Computing the deficiency of housing markets with duplicate houses

Abstract

The model of a housing market, introduced by Shapley and Scarf in 1974 [14], captures a fundamental situation in an economy where each agent owns exactly one unit of some indivisible good: a house. We focus on an extension of this model where duplicate houses may exist. As opposed to the classical setting, the existence of an economical equilibrium is no longer ensured in this case. Here, we study the deficiency of housing markets with duplicate houses, a notion measuring how close a market can get to an economic equilibrium. We investigate the complexity of computing the deficiency of a market, both in the classical sense and also in the context of parameterized complexity. We show that computing the deficiency is NP-hard even under several severe restrictions placed on the housing market, and thus we consider different parameterizations of the problem. We prove W[1]-hardness for the case where the parameter is the value of the deficiency we aim for. By contrast, we provide an FPT algoritm for computing the deficiency of the market, if the parameter is the number of different house types. © 2010 Springer-Verlag.