Random paths to stability in the roommate problem

This paper studies whether a sequence of myopic blockings leads to a stable matching in the roommate problem. We prove that if a stable matching exists and preferences are strict, then for any unstable matching, there exists a finite sequence of successive myopic blockings leading to a stable matching. This implies that, starting from any unstable matching, the process of allowing a randomly chosen blocking pair to form converges to a stable matching with probability one. This result generalizes those of Roth and Vande Vate (1990) and Chung (2000) under strict preferences

On roommate problem with weak preferences.

Wong, Tak Yuen.Thesis (M.Phil.)--Chinese University of Hong Kong, 2008.Includes bibliographical references (leaves 29-30).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 2 --- Literature Review --- p.6Chapter 3 --- The Roommate Problem --- p.8Chapter 4 --- The Existence of Stable Matchings --- p.11Chapter 5 --- Random Paths to Stability --- p.22Chapter 6 --- Concluding Remarks --- p.2

Stochastic Stability for Roommate Markets

We show that for any roommate market the set of stochastically stable matchings coincideswith the set of absorbing matchings. This implies that whenever the core is non-empty (e.g.,for marriage markets), a matching is in the core if and only if it is stochastically stable, i.e., stochastic stability is a characteristic of the core. Several solution concepts have beenproposed to extend the core to all roommate markets (including those with an empty core).An important implication of our results is that the set of absorbing matchings is the onlysolution concept that is core consistent and shares the stochastic stability characteristic withthe core.Economics (Jel: A)

The Evolution of Roommate Networks: A Comment on Jackson and Watts JET (2002)

We extend Jackson and Watts's (2002) result on the coincidence of S-stochastically stable and core stable networks from marriage problems to roommate problems. In particular, we show that the existence of a side-optimal core stable network, on which the proof of Jackson and Watts (2002) hinges, is not crucial for their result.core, networks, roommate problems, stochastic stability

The Evolution of Roommate Networks: A Comment on Jackson and Watts JET (2002)

In this note we extend Jackson and WattsJET2002''s result on the coincidence of S-stochastically stable and core stable networks from the marriage problem to the solvable roommate problem. In particular, we show that the polarization structure of the marriage problem on which the proof of Jackson and WattsJET2002 hinges, is not crucial for their result.microeconomics ;

Adjusting Prices in the Many-to-many Assignment Game

Starting with an initial price vector, prices are adjusted in order to eliminate the demand excess and at the same time to keep the transfers to the sellers as low as possible. In each step of the auction, to which sellers should those transfers be made (minimal overdemanded sets) is the key definition in the description of the algorithm. Such approach was previously used by several authors. We introduce a novel distinction by considering multiple sellers owing multiple identical objects and multiple buyers with a quota greater than one consuming at most one unit of each seller’s good. This distinction induces a necessarily more complicated construction of the overdemanded sets than the constructions existing in the literature, even in the simplest case of additive utilities considered here. As the previous papers, our mechanism yields the minimum competitive equilibrium price vector. A procedure to find the maximum competitive equilibrium price vector is also provided.matching; stable payoff; competitive equilibrium payoff; optimal stable payoff; lattice social costs; pure comparative vigilance; super-symmetric rule

Farsighted Stability for Roommate Markets

Using a bi-choice graph technique (Klaus and Klijn, 2009), we show that a matching for a roommate market indirectly dominates another matching if and only if no blocking pair of the former is matched in the latter (Proposition 1). Using this characterization of indirect dominance, we investigate von Neumann-Morgenstern farsightedly stable sets. We show that a singleton is von Neumann-Morgenstern farsightedly stable if and only if the matching is stable (Theorem 1). We also present roommate markets with no and with a non-singleton von Neumann-Morgenstern farsightedly stable set (Examples 1 and 2).core, farsighted stability, one- and two-sided matching, roommate markets, von Neumann-Morgenstern stability.

The Stability of the Roommate Problem Revisited

The lack of stability in some matching problems suggests that alternative solution concepts to the core might be applied to find predictable matchings. We propose the absorbing sets as a solution for the class of roommate problems with strict preferences. This solution, which always exists, either gives the matchings in the core or predicts some other matchings when the core is empty. Furthermore, it satisfies an interesting property of outer stability. We also characterize the absorbing sets, determine their number and, in case of multiplicity, we find that they all share a similar structure.roommate problem, core, absorbing sets

The stability of the roommate problem revisited

The lack of stability in some matching problems suggests that alternative solution concepts to the core might be a step towards furthering our understanding of matching market performance. We propose absorbing sets as a solution for the class of roommate problems with strict preferences. This solution, which always exists, either gives the matchings in the core or predicts other matchings when the core is empty. Furthermore, it satisfies the interesting property of outer stability. We also determine the matchings in absorbing sets and find that in the case of multiple absorbing sets a similar structure is shared by all.roommate problem, core, absorbing sets