Generalizing Top Trading Cycles for Housing Markets with Fractional Endowments

The housing market setting constitutes a fundamental model of exchange economies of goods. In most of the work concerning housing markets, it is assumed that agents own and are allocated discrete houses. The drawback of this assumption is that it does not cater for randomized assignments or allocation of time-shares. Recently, house allocation with fractional endowment of houses was considered by Athanassoglou and Sethuraman (2011) who posed the open problem of generalizing Gale’s Top Trading Cycles (TTC) algorithm to the case of housing markets with fractional endowments. In this paper, we address the problem and present a generalization of TTC called FTTC that is polynomialtime as well as core stable and Pareto optimal with respect to stochastic dominance even if there are indifferences in the preferences. We prove that if each agent owns one discrete house, FTTC coincides with a state of the art strategyproof mechanism for housing markets with discrete endowments and weak preferences. We show that FTTC satisfies a maximal set of desirable properties by proving two impossibility theorems. Firstly, we prove that with respect to stochastic dominance, core stability and no justified envy are incompatible. Secondly, we prove that there exists no individual rational, Pareto optimal and weak strategyproof mechanism, thereby answering another open problem posed by Athanassoglou and Sethuraman (2011). The second impossibility implies a number of results in the literature.