Jorge Oviedo

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.

Core many-to-one matchings by fixed-point methods

We characterize the core many-to-one matchings as fixed points of a map. Our characterization gives an algorithm for finding core allocations; the algorithm is efficient and simple to implement. Our characterization does not require substitutable preferences, so it is separate from the structure needed for the non-emptiness of the core. When preferences are substitutable, our characterization gives a simple proof of the lattice structure of core matchings, and it gives a method for computing the join and meet of two core matchings.

Lexicographic optimality in the multiple objective linear programming: The nucleolar solution

Abstract In this paper we study the problem of multiple objective linear programming (MOLP). We introduce a new solution concept which is related to that of the nucleolus of n -person cooperative game theory. We prove that a general MOLP problem always has a solution in the new sense. The points in the nucleolus are efficient in the classic way. We prove existence and at the same time we introduce a constructing algorithm for computing it.

On group strategy-proof mechanisms for a many-to-one matching model

Abstract.For the many-to-one matching model in which firms have substitutable and quota q−separable preferences over subsets of workers we show that the workers-optimal stable mechanism is group strategy-proof for the workers. Therefore, in centralized markets like entry-level professional labor markets if the proposed matching is the workers-optimal stable matching then, no group of workers can never benefit by reporting untruthfully their preference relations. We exhibit an example showing that this property fails if the preferences of firms are substitutable but not quota q−separable.

An algorithm to compute the full set of many-to-many stable matchings

The paper proposes an algorithm to compute the full set of many-to-many stable matchings when agents have substitutable preferences. The algorithm starts by calculating the two optimal stable matchings using the deferred-acceptance algorithm. Then, it computes each remaining stable matching as the firm-optimal stable matching corresponding to a new preference profile, which is obtained after modifying the preferences of a previously identified sequence of firms.

The core of a repeated n-person cooperative game

Abstract We study the core of a repeated cooperative game. We define the repeated cooperative game as a repeated game where in each round the agents play a cooperative game. We introduce an imputation sequence and a dominated imputation sequence. We define the core of the repeated game as the set of all undominated imputation sequences. We show that the core of a repeated cooperative game contains the core of the original cooperative game.

On the Invariance of the Set of Core Matchings with Respect to Preference Profiles

We consider the general many-to-one matching model with ordinal preferences and give a procedure to partition the set of preference profiles into subsets with the property that all preference profiles in the same subset have the same Core. We also show how to identify a profile of (incomplete) binary relations containing the minimal information needed to generate as strict extensions all the (complete) preference profiles with the same Core. This is important for applications since it reduces the amount of information that agents have to reveal about their preference relations to centralized Core matching mechanisms; moreover, this reduction is maximal.

SHAPLEY VALUE IN A MODEL OF INFORMATION TRANSFERAL

In this paper we analyze the value of the information in a cooperative model. There is an agent (the innovator), having relevant information which can be sold to some potential buyers. The n potential users of the information share a market. The expected utility of each of them can be improved by obtaining the information. The whole situation is modelled as a (n + 1)–person cooperative game.We study the properties of the characteristic function of this game. It fulfills a weak version of the superadditivity property, namely 0-monotonicity. The game is proved to be monotonic.We compute the Shapley value and we prove it is an imputation for the game. We compare the Shapley value with the equilibrium studied in a noncooperative model by Quintas (1995). We also study some limit cases when the potential buyers are completely informed or uninformed. It includes Big Boss Games (Muto et al. (1988)) and other limit cases.We conclude that in a cooperative environment the buyers avoid being exploited by the innovator. Conversely the innovator obtains a higher utility in a noncooperative game.

On the invariance of the set of stable matchings with respect to substitutable preference profiles

For the many-to-one matching model we give a procedure to partition the set of substitutable preference profiles into equivalence classes with the property that all profiles in the same class have the same set of stable matchings. This partition allows to reduce the amount of information required by centralized stable mechanisms.

The core of a further m-sided assignment game

Abstract A further generalization of the Shapley-Shubik housing market is considered in which there are m types of individuals instead of two. This is different from the generalization of Quint. These games can have empty cores.