Moreover, Demange ( 1982) and Leonard ( 1983) prove that there exists a core allocation at which each buyer attains his/her marginal contribution to the grand coalition (the buyer-optimal core allocation) and there exists a core allocation at which each seller attains his/her marginal contribution to the grand coalition (the seller-optimal core allocation). They showed that the core of an assignment game is always non-empty and has a lattice structure. They studied a solution concept, the core, which is the set of allocations that cannot be improved upon by any coalition. Shapley and Shubik coined the term “assignment game” to describe the TU game associated with their two-sided market. The valuation matrix represents the joint surplus generated by each pair formed by a buyer and a seller. Each buyer has n valuations (one for each house) each seller has a reservation value for her house. In their setting, each buyer is interested in buying at most one house and each seller has one house for sale. In their seminal paper, Shapley and Shubik ( 1971) consider a two-sided housing market with m buyers and n sellers. Johri and Leach ( 2002) study a model in which sellers and buyers have heterogeneous tastes and they show that middlemen are better off if they have a multi-unit inventory of differentiated products. Allowing for search frictions and a monopolistic middleman, Bloch and Ryder ( 2000) study a market where buyers and sellers bargain over the surplus. He shows that direct trade has a negative effect on the market power of middlemen. Fingleton ( 1997) investigates competition between middlemen when direct trade between buyers and sellers is available. In Yavaş ( 1994) agents can search for matches on their own, or they can resort to a middleman who mediates between agents of opposite sides to facilitate their pairing. The work by Rubinstein and Wolinsky ( 1987) is the first one to study the activity of middlemen in search markets. Markets with middlemen have been studied in different contexts (search and matching models, general equilibrium model, etc.). As is common in these applications, we assume that a middleman may serve multiple buyer–seller pairs. In financial markets, brokers provide their service to investors (in exchange for a fee) and each investor may or may not hire a broker. For example, in the real estate market, a seller may or may not use a realtor facilitating the sale of her house. A given buyer and a given seller may trade directly, or they may use the services of a middleman. We assume that utility is transferable between all agents and this allows the use of cooperative games with transferable utility (or TU games, for short). Units need not be homogeneous, i.e., a buyer may have different valuations for the respective units owned by two distinct sellers. Each seller owns one indivisible unit and each buyer seeks to purchase one unit (from any of the sellers) in exchange for money. Finally, we establish the coincidence between the core and the set of competitive equilibrium payoff vectors.Ĭonsider a commodity whose market exhibits three types of agents: buyers, sellers, and middlemen. However, we prove that in these small markets the maximum core payoff to each middleman is her marginal contribution. In general, the core does not exhibit a middleman-optimal allocation, not even when there are only two buyers and two sellers. 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. 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. For each such market, we examine the associated TU game. 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. This paper studies matching markets in the presence of middlemen.
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