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The number of stable matchings in models of the Gale-Shapley type with preferences given by partial orders

100%
DRGAS-BURCHARDT E.
Operations Research and Decisions
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2015
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vol. 25
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issue 1
5-15
EN
From the famous Gale–Shapley theorem we know that each classical marriage problem admits at least one stable matching. This fact has inspired researchers to search for the maximum number of possible stable matchings, which is equivalent to finding the minimum number of unstable matchings among all such problems of size n. In this paper, we deal with this issue for the Gale–Shapley model with preferences represented by arbitrary partial orders. Also, we discuss this model in the context of the classical Gale–Shapley model.
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Stability and Generalized Competitive Equilibria in a Many-to-Many Gale-Shapley Market Model

51%
Świtalski Z.
Przegląd Statystyczny
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2016
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vol. 63
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issue 3
237-254
PL
W artykule zdefiniowano, dla pewnego wariantu modelu rynku Gale’a-Shapleya (typu „many-to-many”), pojęcie uogólnionej równowagi konkurencyjnej i pokazano, że przy odpowiednich założeniach, skojarzenia stabilne w tym modelu mogą być reprezentowane jako alokacje równowag konkurencyjnych (i vice versa). Przedstawione wyniki są daleko idącymi uogólnieniami „lematu o podaży i popycie” z pracy Azevedo, Leshno (2011) dotyczącego modelu rekrutacji kandydatów do szkół. Wykorzystując wyniki Alkana, Gale’a (2003), udowodniono również twierdzenie o istnieniu uogólnionych równowag dla podanego modelu.
EN
We define, for some variant of a many-to-many market model of Gale-Shapley type, a concept of generalized competitive equilibrium and show that, under suitable conditions, stable matchings in such a model can be represented as competitive equilibria allocations (and vice versa). Our results are far-reaching generalizations of the “discrete supply and demand lemma” of Azevedo, Leshno (2011) for the college admissions market. Using the results of Alkan, Gale (2003), we also prove a theorem on existence of generalized equilibria in our model.
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