The exclusive partnership condition here models the fact that players only get at most one item, and each item y can only be sold to a single player. To see how this notion captures a matching market, consider a bipartite G-the “left nodes” correspond to buyers and the “right nodes” correspond to the items the value of an edge e between a buyer i and an item y is simply the value v player i has for the item y. (That is, for every edge ( x, y ), the amount v ( x, y ) gets divided between x and y.) d : V → ℕ is a “division” (or split) of amounts to nodes such that for every e = ( x, y ) ∈ M, d ( x ) + d ( y ) = v ( e ), and for every node x not part of an edge in M (i.e., M ( x ) = ⊥), d ( x ) = 0.M is a matching in G (i.e., a subset of E where no two edges share a common endpoint).An outcome of an exchange network ( G, v ) is a pair ( M, d ) where: An exchange network is a pair ( G, v ), where G = ( V, E ) is a graph and v : E → ℕ is a function. The outcome of such an exchange network is thus a matching in G, and assignments of values to players representing the players’ share of the value of the partnership they (potentially) participate in:ĭefinition 9.1. Nodes can participate in at most one exchange (i.e., they form exclusive partnerships) in other words, the partnerships form a matching in G. Each edge e is associated with some amount v ( e ) that can be split between its endpoints if an exchange between the endpoints takes place-think of this value as the value of a potential partnership between the endpoints. We first describe the problem and then relate it back to matching markets.Īssume we have an undirected graph G = ( V, E ) with n nodes the nodes in this graph represent the players, and an edge between two players means that the players may perform “an exchange” with each other. Roughly speaking, rather than just considering bipartite graphs where the left nodes are buyers and the right nodes are items/sellers, we will now consider exchanges over arbitrary networks. In this chapter, we will consider a generalized form of matching markets, referred to as exchange networks.
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