6 research outputs found
Fractional solutions for capacitated NTU-games, with applications to stable matchings
Abstract. In this paper we investigate some new applications of Scarf’s Lemma. First, we introduce the notion of fractional core for NTU-games, which is always nonempty by the Lemma. Stable allocation is a general solution concept for games where both the players and their possible cooperations can have capacities. We show that the problem of finding a stable allocation, given a finitely generated NTU-game with capacities, is always solvable by a variant of Scarf’s Lemma. Then we describe the interpretation of these results for matching games. Finally we consider an even more general setting where players ’ contributions in a joint activity may be different. We show that a stable allocation can be found by the Scarf algorithm in this case as well, and we demonstrate the usage of this method for the hospitals resident problem with couples. This problem is relevant in many practical applications, such as NRM
Matching couples with Scarf’s algorithm
Scarf's algorithm [20] provides fractional core elements for NTU-games. Bir�o and
Fleiner [3] showed that Scarf's algorithm can be extended for capacitated NTU-games. In
this setting agents can be involved in more than one coalition at a time, cooperations may be
performed with di�erent intensities up to some limits, and the contribution of the agents can
also di�er in a coalition. The fractional stable solutions for the above model, produced by the
extended Scarf algorithm, are called stable allocations. In this paper we apply this solution
concept for the Hospitals / Residents problem with Couples (HRC). This is one of the most
important general stable matching problems due to its relevant applications, also well-known
to be NP-hard. We show that if a stable allocation yielded by the Scarf algorithm turns out
to be integral then it provides a stable matching for an instance of HRC, so this method
can be used as a heuristic. In an experimental study, we compare this method with other
heuristics constructed for HRC that are applied in practice in the American and Scottish
resident allocation programs, respectively. Our main �nding is that the Scarf algorithm
outperforms all the other known heuristics when the proportion of couples is high
Matching with couples: a Multidisciplinary Survey
This survey deals with two-sided matching markets where one set of agents (workers/residents) has to be matched with another set of agents (firms/hospitals). We first give a short overview of a selection of classical results. Then, we review recent contributions to a complex and representative case of matching with complementarities, namely matching markets with couples. We discuss contributions from computer scientists, economists, and game theorists. © 2013 World Scientific Publishing Company