Complements and substitutes in generalized multisided assignment economies

We consider a finitely populated economy in which there are different types of agent, each agent is of exactly one type, and profit is created by coalitions containing at most one agent of each type (or side). The surplus of a so-called generalized multisided assignment economy is defined as the maximum aggregate profit that can be attained by matching agents into pairwise disjoint coalitions of the above kind. We present negative results that establish that when the economy consists of more than two sides (i) agents on different sides may not be complements, i.e., they do not necessarily reinforce each other’s influence on the surplus and (ii) agents on the same side may not be substitutes, i.e., they do not necessarily interfere with each other’s influence on the surplus. These findings are in marked contrast with the results for two-sided assignment economies (Shapley, 1962). We propose novel notions for the complementarity and the substitutability of disjoint subsets of agents and we find conditions that ensure that the former are satisfied.