Housing Markets with Indifferences: A Tale of Two Mechanism

The (Shapley-Scarf) housing market is a well-studied and fundamental model of an exchange economy. Each agent owns a single house and the goal is to reallocate the houses to the agents in a mutually beneficial and stable manner. Recently, Alcalde-Unzu and Molis (2011) and Jaramillo and Manjunath (2011) independently examined housing markets in which agents can express indifferences among houses.They proposed two important families of mechanisms, known as TTAS and TCR respectively. We formulate a family of mechanisms which not only includes TTAS and TCR but also satisfies many desirable properties of both families. As a corollary, we show that TCR is strict core selecting (if the strict core is non-empty). Finally, we settle an open question regarding the computational complexity of the TTAS mechanism. Our study also raises a number of interesting research questions. Introduction Housing markets are fundamental models of exchange economies of goods where the goods could range from dormitories to kidneys (Sönmez and Ünver 2011). The classic housing market (also called the Shapley-Scarf Market) consists of a set of agents each of which owns a house and has strict preferences over the set of all houses. The goal is to redistribute the houses to the agents in the most desirable fashion. Shapley and Scarf (1974) showed that a simple yet elegant mechanism called Gale’s Top Trading Cycle (TTC) is strategy-proof and finds an allocation which is in the core. TTC is based on multi-way exchanges of houses between agents. Since the basic assumption in the model is that agents have strict preferences over houses, TTC is also strict core selecting and therefore Pareto optimal. Indifferences in preferences are not only a natural relaxation but are also a practical reality in many cases. Many new challenges arise in the presence of indifferences: the core does not imply Pareto optimality; the strict core can be empty (Quint and Wako 2004); and strategic issues need to be re-examined. In spite of these challenges, Alcalde-Unzu and Molis (2011) and Jaramillo and Manjunath (2011) proposed desirable mechanisms for housing markets with indifferences. Alcalde-Unzu and Molis (2011) presented the Top Copyright c © 2012, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. Trading Absorbing Sets (TTAS) family of mechanisms which are strategy-proof, core selecting (and therefore individually rational), Pareto optimal, and strict core selecting (if the strict core is non-empty). Independently, Jaramillo and Manjunath (2011) came up with a different family of mechanisms called Top Cycle Rules (TCR) which are strategyproof, core selecting, and Pareto optimal. Whereas it was shown in (Jaramillo and Manjunath 2011) that each TCR mechanism runs in polynomial time, the time complexity of TTAS was raised as an open problem in (Alcalde-Unzu and Molis 2011). We first highlight the commonality of TCR and TTAS by describing a simple class of mechanisms called Generalized Absorbing Top Trading Cycle (GATTC) which encapsulates the TTAS and TCR families. It is proved that each GATTC mechanism is core selecting, strict core selecting, and Pareto optimal. As a corollary, TCR is strict core selecting. We note that whereas a GATTC mechanism satisfies a number of desirable properties, the strategy-proofness of a particular GATTC mechanism hinges critically on the order and way of choosing trading cycles. Finally, we settle the computational complexity of TTAS. By simulating a binary counter, it is shown that a TTAS mechanism can take exponential time to terminate. Preliminaries Let N be a set of n agents and H a set of n houses. The endowment function ω : N → H assigns to each agent the house he originally owns. Each agent has complete and transitive preferences %i over the houses and %= (%1, . . . %n) is the preference profile of the agents. The housing market is a quadruple M = (N,H, ω,%). For S ⊆ N, we denote ⋃ i∈S ω(i) by ω(S ). A function x : S → H is an allocation on S ⊆ N if there exists a bijection π on S such that x(i) = ω(π(i)) for each i ∈ S . The goal in housing markets is to re-allocate the houses in a mutually beneficial and efficient way. An allocation is individually rational (IR) if x(i) %i ω(i). A coalition S ⊆ N blocks an allocation x on N if there exists an allocation y on S such that for all i ∈ S , y(i) ∈ ω(S ) and y(i) i x(i). An allocation x on N is in the core (C) of market M if it admits no blocking coalition. An allocation that is in the core is also said to be core stable. An allocation is weakly Pareto optimal (w-PO) if N is not a blocking coalition. A coalition S ⊆ N weakly blocks an Proceedings of the Twenty-Sixth AAAI Conference on Artificial Intelligence