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This paper studies matching markets in the presence of middlemen. In our framework, a buyer-seller pair may either trade directly or use the services of a middleman; and a middleman may serve multiple buyer-seller pairs. For each such market, we examine the associated TU game. We first show that, in our context, an optimal matching can be obtained by considering the two-sided assignment market where each buyer-seller pair is allowed to use the mediation services of any middleman free of charge. Second, we prove that matching markets with middlemen are totally balanced: in particular, we show the existence of a buyer-optimal (seller-optimal) core allocation where each buyer (seller) receives her marginal contribution to the grand coalition. In general, the core does not exhibit a middleman-optimal allocation, not even when there are only two buyers and two sellers. However, we prove that in these small markets the maximum core payoff to each middleman is her marginal contribution. Finally, we establish the coincidence between the core and the set of competitive equilibrium payoff vectors. Hungarian National Research,Development and Innovation Office [NKFIH K-119930] Published version Ata Atay is a Serra Hunter Fellow (Professor Lector Serra Hunter) under the Serra Hunter Plan (Pla SerraHunter). Any opinions, findings, interpretations, conclusions or recommendations expressed in this paper arethose of its authors and do not necessarily reflect the views of the National Science Foundation. Ata Ataygratefully acknowledges financial support from the University of Barcelona through Grant AS017672.Tamas Solymosi gratefully acknowledges financial support from the Hungarian National Research,Development and Innovation Office through Grant NKFIH K-119930.

Since stable matchings may not exist, we propose a weaker notion of stability based on the credibility of blocking pairs. We adopt the weak stability notion of Klijn and Masso (2003) for the marriage problem and we extend it to the roommate problem. We first show that although stable matchings may not exist, a weakly stable matching always exists in a roommate problem. Then, we adopt a solution concept based on the credibility of the deviations for the roommate problem: the bargaining set. We show that weak stability is not sufficient for a matching to be in the bargaining set. We generalize the coincidence result for marriage problems of Klijn and Masso (2003) between the bargaining set and the set of weakly stable and weakly efficient matchings to roommate problems. Finally, we prove that the bargaining set for roommate problems is always non-empty by making use of the coincidence result.

This paper takes a game theoretical approach to open shop scheduling problems to minimize the sum of completion times. We assume that there is an initial schedule to process the jobs (consisting of a number of operations) on the machines and that each job is owned by a different player. Thus, we can associate a cooperative TU-game to any open shop scheduling problem, assigning to each coalition the maximal cost savings it can obtain through admissible rearrangements of jobs’ operations. A number of different approaches to admissible schedules for a coalition are introduced and, in the main result of the paper, a core allocation rule is provided for games arising from unit (execution times and weights) open shop scheduling problems for the most of these approaches. To sharpen the bounds of the set of open shop scheduling problems that result in games that are balanced, we provide two counterexamples: one for general open shop problems and another for further relaxations of the definition of admissible rearrangements for a coalition.

A generalization of the classical three-sided assignment market is considered, where value is generated by pairs or triplets of agents belonging to different sectors, as well as by individuals. For these markets we represent the situation that arises when some agents leave the market with some payoff by means of a generalization of Owen (Ann Econ Stat 25-26:71-79, 1992) derived market. Consistency with respect to the derived market, together with singleness best and individual anti-monotonicity, axiomatically characterize the core for these generalized three-sided assignment markets. When one sector is formed by buyers and the other by two different type of sellers, we show that the core coincides with the set of competitive equilibrium payoff vectors.

We study a special three-sided matching game, the so-called supplier-firm-buyer game, in which buyers and sellers (suppliers) trade indirectly through middlemen (firms). Stuart (1997) showed that all supplier-firm-buyer games have non-empty core. We show that for these games the core coincides with the classical bargaining set (Davis and Maschler, 1967), and also with the Mas-Colell bargaining set (Mas-Colell, 1989).

We consider a multi-sided assignment game with the following characteristics : -a- the agents are organized in m sectors that are connected by a graph that induces a weighted m-partite graph on the set of agents, -b- a basic coalition is formed by agents from different connected sectors, and –c- the worth of a basic coalition is the addition of the weights of all its pairs that belong to connected sectors. We provide a sufficient condition on the weights to guarantee balancedness of the related multi-sided assignment game. Moreover, when the graph on the sectors is cycle-free, we prove the game is strongly balanced and the core is described by means of the cores of the underlying two-sided assignment games associated with the edges of this graph. Moreover, once selected a spanning tree of the cycle-free graph on the sectors, the equivalence between core and competitive equilibria is established.