Computing the Deficiency of Housing Markets with Duplicate Houses

The model of housing market is due to Shapley and Scarf (1974). Their paper contains the celebrated result stating that each housing market admits an economic equilibrium. However, in a slight modification of this model which admits duplicate houses the existence of equilibrium is no longer ensured. In this paper we study the computational complexity of the deficiency problem for housing markets with duplicate houses: what is the minimum number of agents who cannot get a most preferred house in their budget set? We show that this problem is NP-hard even under several severe restrictions placed on the housing market, e.g. if the maximum number of houses of one type is two and the maximum number of preferred houses for each agent is one. We further prove that the problem is fixed-parameter intractable if 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.