We develop a theory of stability in many-to-many matching markets. We give conditions under wich ... more We develop a theory of stability in many-to-many matching markets. We give conditions under wich the setwisestable set, a core-like concept, is nonempty and can be approached through an algorithm. 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.
We characterize the core many-to-one matchings as fixed points of a map. Our characterization giv... more 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.
We develop a theory of stability in many-to-many matching markets. We give conditions under which... more 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.
European Journal of Operational Research, Dec 1, 2000
Consider the following cooperative game 1 in characteristic function form with transferable utili... more Consider the following cooperative game 1 in characteristic function form with transferable utility, N={1,2} and v({1})=v({2})=0, v({1,2})=1. We repeat this game two times. If we consider this repeated game as a cooperative game: we have four players N 2 ={1 0 ,2 0 ,1 1 ,2 1 }, ...
We develop a theory of stability in many-to-many matching markets. We give conditions under which... more 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.
This work is addressed to the problem of constructing birational games with predetermined equilib... more This work is addressed to the problem of constructing birational games with predetermined equilibrium points. We develop techniques which generalize those introduced for bimatrix games. A necessary and sufficient condition for a pair of strategies to be a unique equilibrium point of a birational game is given. KEY WORDS. Two-person games. Birational games. Equilibr iu� points. Constructing birational games. Uniqueness.
Journal of the Operations Research Society of China, Aug 4, 2020
For the marriage model with indi¤erences, we de…ne an equivalence relation over the stable matchi... more For the marriage model with indi¤erences, we de…ne an equivalence relation over the stable matching set. We identify a su¢ cient condition, the closing property, under which we can extend results of the classical model (without indi¤erences) to the equivalence classes of the stable matching set. This condition allows us to extend the lattice structure over classes of equivalences and the rural hospital theorem.
DOAJ (DOAJ: Directory of Open Access Journals), 2004
Baïou y Balinski (2002) generalizaron el modelo de asignación empresastrabajadores a uno donde se... more Baïou y Balinski (2002) generalizaron el modelo de asignación empresastrabajadores a uno donde se planifica la asignación determinando además de la asignación de los trabajadores a las empresas cuánto tiempo los trabajadores le dedicarán a la empresa. Una planificación es estable si ningún par empresa-trabajador puede incrementar sus horas de trabajo juntos, perjudicando a algún agente menos deseable. Este artículo estudia la relación que existe entre este problema y un problema de matching y se muestra que cada planificación estable es equivalente a cierto matching estable.
We also construct the extended replicator dynamics for these games and we study an application to... more We also construct the extended replicator dynamics for these games and we study an application to a model of strategic planning of investment.
European Journal of Operational Research, May 1, 1997
A further generalization of the Shapley-Shubik housing market is considered in which there are m ... more 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.
In a many-to-one matching model with responsive preferences in which indifferences are allowed, w... more In a many-to-one matching model with responsive preferences in which indifferences are allowed, we study three notions of core, three notions of stability, and their relationships. We show that (i) the core contains the stable set, (ii) the strong core coincides with the strongly stable set, and (iii) the super core coincides with the super stable set. We also show how the core and the strong core in markets with indifferences relate to the stable matchings of their associated tie-breaking strict markets.
... a minimal connected graph that contains to G(x, y) G(.x) G(y). In pardcu!ar, if G(.u,y) is I ... more ... a minimal connected graph that contains to G(x, y) G(.x) G(y). In pardcu!ar, if G(.u,y) is I connectedgraph then the ... 1981) "4 Conjecture of Bolker ED", Journal of Combinntorid Theory, Series B 31, 1-8 [3] Balinski, M. L. (1974) "On two special classes of Transportation Polytopes". ...
The Blocking Lemma identi…es a particular blocking pair for each nonstable and individually ratio... more The Blocking Lemma identi…es a particular blocking pair for each nonstable and individually rational matching that is preferred by some agents of one side of the market to their optimal stable matching. Its interest lies in the fact that it has been an instrumental result to prove key results on matching. For instance, the fact that in the college admissions problem the workers-optimal stable mechanism is group strategy-proof for the workers and the strong stability theorem in the marriage model follow directly from the Blocking Lemma. However, it is known that the Blocking Lemma and its consequences do not hold in the general many-to-one matching model in which …rms have substitutable preference relations. We show that the Blocking Lemma holds for the many-to-one matching model in which …rms'preference relations are, in addition to substitutable, quota q separable. We also show that the Blocking We are grateful to Flip Klijn, Howard Petith, William Thomson, an associate editor, and two referees for very helpful comments.
European Journal of Operational Research, Mar 1, 1992
In this paper we study the problem of multiple objective linear programming (MOLP). We introduce ... more 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.
We characterize the core many-to-one matchings as fixed points of a map. Our characterization giv... more 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.
In a many-to-one matching market with substitutable preferences, we analyze the game induced by a... more In a many-to-one matching market with substitutable preferences, we analyze the game induced by a stable matching rule. We show that any stable matching rule implements the individually rational correspondence in Nash equilibrium when both sides of the market play strategically. We also show that when only workers play strategically in Nash equilibrium and when firms' preferences satisfy the law of aggregate demand, any stable matching rule implements the stable correspondence in Nash equilibrium.
We consider the general many-to-one matching model with ordinal preferences and give a procedure ... more We consider the general many-to-one matching model with ordinal preferences and give a procedure to partition the set of preference pro…les into subsets with the property that all preference pro…les in the same subset have the same Core. We also show how to identify a pro…le of (incomplete) binary relations containing the minimal information needed to generate as strict extensions all the (complete) preference pro…les 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.
We develop a theory of stability in many-to-many matching markets. We give conditions under wich ... more We develop a theory of stability in many-to-many matching markets. We give conditions under wich the setwisestable set, a core-like concept, is nonempty and can be approached through an algorithm. 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.
We characterize the core many-to-one matchings as fixed points of a map. Our characterization giv... more 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.
We develop a theory of stability in many-to-many matching markets. We give conditions under which... more 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.
European Journal of Operational Research, Dec 1, 2000
Consider the following cooperative game 1 in characteristic function form with transferable utili... more Consider the following cooperative game 1 in characteristic function form with transferable utility, N={1,2} and v({1})=v({2})=0, v({1,2})=1. We repeat this game two times. If we consider this repeated game as a cooperative game: we have four players N 2 ={1 0 ,2 0 ,1 1 ,2 1 }, ...
We develop a theory of stability in many-to-many matching markets. We give conditions under which... more 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.
This work is addressed to the problem of constructing birational games with predetermined equilib... more This work is addressed to the problem of constructing birational games with predetermined equilibrium points. We develop techniques which generalize those introduced for bimatrix games. A necessary and sufficient condition for a pair of strategies to be a unique equilibrium point of a birational game is given. KEY WORDS. Two-person games. Birational games. Equilibr iu� points. Constructing birational games. Uniqueness.
Journal of the Operations Research Society of China, Aug 4, 2020
For the marriage model with indi¤erences, we de…ne an equivalence relation over the stable matchi... more For the marriage model with indi¤erences, we de…ne an equivalence relation over the stable matching set. We identify a su¢ cient condition, the closing property, under which we can extend results of the classical model (without indi¤erences) to the equivalence classes of the stable matching set. This condition allows us to extend the lattice structure over classes of equivalences and the rural hospital theorem.
DOAJ (DOAJ: Directory of Open Access Journals), 2004
Baïou y Balinski (2002) generalizaron el modelo de asignación empresastrabajadores a uno donde se... more Baïou y Balinski (2002) generalizaron el modelo de asignación empresastrabajadores a uno donde se planifica la asignación determinando además de la asignación de los trabajadores a las empresas cuánto tiempo los trabajadores le dedicarán a la empresa. Una planificación es estable si ningún par empresa-trabajador puede incrementar sus horas de trabajo juntos, perjudicando a algún agente menos deseable. Este artículo estudia la relación que existe entre este problema y un problema de matching y se muestra que cada planificación estable es equivalente a cierto matching estable.
We also construct the extended replicator dynamics for these games and we study an application to... more We also construct the extended replicator dynamics for these games and we study an application to a model of strategic planning of investment.
European Journal of Operational Research, May 1, 1997
A further generalization of the Shapley-Shubik housing market is considered in which there are m ... more 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.
In a many-to-one matching model with responsive preferences in which indifferences are allowed, w... more In a many-to-one matching model with responsive preferences in which indifferences are allowed, we study three notions of core, three notions of stability, and their relationships. We show that (i) the core contains the stable set, (ii) the strong core coincides with the strongly stable set, and (iii) the super core coincides with the super stable set. We also show how the core and the strong core in markets with indifferences relate to the stable matchings of their associated tie-breaking strict markets.
... a minimal connected graph that contains to G(x, y) G(.x) G(y). In pardcu!ar, if G(.u,y) is I ... more ... a minimal connected graph that contains to G(x, y) G(.x) G(y). In pardcu!ar, if G(.u,y) is I connectedgraph then the ... 1981) "4 Conjecture of Bolker ED", Journal of Combinntorid Theory, Series B 31, 1-8 [3] Balinski, M. L. (1974) "On two special classes of Transportation Polytopes". ...
The Blocking Lemma identi…es a particular blocking pair for each nonstable and individually ratio... more The Blocking Lemma identi…es a particular blocking pair for each nonstable and individually rational matching that is preferred by some agents of one side of the market to their optimal stable matching. Its interest lies in the fact that it has been an instrumental result to prove key results on matching. For instance, the fact that in the college admissions problem the workers-optimal stable mechanism is group strategy-proof for the workers and the strong stability theorem in the marriage model follow directly from the Blocking Lemma. However, it is known that the Blocking Lemma and its consequences do not hold in the general many-to-one matching model in which …rms have substitutable preference relations. We show that the Blocking Lemma holds for the many-to-one matching model in which …rms'preference relations are, in addition to substitutable, quota q separable. We also show that the Blocking We are grateful to Flip Klijn, Howard Petith, William Thomson, an associate editor, and two referees for very helpful comments.
European Journal of Operational Research, Mar 1, 1992
In this paper we study the problem of multiple objective linear programming (MOLP). We introduce ... more 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.
We characterize the core many-to-one matchings as fixed points of a map. Our characterization giv... more 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.
In a many-to-one matching market with substitutable preferences, we analyze the game induced by a... more In a many-to-one matching market with substitutable preferences, we analyze the game induced by a stable matching rule. We show that any stable matching rule implements the individually rational correspondence in Nash equilibrium when both sides of the market play strategically. We also show that when only workers play strategically in Nash equilibrium and when firms' preferences satisfy the law of aggregate demand, any stable matching rule implements the stable correspondence in Nash equilibrium.
We consider the general many-to-one matching model with ordinal preferences and give a procedure ... more We consider the general many-to-one matching model with ordinal preferences and give a procedure to partition the set of preference pro…les into subsets with the property that all preference pro…les in the same subset have the same Core. We also show how to identify a pro…le of (incomplete) binary relations containing the minimal information needed to generate as strict extensions all the (complete) preference pro…les 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.
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Papers by Jorge Oviedo