A theory of stability in many-to-many matching markets

We develop a theory of stability in many-to-many matching markets. We give conditions under which the setwise-stable set, a core-like concept, is nonempty and can be approached through an algorithm. The usual core may be empty. The setwise-stable set coincides with the pairwise-stable set and with the predictions of a non-cooperative bargaining model. The setwise-stable set possesses the conflict/coincidence of interest properties from many-to-one, and one-to-one models. The theory parallels the standard theory of stability for many-to-one, and one-to-one, models. We provide results for a number of core-like solutions, besides the setwise-stable set

On preferences over subsets and the lattice structure of stable matchings

This paper studies the structure of stable multipartner matchings in two-sided markets where choice functions are quotafilling in the sense that they satisfy the substitutability axiom and, in addition, fill a quota whenever possible. It is shown that (i) the set of stable matchings is a lattice under the common revealed preference orderings of all agents on the same side, (ii) the supremum (infimum) operation of the lattice for each side consists componentwise of the join (meet) operation in the revealed preference ordering of the agents on that side, and (iii) the lattice has the polarity, distributivity, complementariness and full-quota properties

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

A Theory of Stability in Many-to-Many Matching Markets

Matching market, Core, Setwise stable set

A Theory of Stability in Many-to-many Matching Markets

We develop a theory of stability in many-to-many matching markets. We give conditions under which the setwise-stable set, a core-like concept, is nonempty and can be approached through an algorithm. The usual core may be empty. The setwise-stable set coincides with the pairwise-stable set, and with the predictions of a non-cooperative bargaining model. The set-wise stable set possesses the canonical conflict/coincidence of interest properties from many-to-one, and one-to-one models. The theory parallels the standard theory of stability for many-to-one, and one-to- one, models. We provide results for a number of core-like solutions, besides the setwise-stable set.many-to-many matchings, substitutability, tarski fixed point theorem, setwise stability, core

A Many-to-Many 'Rural Hospital Theorem'

We show that the full version of the so-called 'rural hospital theorem' (Roth, 1986) generalizes to many-to-many matching where agents on both sides of the market have separable and substitutable preferences.matching, many-to-many, stability, rural hospital theorem.

Incomplete Information and Small Cores in Matching Markets

We study Bayesian Nash equilibria of stable mechanisms in centralized matching markets under incomplete information. We show that truth-telling is a Bayesian Nash equilibrium of the revelation game induced by a common belief and a stable mechanism if and only if all the profiles in the support of the common belief have singleton cores. Our result matches the observations of Roth and Peranson (1999) in the National Resident Matching Program (NRMP) in the United States: (i) the cores of the profiles submitted to the clearinghouse are small and (ii) while truth-telling is not a dominant strategy most participants of the NRMP truthfully reveal their preferences.Matching Market, Incomplete Information, Small Core

Two-sided matching theory

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any:2015, Director: Marina NúñezThe purpose of this degree project is to study two-sided matchings where money is not involved. Matching theory is a branch of discrete mathematics belonging to game theory. This theory considers markets with two disjoint sets, such as men and women, firms and workers or colleges and students. Each agent on one sector has preferences (a complete and transitive binary relation) over the set of agents on the opposite side. Then, a matching is a set of pairs formed by agents of different side, in such a way that one agent can take part in at most one pair. We can situate its origin in the article of Gale and Shapley (1962) "College admissions and the stability of marriage" followed by the book of Knuth (1976), which first edition in French had the title of "Mariages stables". The first chapter of this monograph focuses on the theory of one-to-one matching, that is known as the marriage problem. This chapter provides the theoretical basis to develop two-sided matching theory, since the notions of stability and optimality for matchings are studied in depth. Chapter 2 is devoted to many-to-one matching problems, say the college admission problem, to analyse until which extent the results obtained for one-to-one markets still hold. In these two chapters the existence of stable matchings, their properties and the structure of the set of stable matchings are studied. Chapter 3 is a real-life application of the theory of matchings: the school choice problem. Here, we are going to analyse which algorithms have been used to fairly assign children to schools. This problem is currently under study, approached from the fields of mathematics, economics, operations research or computer science

Credible Group Stability in Many-to-Many Matching Problems

It is known that in two-sided many-to-many matching problems, pairwise stable matchings may not be immune to group deviations, unlike in many- to-one matching problems (Blair 1988). In this paper, we show that pairwise stability is equivalent to credible group stability when one side has responsive preferences and the other side has categorywise- responsive preferences. A credibly group-stable matching is immune to any “executable” group deviations with an appropriate definition of executability. Under the same preference restriction, we also show the equivalence between the set of pairwise-stable matchings and the set of matchings generated by coalition-proof Nash equilibria of an appropriately defined strategic-form game.