, which capture agents\u27 qualities. Specifically, every agent within a tier has the same public score, and agents on each side have preferences independently generated proportionally to the public scores of the other side. We compute the expected average rank which agents in each tier have for their partners in the man-optimal stable matching, and prove concentration results for the average rank in asymptotically large markets. Furthermore, despite having a significant effect on ranks, public scores do not strongly influence the probability of an agent matching to a given tier of the other side. This generalizes the results by Pittel [Pittel, 1989], which analyzed markets with uniform preferences. The results quantitatively demonstrate the effect of competition due to the heterogeneous attractiveness of agents in the market

A Stable Marriage Requires Communication

The Gale-Shapley algorithm for the Stable Marriage Problem is known to take $\Theta(n^2)$ steps to find a stable marriage in the worst case, but only $\Theta(n \log n)$ steps in the average case (with $n$ women and $n$ men). In 1976, Knuth asked whether the worst-case running time can be improved in a model of computation that does not require sequential access to the whole input. A partial negative answer was given by Ng and Hirschberg, who showed that $\Theta(n^2)$ queries are required in a model that allows certain natural random-access queries to the participants' preferences. A significantly more general - albeit slightly weaker - lower bound follows from Segal's general analysis of communication complexity, namely that $\Omega(n^2)$ Boolean queries are required in order to find a stable marriage, regardless of the set of allowed Boolean queries. Using a reduction to the communication complexity of the disjointness problem, we give a far simpler, yet significantly more powerful argument showing that $\Omega(n^2)$ Boolean queries of any type are indeed required for finding a stable - or even an approximately stable - marriage. Notably, unlike Segal's lower bound, our lower bound generalizes also to (A) randomized algorithms, (B) allowing arbitrary separate preprocessing of the women's preferences profile and of the men's preferences profile, (C) several variants of the basic problem, such as whether a given pair is married in every/some stable marriage, and (D) determining whether a proposed marriage is stable or far from stable. In order to analyze "approximately stable" marriages, we introduce the notion of "distance to stability" and provide an efficient algorithm for its computation

Strategyproofness-Exposing Mechanism Descriptions

A menu description presents a mechanism to player $i$ in two steps. Step (1) uses the reports of other players to describe $i$'s menu: the set of $i$'s potential outcomes. Step (2) uses $i$'s report to select $i$'s favorite outcome from her menu. Can menu descriptions better expose strategyproofness, without sacrificing simplicity? We propose a new, simple menu description of Deferred Acceptance. We prove that -- in contrast with other common matching mechanisms -- this menu description must differ substantially from the corresponding traditional description. We demonstrate, with a lab experiment on two elementary mechanisms, the promise and challenges of menu descriptions