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.

Unilateral Stability in Matching Problems

The canonical solution concept used in matching problems is pairwise stability, whose premise is that harmony is disrupted by any two agents intentionally leaving their partners to be with each other. We instead focus on scenarios in which harmony is disrupted merely by a single agent unilaterally initiating contact with a member of a different pair, whether or not his approach is reciprocated. A variety of solution concepts are proposed in which taboos, status, or power systematically limit such initiatives in order to achieve harmony

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)

Stable marriage and roommates problems with restricted edges: complexity and approximability

In the Stable Marriage and Roommates problems, a set of agents is given, each of them having a strictly ordered preference list over some or all of the other agents. A matching is a set of disjoint pairs of mutually acceptable agents. If any two agents mutually prefer each other to their partner, then they block the matching, otherwise, the matching is said to be stable. We investigate the complexity of finding a solution satisfying additional constraints on restricted pairs of agents. Restricted pairs can be either forced or forbidden. A stable solution must contain all of the forced pairs, while it must contain none of the forbidden pairs. Dias et al. (2003) gave a polynomial-time algorithm to decide whether such a solution exists in the presence of restricted edges. If the answer is no, one might look for a solution close to optimal. Since optimality in this context means that the matching is stable and satisfies all constraints on restricted pairs, there are two ways of relaxing the constraints by permitting a solution to: (1) be blocked by as few as possible pairs, or (2) violate as few as possible constraints n restricted pairs. Our main theorems prove that for the (bipartite) Stable Marriage problem, case (1) leads to View the MathML source-hardness and inapproximability results, whilst case (2) can be solved in polynomial time. For non-bipartite Stable Roommates instances, case (2) yields an View the MathML source-hard but (under some cardinality assumptions) 2-approximable problem. In the case of View the MathML source-hard problems, we also discuss polynomially solvable special cases, arising from restrictions on the lengths of the preference lists, or upper bounds on the numbers of restricted pairs

Mobility Sharing as a Preference Matching Problem

Traffic congestion, dominated by single-occupancy vehicles, reflects not only transportation system inefficiency and negative externalities but also a sociological state of human isolation. Advances in information and communication technology are enabling the growth of real-time ridesharing to improve system efficiency. While most ridesharing algorithms optimize fellow passenger matching based on efficiency criteria (maximum number of paired trips, minimum total vehicle-time, or vehicle-distance traveled), very few explicitly consider passengers' preference for their peers as the matching objective. The existing literature either considers the bipartite driver-passenger matching problem, which is structurally different from the monopartite passenger-passenger matching, or only considers the passenger-passenger problem in a simplified one-origin-multiple-destination setting. We formulate a general monopartite passenger matching model in a road network and illustrate the model by pairing 301,430 taxi trips in Manhattan in two scenarios: one considering 1000 randomly generated preference orders and the other considering four sets of group-based preference orders. In both scenarios, compared with efficiency-based matching models, preference-based matching improves the average ranking of paired fellow passenger to the near-top position of people's preference orders with only a small efficiency loss at the individual level and a moderate loss at the aggregate level. The near-top-ranking results fall in a narrow range even with the random variance of passenger preference as inputs.Singapore-MIT Alliance (Future Mobility Program)Massachusetts Institute of Technology. Institute for Data, Systems, and Society. Seed Fun

Pairing games and markets

Pairing Games or Markets studied here are the non-two-sided NTU generalization of assignment games. We show that the Equilibrium Set is nonempty, that it is the set of stable allocations or the set of semistable allocations, and that it has has several notable structural properties. We also introduce the solution concept of pseudostable allocations and show that they are in the Demand Bargaining Set. We give a dynamic Market Procedure that reaches the Equilibrium Set in a bounded number of steps. We use elementary tools of graph theory and a representation theorem obtained here

A new solution concept for the roommate problem

Abstract The aim of this paper is to propose a new solution concept for the roommate problem with strict preferences. We introduce maximum irreversible matchings and consider almost stable matchings (Abraham et al., 2006) and maximum stable matchings (Tan 1990, 1991b). These solution concepts are all core consistent. We find that almost stable matchings are incompatible with the other two concepts. Hence, to solve the roommate problem we propose matchings that lie at the intersection of the maximum irreversible matchings and maximum stable matchings, which we call Q -stable matchings. We construct an efficient algorithm for computing one element of this set for any roommate problem. We also show that the outcome of our algorithm always belongs to an absorbing set (Inarra et al., 2013)

A new solution for the roommate problem

The aim of this paper is to propose a new solution for the roommate problem with strict references. We introduce the solution of maximum ir reversibility and consider almost stable matchings (Abraham et al. [2]) and maximum stable m atchings (Tan [30] [32]). We find that almost stable matchings are incompatible with the o ther two solutions. Hence, to solve the roommate problem we propose matchings that lie at t he intersection of the maximum irreversible matchings and maximum stable matchings , which are called Q-stable matchings. These matchings are core consistent and we offer an efficient algorithm for computing one of them. The outcome of the algorithm belongs to an ab sorbing set

Matching theory: the roommates problem

Trabajo de fin de Grado. Grado en Economía. Curso académico 2020-2021.[ES] El problema de los compañeros de cuarto (roommates problem) se incluye en la teoría de matching y persigue la formación estable de parejas en un grupo determinado. Este trabajo de fin de grado se centra en el estudio de dicho problema en comparación con el de emparejamiento bilateral uno a uno. Además,se analiza el conocido como algoritmo de Irving, que conduce a una solución estable, siempre que exista