Housing Markets with Indifferences: A Tale of Two Mechanisms

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 © 2013, 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 allocation x on N if there exists an allocation y on S such that for all i ∈ S , y(i) ∈ ω(S ), y(i) %i x(i), and there exists an i ∈ S such that y(i) i x(i). An allocation x on N is in the strict core (SC) of market M if it admits no weakly blocking coalition. An allocation that is in the strict core is also said to be strict core stable. An allocation is Pareto optimal (PO) if N is not a weakly blocking coalition. It is clear that strict core implies core and also Pareto optimality. Core implies weak Pareto optimality and also individual rationality. A mechanism that always returns a Pareto optimal and (strict) core stable allocation is said to be Pareto optimal and (strict) core-selecting respectively. A mechanism is strategy-proof if for each agent, reporting false preferences to the mechanism will not be beneficial to the agent (i.e., when the agent reports false preferences, he will not end up with a house that he prefers more than the house he would get when he reports his true preferences to the mechanism). Desirable allocations of housing markets can be computed via a graph-theoretic approach to housing markets. Each housing market M = (N,H, ω,%) has a corresponding simple digraph G(%) = (N ∪ H, E) such that for each i ∈ N and h ∈ H, (i, h) ∈ E if h % h′ for all h′ ∈ H, and (h, i) if h = ω(i). In other words, each agent points to his maximally preferred houses and each house points to his owner. An absorbing set of a digraph is a strongly connected component from which there are no outgoing edges. Two nodes constitute a symmetric pair if there are edges from each node to the other. Both nodes are then called paired-symmetric. An absorbing set is paired-symmetric if each node belongs to a symmetric pair. GATTC In this section, we formulate a simple family of mechanisms called Generalized Absorbing Top Trading Cycle (GATTC) which is designed for housing markets with indifferences and extends not only TTC but also includes the two families TTAS and TCR. It is based on multi-way exchanges of houses between agents. We will show that GATTC satisfies many desirable properties of housing mechanisms such as being core-selecting and Pareto optimal. Before we describe GATTC, we will introduce the original TTC mechanism which is for the domain of housing markets with strict preferences. TTC works as follows. For a housing market M with strict preferences, we first construct the corresponding graph G(%) as defined above. Then, we start from an agent and walk arbitrarily along the edges until a cycle is completed. A cycle starting from any agent is of course guaranteed to exist as each node in G(%) has positive outdegree. This cycle is removed from G(%). Within the removed cycle, each agent gets the house he was pointing to. The graph G(%) is adjusted so that the remaining agents point to the most preferred houses among the remaining houses. The process is repeated until all the houses and agents are deleted from the graph.1 For a housing market with indifferences, TTC can still be used to return a core selecting allocation: break ties arbi1Please see Section 2.2 of (Sönmez and Ünver, 2011) for an elegant illustration of how TTC works. trarily and then run TTC. However such an allocation may not be Pareto optimal (see e.g., Alcalde-Unzu and Molis, 2011; Jaramillo and Manjunath, 2011). GATTC achieves Pareto optimality and is based on absorbing sets and the concept of a ‘good cycle’. A good cycle is any cycle in G(%) which contains at least one node that is not pairedsymmetric. By implementing a cycle we mean reallocating the houses along the cycle. For example consider the cycle a0, h1, a1, . . . , hm, am, h0, a0. Then for all i ∈ {0, . . . ,m}, house hi+1 mod m is made to point to ai. The following is the description of a GATTC mechanism.