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

Matching under Preferences

Matching theory studies how agents and/or objects from different sets can be matched with each other while taking agents\u2019 preferences into account. The theory originated in 1962 with a celebrated paper by David Gale and Lloyd Shapley (1962), in which they proposed the Stable Marriage Algorithm as a solution to the problem of two-sided matching. Since then, this theory has been successfully applied to many real-world problems such as matching students to universities, doctors to hospitals, kidney transplant patients to donors, and tenants to houses. This chapter will focus on algorithmic as well as strategic issues of matching theory. Many large-scale centralized allocation processes can be modelled by matching problems where agents have preferences over one another. For example, in China, over 10 million students apply for admission to higher education annually through a centralized process. The inputs to the matching scheme include the students\u2019 preferences over universities, and vice versa, and the capacities of each university. The task is to construct a matching that is in some sense optimal with respect to these inputs. Economists have long understood the problems with decentralized matching markets, which can suffer from such undesirable properties as unravelling, congestion and exploding offers (see Roth and Xing, 1994, for details). For centralized markets, constructing allocations by hand for large problem instances is clearly infeasible. Thus centralized mechanisms are required for automating the allocation process. Given the large number of agents typically involved, the computational efficiency of a mechanism's underlying algorithm is of paramount importance. Thus we seek polynomial-time algorithms for the underlying matching problems. Equally important are considerations of strategy: an agent (or a coalition of agents) may manipulate their input to the matching scheme (e.g., by misrepresenting their true preferences or underreporting their capacity) in order to try to improve their outcome. A desirable property of a mechanism is strategyproofness, which ensures that it is in the best interests of an agent to behave truthfully

Coalitions and Cliques in the School Choice Problem

The school choice mechanism design problem focuses on assignment mechanisms matching students to public schools in a given school district. The well-known Gale Shapley Student Optimal Stable Matching Mechanism (SOSM) is the most efficient stable mechanism proposed so far as a solution to this problem. However its inefficiency is well-documented, and recently the Efficiency Adjusted Deferred Acceptance Mechanism (EADAM) was proposed as a remedy for this weakness. In this note we describe two related adjustments to SOSM with the intention to address the same inefficiency issue. In one we create possibly artificial coalitions among students where some students modify their preference profiles in order to improve the outcome for some other students. Our second approach involves trading cliques among students where those involved improve their assignments by waiving some of their priorities. The coalition method yields the EADAM outcome among other Pareto dominations of the SOSM outcome, while the clique method yields all possible Pareto optimal Pareto dominations of SOSM. The clique method furthermore incorporates a natural solution to the problem of breaking possible ties within preference and priority profiles. We discuss the practical implications and limitations of our approach in the final section of the article

Some things couples always wanted to know about stable matchings (but were afraid to ask)

It is well-known that couples that look jointly for jobs in the same centralized labor market may cause instabilities. We demonstrate that for a natural preference domain for couples, namely the domain of responsive preferences, the existence of stable matchings can easily be established. However, a small deviation from responsiveness in one couple's preference relation that models the wish of a couple to be closer together may already cause instability. This demonstrates that the nonexistence of stable matchings in couples markets is not a singular theoretical irregularity. Our nonexistence result persists even when a weaker stability notion is used that excludes myopic blocking. Moreover, we show that even if preferences are responsive there are problems that do not arise for singles markets. Even though for couples markets with responsive preferences the set of stable matchings is nonempty, the lattice structure that this set has for singles markets does not carry over. Furthermore we demonstrate that the new algorithm adopted by the National Resident Matching Program to fill positions for physicians in the United States may cycle, while in fact a stable matchings does exist, and be prone to strategic manipulation if the members of a couple pretend to be single.matching, couples, stability

The strategy structure of some coalition formation games

In coalitional games with side payments, the core predicts which coalitions form and how benefits are shared. The predictions however run into difficulties if the core is empty or if some coalitions benefit from not blocking truthfully. These difficulties are analyzed in games in which an a priori given collection of coalitions can form, as the collection of pairs of buyer-seller in an assignment game. The incentive properties of the core and of its selections are investigated in function of the collection. Furthermore the relationships with Vickrey-Clarke-Groves mechanisms are drawn.coalition formation ; assignment ; manipulability ; substitutes ; incremental value ; Vickrey-Clarke-Groves mechanism

Researching with Whom? Stability and Manipulation

This paper explores the existence of stable research teams, when each agent's preferences depend on the set of researchers collaborating with her. We introduce a property over researchers' preferences, called tops responsiveness guaranteeing the existence of stable research teams configurations. We also provide a stable mechanism, induced by the so-called tops covering algorithm, which is strategy-proof when researchers preferences satisfy tops responsiveness. Furthermore, we also find out that, in this framework the tops covering mechanism is the unique strategy-proof mechanism that always selects stable allocations.Coalition Formation, Research Teams Configurations, Stability, Strategy-Proofness.