Authors: Ata Atay, Ana Mauleon, Vincent Vannetelbosch
Abstract Since stable matchings may not exist, we adopt a weaker notion of stability for solving the roommate problem: the bargaining set. Klijn and Massó (2003) show that the bargaining set coincides with the set of weakly stable and weakly efficient matchings in the marriage problem. First, we show that a weakly stable matching always exists in the roommate problem. However, weak stability is not sufficient for a matching to be in the bargaining set. Second, we prove that the bargaining set is always non-empty. Finally, as Klijn and Massó (2003) get for the marriage problem, we show that the bargaining set coincides with the set of weakly stable and weakly efficient matchings in the roommate problem. Keywords: Roommate problem, matching, (weak) stability, bargaining set JEL Classifications: C71, C78 Mathematics Subject Classification: 91B68
Abstract This paper takes a game theoretical approach to open shop scheduling problems with unit execution times to minimize the sum of completion times. By supposing an initial schedule and associating each job (consisting in a number of operations) to a different player, we can construct a cooperative TU-game associated with any open shopscheduling problem. We assign to each coalition the maximal cost savings it can obtain through admissible rearrangements of jobs’ operations. By providing a core allocation, we show that the associated games are balanced. Finally, we relax the definition of admissible rearrangements for a coalition to study to what extend balancedness still holds. Keywords: Open shop, scheduling, cooperative game theory, core, balancedness JEL Classification: C44, C71 Mathematics Subject Classification (2010): 90B35, 91A12
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