Core and Top Trading Cycles in a Market with Indivisible Goods and Externalities

Transcription

1 Core and Top Trading Cycles in a Market with Indivisible Goods and Externalities Miho Hong Jaeok Park August 2, 2018 Abstract In this paper, we incorporate externalities into Shapley-Scarf housing markets. Agents preferences are defined over allocations rather than houses, and we focus on preferences that are egocentric in the sense that agents primarily care about their own allotments. When preferences are egocentric, we can apply the top trading cycles (TTC) algorithm using the associated preferences over houses. We establish that the allocation generated by the TTC algorithm is a stable allocation in the core, and we present a further preference restriction under which it is the unique such allocation. We also investigate the properties of the TTC algorithm as a mechanism. Our results extend the existing results on the TTC algorithm to the case of egocentric preferences, and they suggest that the TTC algorithm is useful and has desirable properties even in the presence of externalities. Keywords: Core, Externalities, Housing markets, Indivisibility, Stability, Top trading cycles. JEL Classification: C71, C78, D62. 1 Introduction In this paper, we study a market with indivisibilities and externalities. In their pioneering work, Shapley and Scarf (1974) introduce an exchange economy with a finite number of We are grateful to Takashi Akahoshi, Benjamin Casner, Yeon-Koo Che, Eun Jeong Heo, Daisuke Hirata, Jinyong Jeong, Seungwon (Eugene) Jeong, Jinwoo Kim, Bettina Klaus, Fuhito Kojima, Peng Liu, Eiichi Miyagawa, William Phan, Tayfun Sönmez, Takashi Ui, and seminar and conference participants at Kyoto University, Hitotsubashi University, Kobe University, the 14th Meeting of the Society for Social Choice and Welfare, the 2018 Asian Meeting of the Econometric Society, and the 29th International Conference on Game Theory at Stony Brook for valuable comments and suggestions. School of Economics, Yonsei University ( miho.m.hong@gmail.com) Corresponding author. School of Economics, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul 03722, Republic of Korea ( jaeok.park@yonsei.ac.kr) 1

2 agents in which each agent owns an indivisible object (say, a house), has use for exactly one object, and looks for trading with other agents. Such an economy is referred to as a housing market in the literature. It is one of the simplest kinds of exchange economies one can imagine, and together with a house allocation problem (Hylland and Zeckhauser, 1979) it has served as a basic model for indivisible goods allocation problems and one-sided matching problems (see, for example, Moulin, 1995; Abdulkadiroğlu and Sönmez, 2013). Externalities are abundant in the real world, and a market with indivisible goods is not an exception. For example, in residential areas, a resident of a house cares about the demographics of his neighbors, and in universities, a faculty member s preferences regarding offices may depend on his relationships with colleagues using nearby offices. Despite the ubiquity of externalities, most existing works on housing markets have not taken externalities into consideration, assuming that each agent cares only about his own allotment but not others. The goal of this paper is to provide an analysis of housing markets with externalities. Shapley and Scarf (1974) show that the core (defined by strong domination) of a housing market is nonempty. They also introduce an algorithm called the top trading cycles (TTC) algorithm that produces an allocation in the core. Roth and Postlewaite (1977) complement Shapley and Scarf s (1974) results by proving that the core (defined by weak domination) of a housing market is a singleton when agents have strict preferences over houses and that the unique allocation in the core can be obtained by the TTC algorithm. As the core is the main solution concept for housing markets and the TTC algorithm provides a method to find an allocation in the core, we aim to investigate whether the tight connection between the core and the TTC algorithm continues to hold in the presence of externalities. The presence of externalities creates difficulties in applying the TTC algorithm as well as defining the core. At each step of the TTC algorithm, each remaining agent points to the owner of the house he prefers most among the remaining houses. However, when there are externalities, an agent s preferences over houses may depend on how the rest of the houses are allocated to the other agents, and thus his favorite house may not be well-defined. In order to avoid this complication, we focus on a class of preferences where an agent is primarily interested in his own allotment. That is, each agent has a strict preference order over houses, and whenever any two allocations assign distinct houses to him, his ranking between the two allocations is determined by that between the houses he receives at those allocations. 1 An agent is subject to externalities in the sense that he may have a strict ranking between two allocations that assign the same house to him. We call this kind of 1 Dutta and Massó (1997) propose similar preference restrictions in two-sided many-to-one matching markets. They consider firm-lexicographic (resp. worker-lexicographic ) preferences where a worker s preferences over firms (resp. colleagues) dictate his overall preferences over firm-colleague pairs. In the context of two-sided one-to-one matching markets with contracts, Li (1993) uses weak externalities to refer to a situation where an agent cares about his own partner and contract prior to others. Sasaki and Toda (1996) call such preferences order preserving. 2

3 preferences egocentric preferences. Though restrictive, it is a reasonable class of preferences over allocations. It expresses the idea that one s own allotment is of higher significance than others allotments in one s utility. In other words, it reflects a situation in which the size of externalities is small relative to that of utility from one s own allotment. When an agent has egocentric preferences, he has well-defined preferences over houses. Thus, any housing market with egocentric preferences is associated with a housing market without externalities, and we can apply the TTC algorithm to the associated housing market. The core is the set of allocations that are immune to coalitional deviations, and thus it depends on what kind of outcomes a coalition can attain by deviating from a proposed allocation. The presence of externalities creates two issues. First, when there are externalities, it matters to a coalition how the residual agents react to its deviation. Thus, it needs to form a belief about the reaction of the residual agents, and different kinds of expectation lead to various concepts of the core. Second, in the presence of externalities, a coalition may benefit from the reallocation of the outside agents houses even when its members keep the same houses as in the proposed allocation, and thus some agents may form a coalition just to influence others houses. However, it is not so natural that the outside agents adjust their behavior even when a coalition makes no change in its behavior. So we deal with the second issue by requiring in our definition of domination that a deviating coalition should change its behavior from the proposed allocation. Given our concept of domination, we consider two extreme concepts of the core. At one extreme, deviating agents have optimistic prediction that the outside agents will behave in a way that benefits the deviators. At the other extreme, deviating agents have pessimistic expectation that the outside agents aim to deter the deviation. We call the core based on optimistic and pessimistic expectation the ω-core (Kóczy, 2007) and the α-core (Aumann, 1961), respectively. The ω-core is the smallest and the α-core is the largest among cores based on various kinds of expectation. So we resolve the first issue by providing results in the strongest form with an extreme core concept. In particular, we present an existence result using the ω-core and a uniqueness result using the α-core, so that the results continue to hold with any other core notion. In other words, our results hold independently of the core notion we adopt. We also introduce two intermediate notions of the core, called the γ-core and the δ-core (Hart and Kurz, 1983), and study their logical relationship. The γ-core is based on the belief that the remaining agents end up with their own endowments when a coalition deviates, whereas the δ-core is based on the belief that they keep trading their houses as in the proposed allocation whenever possible. 2 Another solution concept that plays a significant role in the presence of externalities is stable allocations. Suppose that agents agree on an allocation and execute trade to 2 Existing studies on two-sided matching markets with externalities have made similar assumptions: when some agents deviate, other unaffected agents keep their matches. See, for example, Mumcu and Saglam (2010) for one-to-one matching and Bando (2012, 2014) for many-to-one matching. 3

4 achieve the allocation. Without externalities (and strict preferences over houses), no further improvement by a coalition is possible if the allocation is in the core (defined by weak domination). However, with externalities, a group of agents may get better off by reallocating their allotments among themselves even if the allocation is in the ω-core. This is because reallocation by a coalition can make an outside agent worse off and thus the new allocation does not necessarily dominate the original endowment allocation. We call an allocation that is immune to further coalitional deviations a stable allocation, following the terminology of Roth and Postlewaite (1977). We can expect that an allocation that is not stable would not persist, and thus we focus on stable allocations. In sum, we take stable allocations in the core as our solution concept. An allocation being in the core means that no coalition has a profitable deviation from the allocation before it is implemented, while an allocation being stable indicates that no coalition has an incentive to deviate from the allocation after it is implemented. Thus, our solution concept checks that an allocation is not blocked before and after trade is executed according to the allocation. In our main results, we study the relationship between the TTC algorithm and stable allocations in the core. We first show that the allocation generated by the TTC algorithm is a stable allocation in the ω-core (Theorem 1). Since the ω-core is the smallest core, a solution is guaranteed to exist regardless of the core concept, and it can be obtained by applying the TTC algorithm. We next show that, when externality creators have aligned interests, a stable allocation in the α-core is unique and coincides with the allocation obtained by the TTC algorithm (Theorem 2). Since the α-core is the largest core and there exists a stable allocation in any smaller core, the uniqueness result still holds with any other core concept. This result generalizes the existing result of Roth and Postlewaite (1977) that an allocation in the core of a housing market without externalities is unique and can be obtained by the TTC algorithm. There are other desirable properties of the TTC algorithm established in the existing literature. For example, Abdulkadiroğlu and Sönmez (1998) show that one can generate all Pareto efficient allocations from the TTC algorithm by varying agents endowments, Roth (1982) and Moulin (1995) prove that the TTC algorithm as a mechanism is (coalitionally) strategy-proof, and Ma (1994) establishes that a mechanism is individually rational, Pareto efficient, and strategy-proof if and only if it is the TTC mechanism. In Propositions 1 3, we show that these existing results can be generalized to the case of egocentric preferences when we replace Pareto efficiency with stability. This suggests that stability plays a more important role than Pareto efficiency in the presence of externalities, while without externalities these two concepts coincide. Our results also indicate that it is desirable to select the allocation generated by the TTC algorithm when there are multiple stable allocations in the core. So our analysis implies that, when externalities are not so large, the TTC algorithm continues to provide the best solution for housing markets. In 4

5 other words, the desirable properties of the TTC algorithm are robust to the introduction of small externalities. 3 There is a growing body of literature on matching markets with externalities, especially in two-sided settings (see Bando et al., 2016, for a recent survey). Sasaki and Toda (1996) provide an early analysis on two-sided one-to-one matching problems with externalities. They consider a scenario where a deviating pair holds a pessimistic belief about the reaction of non-deviating agents, and they show that a stable matching exists for all preference profiles if and only if agents estimate that all possible matchings can arise after a deviation. Mumcu and Saglam (2010) also consider two-sided one-to-one matching problems with externalities but use a different stability notion analogous to the δ-core, as explained in footnote 2. They provide a sufficient condition on the preference profile for the existence of stable matchings. Bando (2012, 2014) studies two-sided many-to-one matching problems with externalities among firms, and he presents preference restrictions for the existence of quasi-stable matchings as well as a modified deferred acceptance algorithm that generates the worker-optimal quasi-stable matching. Dutta and Massó (1997) analyze two-sided many-to-one matching problems with preferences over colleagues. In this setting, externalities exist only among workers hired by the same firm, and thus, unlike in our model and the aforementioned models, the core concept does not depend on the reaction of residual agents. Dutta and Massó (1997) show among other results that, when preferences are firm-lexicographic, a matching in the core exists and can be obtained by finding a stable matching under the associated preference profile without externalities. Our work is closely related to theirs in that not only we adopt a similar preference restriction but also we establish the existence of a solution by using the allocation in the core of the associated housing market without externalities. There are not many studies on one-sided matching problems with externalities. Mumcu and Saglam (2007) study housing markets with externalities and show that there can be no or more than one allocation in the core. Baccara et al. (2012) examine the problem of assigning offices to faculty members, which can be modeled as a house allocation problem, and quantify the effects of network externalities using field data. Sönmez (1999) and Ehlers (2018) consider a class of indivisible goods allocation problems, which includes housing markets as a special case, while allowing externalities. Their main results and comparison with ours are discussed in Section 4. Recently, Graziano et al. (2018) study housing markets with externalities and present two preference restrictions under which the set of stable allocations is a von Neumann Morgenstern stable set. Their first preference restriction corresponds to our egocentricity assumption but is more restrictive in that they assume strict preferences over allocations. So we establish 3 In a similar vein, Fisher and Hafalir (2016) consider two-sided one-to-one matching markets with small aggregate externalities and provide a sufficient condition for the existence of stable matchings. 5

6 that stable allocations exist (Theorem 1) and form a stable set (Remark 1) even when an agent can be indifferent between two allocations where he receives the same house. The rest of this paper is organized as follows. In Section 2, we present the model of housing markets with egocentric preferences, and introduce some desirable properties of allocations. In Section 3, we describe the TTC algorithm and study the properties of allocations it produces. In Section 4, we examine the properties of the TTC algorithm as a mechanism. In Section 5, we discuss some intermediate core concepts and the case of non-egocentric preferences. In Section 6, we conclude. 2 The Model 2.1 Housing Markets with Egocentric Preferences We consider an exchange economy with a finite number of agents in which each agent owns a house initially, has use for one and only one house, and seeks to trade houses with other agents. Let N and H be finite sets of agents and houses, respectively, with N = H 3. An allocation describes the house assigned to each agent and is represented by a one-to-one function from N onto H. Let A be the set of all allocations. For any allocation a A, a(i) denotes the house received by agent i, and we call it the allotment of agent i at allocation a. We allow consumption externalities in that an agent cares not only about his own allotment but also about others. So each agent s preferences are defined over the set of allocations instead of houses. For each agent i N, R i denotes his preference relation, which is a complete and transitive binary relation on A. For any a, b A, ar i b means that agent i likes allocation a at least as much as allocation b. Given R i, we denote its asymmetric and symmetric parts by P i and I i, respectively. That is, for any a, b A, we write ap i b if ar i b and not br i a, and we write ai i b if ar i b and br i a. A profile of agents preference relations is denoted by R = (R i ) i N. Agents ownership is described by an initial endowment allocation e A so that e(i) is the house initially owned by agent i. A housing market (with externalities) is a 4-tuple N, H, R, e. We assume that each agent is not indifferent between any two allocations where he receives different houses. In other words, ai i b implies a(i) = b(i). We also assume that each agent cares about his own allotment prior to others. More specifically, ap i b and a(i) b(i) implies a P i b for any a and b such that a (i) = a(i) and b (i) = b(i). We call a preference relation satisfying these two assumptions egocentric. Note that, given an egocentric preference relation R i, we can derive a preference relation R i on H by defining h R i h if (i) h = h or (ii) h h and ap i b for all a, b A such that a(i) = h and b(i) = h. By our assumptions, Ri is a complete, transitive, and antisymmetric binary relation on H, expressing agent i s strict preferences over houses as his allotment. We refer to R i as 6

7 the preference relation over own-allotments associated with R i. We write the asymmetric part of R i as P i, which is defined analogously to P i. Given a housing market N, H, R, e with egocentric preferences, we can consider a related housing market N, H, R, e without externalities, where R = ( R i ) i N. In the literature on matching without externalities, it is common to assume strict preferences over one s possible allotments for analytic convenience, and our assumptions generalize it by allowing externalities. In the following, when we mention the case of no externalities, we refer to the case of strict preferences over houses unless otherwise stated. 2.2 Properties of Allocations We discuss several desirable properties that an allocation can possess, first in a housing market and then in a house allocation problem Core The core is a central solution concept for cooperative games, providing a set of allocations that are immune to coalitional deviations. When there are externalities, the preferences of coalition members depend on the allocation outside of the coalition. Hence, when a coalition plans a deviation, it needs to predict how the residual agents react to the deviation, and it implements the deviation only if its members get better off given its expectation. There are different kinds of expectation that a coalition can form about the residual agents behavior, and here we present two extreme kinds while discussing intermediate ones later in Section 5.1. In the most optimistic prediction, a coalition believes that the outside agents will do a favor for its members, and thus it will deviate if it is possible to make its members better off with the help of the outsiders. In contrast, in the most pessimistic prediction, a coalition expects that the residual agents will take action against its members interests, and so it will deviate if the deviation can benefit its members even in the worst possible scenario. Of course, these two kinds of expectation are too extreme and seem unrealistic, but we study them because the core notions based on them provide a lower and an upper bound for any core notion based on a more reasonable and realistic kind of expectation. 4 A nonempty subset of N is called a coalition, while N as a coalition is called the grand coalition. For any a A and any S N, let a(s) = {a(i) : i S}. For a given allocation a, we say that a coalition S is a trading cycle in a (with endowment e) if a(s) = e(s) and a(s ) e(s ) for any nonempty S S with S S. That is, a trading cycle in a is a minimal subset of agents who trade houses among themselves to obtain their allotments at a. 4 For example, the recursive cores proposed by Kóczy (2007), where the residual agents choose a core allocation, lie between the two extreme cores we consider. 7

8 The elements of a trading cycle in a can be indexed as {i 1,..., i m } such that a(i l ) = e(i l+1 ) for all l = 1,..., m 1 and a(i m ) = e(i 1 ), and we sometimes represent the trading cycle by the sequence (i 1,..., i m ). An allocation a uniquely partitions the set of agents into trading cycles in a. An allocation b A dominates an allocation a A via coalition T (or, coalition T blocks a via b) in the market N, H, R, e if (i) b(t ) = e(t ), (ii) b(i) a(i) for some i T, (iii) br i a for all i T, and (iv) bp i a for some i T. The ω-core of the market N, H, R, e is the set of allocations that are not dominated by any allocation in N, H, R, e. For any coalition T, let A e T be the set of all allocations in the submarket consisting of the agents in T. That is, A e T denotes the set of all bijections from T to e(t ). Similarly, let Ae T be the set of all allocations in the submarket consisting of the agents outside of T. When T is the grand coalition, we take A e T as the empty set. For any coalition T, any pair (a T, a T ) A e T Ae T yields an allocation where trade occurs among agents in T and among those outside of T. A coalition T α-blocks an allocation a in the market N, H, R, e if there exists b T A e T such that (b T, b T ) dominates a via T in N, H, R, e for all b T A e T. The α-core of the market N, H, R, e is the set of allocations that are not α-blocked by any coalition in N, H, R, e. When there are externalities, it is possible that an agent benefits from other agents reallocation of their allotments. Hence, some agents may form a coalition in order to affect others allotments while keeping their own allotments. By imposing the requirement b(i) a(i) for some i T in the definition of domination, we preclude this possibility. Given a proposed allocation, if a coalition plans to maintain its members allotments, there is no reason for the outside agents to react and adjust their allotments. Thus, we require that a coalition should make a different arrangement within it in order to initiate a deviation and induce the residual agents reaction. 5 The notion of the ω-core reflects the most optimistic view of deviating agents that trade outside of the coalition can be chosen as they desire. In contrast, the notion of the α-core is based on the most pessimistic belief that the residual agents objective is to deter the deviation. Hence, the ω-core is smaller than the α-core, and any other core notion (given our definition of domination) yields a core that lies between the two. When there are no externalities, it is immaterial what kind of expectation a coalition holds about the residual agents behavior, and thus the two extreme core notions reduce to the usual notion of the core (defined by weak domination) Stability and Pareto Efficiency A house allocation problem is a triple N, H, R, which is the same as a housing market except that agents own houses collectively rather than individually. An allocation a A 5 We further explain the necessity of this requirement in Remark 5 in Section

9 is stable in the problem N, H, R if a belongs to the ω-core of the market N, H, R, a (cf. Roth and Postlewaite, 1977). That is, no coalition can block a stable allocation once it is implemented. Suppose that a is not stable. Then there exists an allocation b and a coalition T such that b dominates a via T in N, H, R, a. Let T = {i T : b(i) a(i)}. Then T T, and br i a for all i T. With egocentric preferences, we have bp i a for all i T, and this holds true even when the agents outside of T reallocate their allotments from b. Note that T consists of trading cycles in b with endowment a, satisfying a(t ) = b(t ). Hence, the concept of stability can be defined equivalently by using the α-core (and thus any other core) instead of the ω-core, and it can be characterized as follows: An allocation a is stable if and only if there exists no allocation b such that b a and b(i) P i a(i) for all i N with b(i) a(i). Thus, stability is a concept that can be checked with preferences over own-allotments. Remark 1. Our notion of stability is slightly different from that of Roth and Postlewaite (1977) in that we use a core concept defined by weak domination while Roth and Postlewaite (1977) adopt strong domination in which every coalition member gains strictly. Kawasaki (2015) shows that the set of stable allocations in the sense of Roth and Postlewaite (1977) is the unique von Neumann Morgenstern (vnm) stable set with respect to some domination relation. Although we use weak domination to define our core concepts, our assumption of egocentric preferences allows us to use strong domination for agents whose allotments differ at the two compared allocations in the characterization of stability, and we can obtain a similar result to that of Kawasaki (2015). For any a, b A, let us define b a if b a and b(i) P i a(i) for all i N with b(i) a(i). Then the set of stable allocations is the unique vnm stable set with respect to the domination relation. 6 An allocation a A is Pareto efficient in the problem N, H, R if there exists no allocation b A such that br i a for all i N and bp i a for some i N. Note that, unlike stable allocations, we need preferences over allocations in order to find Pareto efficient allocations and allocations in any core, as there can be an agent who receives the same allotment at the two compared allocations. Stable allocations as well as allocations in any core are Pareto efficient because the grand coalition cannot block these allocations. When there are no externalities, a Pareto efficient allocation is stable, 7 and thus the concepts of stability and Pareto efficiency coincide. In contrast, stability is stronger than Pareto efficiency when there are externalities, as shown in the following example. 6 The set of stable allocations satisfies internal stability by definition. We can check that the relation is transitive. Hence, if an allocation a is not stable, we can eventually obtain a stable allocation b such that b a, which proves external stability. Furthermore, we can show that there is no other vnm stable set than the set of stable allocations, following the argument in the proof of Theorem 1 of Kawasaki (2015). 7 As noted, if a is not stable, there exists another allocation b such that bp ia for all i N with b(i) a(i). Without externalities, bi ia for all i N with b(i) = a(i). Thus, a is not Pareto efficient. 9

10 Example 1. Consider a housing market with N = {i 1, i 2, i 3 }, H = {h 1, h 2, h 3 }, and e = (h 1, h 2, h 3 ). 8 The agents have egocentric preferences, and their associated preferences over own-allotments are given by P i1 : h 2, h 3, h 1, Pi2 : h 3, h 1, h 2, Pi3 : h 1, h 3, h 2. Agent i 1 s preferences satisfy (h 2, h 1, h 3 )P i1 (h 2, h 3, h 1 ). Let b = (h 2, h 1, h 3 ). Since any reallocation of b makes agent i 1 worse off, b is Pareto efficient. However, b is not stable because agents i 2 and i 3 can get better off by exchanging their allotments from b. Example 1 suggests that there can be too many Pareto efficient allocations when there are externalities: if an agent has a unique most preferred allocation, it is Pareto efficient even when some other agents can become better off by reallocating their allotments from the allocation. Our analysis in the next two sections reveals that stability is a more relevant concept than Pareto efficiency in the presence of externalities. 3 Top Trading Cycles in a Housing Market with Egocentric Preferences 3.1 Top Trading Cycles Algorithm Shapley and Scarf (1974) introduce the top trading cycles (TTC) algorithm, developed by David Gale, as a method to find an allocation in the core of a housing market without externalities. Given a housing market N, H, R, e with egocentric preferences, we can apply the TTC algorithm to the related market N, H, R, e without externalities, where the algorithm proceeds as follows (see, for example, Roth, 1982 and Abdulkadiroğlu and Sönmez, 1998). Step 1. Each agent points to the owner of the house he prefers most. Since there is a finite number of agents, there exists at least one cycle of agents pointing to one another. Each agent in a cycle is assigned the endowment of the agent he points to and removed from the market. If there is at least one remaining agent, proceed to the next step. Otherwise, stop. In general, at: Step k. Each remaining agent points to the owner of the house he prefers most among the remaining houses. Each agent in a cycle is assigned the endowment of the agent he points to and removed from the market. If there is at least one remaining agent, proceed to the next step. Otherwise, stop. 8 When agents are indexed as i 1, i 2, and so on, we often describe an allocation as a list of houses where the first element in the list is assigned to i 1, the second to i 2, and so on. 10

11 Since there is a finite number of agents, the algorithm terminates within a finite number of steps. With strict preferences over houses, each agent at any step has a unique most preferred house. It is possible that multiple cycles form at a step, and in this case agents in all the cycles are removed simultaneously. Let K be the total number of steps in the algorithm, and let S k be the set of agents removed at step k, for k = 1,..., K. The algorithm proceeds in a unique way, producing an allocation as well as a partition {S 1,..., S K } of N. We call the allocation obtained by the TTC algorithm the TTC allocation of the market N, H, R, e. Note that any cycle formed at any step in the TTC algorithm is a trading cycle in the TTC allocation, and thus we call it a top trading cycle. We can regard S k as the union of top trading cycles formed at step k Properties of TTC Allocations When there are no externalities, the TTC algorithm produces the unique allocation in the core. Our first main result presents the properties that the TTC allocation possesses when there are externalities. Theorem 1. Consider a housing market N, H, R, e with egocentric preferences. The TTC allocation of N, H, R, e is stable in N, H, R and is in the ω-core of N, H, R, e. Proof. Let a A be the TTC allocation of N, H, R, e, and let {S 1,..., S K } be the partition of N obtained from the TTC algorithm. First, we show that a is stable in N, H, R. Suppose to the contrary that there exists an allocation b such that b a and b(i) P i a(i) for all i T := {i N : b(i) a(i)}. Let k be the smallest k such that S k T. Choose any i S k T. Then b(i ) P i a(i ). Since a(i ) is agent i s most preferred house at step k, i := b(i ) belongs to S k for some k < k. 10 Since i S k, there is some j S k such that a(j) = i. Since S k T =, we have b(j) = a(j). Then we have b(i ) = i = b(j) and i j, contradicting the one-to-one property of the allocation b. Next, we show that a is in the ω-core of N, H, R, e. Suppose to the contrary that there exists an allocation b and coalition T such that b dominates a via T in N, H, R, e. Let T = {i T : b(i) a(i)}. Then T is nonempty, and let k be the smallest k such that S k T. Choose any i S k T. Since b(i ) a(i ) and br i a, we have b(i ) P i a(i ). Since a(i ) is agent i s most preferred house at step k in the TTC algorithm, i := b(i ) belongs to S k for some k < k. Since b(t ) = e(t ) and i T, we have i T and thus 9 More precisely, a top trading cycle for a housing market can be defined as a trading cycle in which each agent obtains his favorite house. Then S k is the union of top trading cycles for the housing market with the remaining agents at step k. 10 With an abuse of notation, for any allocation a A, we sometimes use a(i) to refer to the owner of a(i), i.e., e 1 (a(i)), when there is no ambiguity about the endowment allocation e. 11

12 b(i ) = a(i ). Starting from i S k T, we can trace the owners of assigned houses along the top trading cycle in S k, where each agent in the cycle belongs to S k T. Hence, there is some j S k T such that a(j) = i. Then b(j) = a(j), and we have b(i ) = i = b(j) and i j, contradicting the one-to-one property of the allocation b. Theorem 1 shows that the TTC allocation is stable and is in the ω-core. We illustrate the result of Theorem 1 with Example 1. The TTC allocation of the housing market in Example 1 is a = (h 2, h 3, h 1 ), which is obtained in one step. It can be checked that both a and b = (h 2, h 1, h 3 ) are in the ω-core. Also, a is stable while b is not. So we can interpret that the TTC algorithm selects a stable allocation among allocations in the ω-core. In other words, there is a stable allocation in the ω-core, while there can be an unstable allocation in it. Since the ω-core is the smallest core given our definition of domination, Theorem 1 remains valid when we adopt any other core concept. When there are no externalities and agents may be indifferent between distinct houses, the core defined by strong domination is nonempty and may contain unstable allocations, but it contains at least one stable allocation, which can be obtained by the TTC algorithm (see Shapley and Scarf, 1974; Roth and Postlewaite, 1977). When externalities are introduced, we obtain similar results even with strict preferences over own-allotments and a core notion defined by weak domination. There can be multiple stable allocations in the ω-core, as shown by the following example. Example 2. Consider a housing market with N = {i 1, i 2, i 3 }, H = {h 1, h 2, h 3 }, and e = (h 1, h 2, h 3 ). The agents have egocentric preferences, and their associated preferences over own-allotments are given by P i1 : h 2, h 3, h 1, Pi2 : h 1, h 3, h 2, Pi3 : h 1, h 2, h 3. Agent i 1 s preferences satisfy (h 2, h 3, h 1 )P i1 (h 2, h 1, h 3 ). The TTC allocation is a = (h 2, h 1, h 3 ), and thus a is stable in N, H, R and is in the ω-core of N, H, R, e by Theorem 1. Let b = (h 2, h 3, h 1 ). It can be checked that b is also stable and is in the ω-core. In Examples 1 and 2, agent i 1 gets better off when agents i 2 and i 3 exchange their allotments from the TTC allocation a to achieve allocation b. In Example 1, both agents i 2 and i 3 get worse off by moving from a to b, and thus b can precluded by imposing stability. In Example 2, however, there are conflicting interests between agents i 2 and i 3 ; agent i 3 benefits from moving from a to b while agent i 2 loses. In this situation, we can preclude neither of a and b because there will be at least one agent who objects to a change, and it creates the possibility of multiple stable allocations in the ω-core. In order to exclude situations like Example 2, we use the following restriction on preference profiles. We say that externality creators have aligned interests under a preference 12

13 profile R if the following condition is met: for any a, b A, if there exists j N such that a(j) = b(j) and ap j b, then either ap i b for all i N with a(i) b(i) or bp i a for all i N with a(i) b(i). The condition says that, if there are agents who can create an externality to another agent by reallocating their allotments, these agents should have common preferences regarding whether to implement the reallocation or not. In other words, if some agents have conflicting interests in reallocating their allotments, then every other agent who receives the same allotment should be indifferent to such a reallocation (that is, no externalities are created by the reallocation). To get a sense of the condition, consider the preference relations over own-allotments in Example 1. Agents i 1 and i 2 have common interests in reallocating h 2 and h 3 between themselves: both prefer (a(i 1 ), a(i 2 )) = (h 2, h 3 ) to (h 3, h 2 ). In this case, agent i 3 can have a strict preference between (h 2, h 3, h 1 ) and (h 3, h 2, h 1 ). In contrast, agents i 1 and i 2 have conflicting interests in reallocating h 1 and h 3 because both prefer h 3 to h 1. Then agent i 3 should be indifferent between (h 1, h 3, h 2 ) and (h 3, h 1, h 2 ). In this way, we can check whether externality creators have aligned interests under a given preference profile. 11 The aligned interests condition limits the extent to which a group of agents exerts externalities to the other agents. 12 For instance, it precludes the preference profile in Example 2: by reallocating their allotments from a = (h 2, h 1, h 3 ), agents i 2 and i 3 can make agent i 1 better off, but they have conflicting interests regarding the change. Our second main result presents a uniqueness result. Theorem 2. Consider a housing market N, H, R, e with egocentric preferences. If externality creators have aligned interests under R, then the TTC allocation of N, H, R, e is the unique allocation that is stable in N, H, R and is in the α-core of N, H, R, e. Proof. Suppose that externality creators have aligned interests under R. Let a A be the TTC allocation of N, H, R, e, and let {S 1,..., S K } be the partition of N obtained from the TTC algorithm. Suppose to the contrary that there is an allocation b, different from a, 11 Let a 1 = (h 1, h 2, h 3), a 2 = (h 1, h 3, h 2), a 3 = (h 2, h 1, h 3), a 4 = (h 2, h 3, h 1), a 5 = (h 3, h 1, h 2), and a 6 = (h 3, h 2, h 1). Consider the following egocentric preferences over allocations: a 3P i1 a 4P i1 a 5I i1 a 6P i1 a 1I i1 a 2, a 4P i2 a 2P i2 a 3P i2 a 5P i2 a 6P i2 a 1, and a 6P i3 a 4P i3 a 1P i3 a 3P i3 a 2I i3 a 5. It can be checked that externality creators have aligned interests under this preference profile, while the associated preference relations over ownallotments are those in Example Roughly speaking, under the aligned interests condition, it is easier to have externalities when agents have diverse preferences over own-allotments. For example, agents i 1 and i 3 have opposite preferences over own-allotments in Example 1, and this allows agent i 2 to have strict preferences over allocations. In the extreme case where every agent has identical preferences over own-allotments, the aligned interests condition permits no externalities. It can also be shown that the condition precludes the case where every agent has strict preferences over allocations. 13

14 that is stable in N, H, R and is in the α-core of N, H, R, e. Let T = {i N : b(i) a(i)}. Since b a, T is nonempty. Let k be the smallest k such that S k T. Choose any i S k T. Since b(i) = a(i) for all i S 1 S k 1, we have a(i ) R i b(i ). Since b(i ) a(i ), we have a(i ) P i b(i ). Suppose that b(i) S k for all i S k. Then S k T blocks b via a in N, H, R, a, which contradicts the stability of b. Hence, there exists i S k such that b(i) / S k. Let a be an arbitrary allocation such that a (i) = a(i) for all i S k. Let T = {i N : b(i) a (i)}. Since a (i) = a(i) for all i S k and S k T, we have S k T. By egocentricity, we know that a P i b for all i S k T. Since there exists i S k such that b(i) / S k, there exists i / S k such that b(i) S k. Since S k is a union of trading cycles in a with endowment e, we have a (i) / S k for all i / S k and thus (S k ) c T. Suppose that a P i b for all i (S k ) c T. Then a P i b for all i T, contradicting the stability of b. Hence, there exists i (S k ) c T such that bp i a. Then, by the aligned interests condition, we have a I i b for all i / T. This implies that S k α-blocks b in N, H, R, e, which contradicts that b is in the α-core. Theorem 2 shows that, under the aligned interests condition, there exists a unique stable allocation in the α-core. Since the α-core is the largest core and there exists a stable allocation in any core, Theorem 2 continues to hold when we use any other core concept. Theorem 2 generalizes the existing result without externalities that the TTC allocation is the unique allocation in the core. Without externalities, the aligned interests condition is satisfied trivially, the α-core coincides with the core, and any allocation in the core is stable. The role of the aligned interests condition in obtaining the uniqueness result can be interpreted as follows. Note that TTC allocations and stable allocations depend only on preferences R over own-allotments while the notion of domination uses preferences R over allocations. So in general, the equivalence between the TTC allocation and a stable allocation in the α-core cannot be expected. But under the aligned interests condition, it suffices to know R to determine whether a stable allocation is dominated or not. Suppose that an allocation a is stable in N, H, R and that an allocation b dominates a via coalition T in N, H, R, e. Let S = {i N : b(i) a(i)}. Note that S T, and thus there exists i S such that bp i a. If bp i a for all i S, then a is not stable, which is a contradiction. Hence, there exists i S such that ap i b. Then by the aligned interests condition, bi i a for all i / S, and only the preference profile R over own-allotments plays a role. The aligned interests condition does not imply that even the ω-core is a singleton. As mentioned following Theorem 1, the two allocations a = (h 2, h 3, h 1 ) and b = (h 2, h 1, h 3 ) are in the ω-core of the housing market in Example 1, while a preference profile over allocations can be constructed to satisfy the aligned interests condition as noted in footnote 11. So imposing stability is indispensable for obtaining a uniqueness result. 14

15 Remark 2. We can obtain the uniqueness result in Theorem 2 using a weaker sufficient condition. We say that the housing market N, H, R, e exhibits aligned interests within trading cycles if the following condition is met: for any a, b A, if S is a trading cycle in a with endowment e such that ap i b for all i S with a(i) b(i) and bp i a for some i / S with a(i) b(i), then ar i b for all i S with a(i) = b(i). This condition is weaker than the previous aligned interests condition in that an agent can be affected by an externality even when the externality creators have conflicting interests, but the direction of the externality should be aligned with the preferences of the externality creators in the same trading cycle. The housing market in Example 2 does not satisfy this condition: For a = (h 2, h 1, h 3 ) and b = (h 2, h 3, h 1 ), we have bp i1 a, ap i2 b, and bp i3 a, while agents i 1 and i 2 constitute a trading cycle in a. The proof of Theorem 2 goes through under the alternative condition, when we take S k as a trading cycle, instead of a union of trading cycles, that has an overlap with T. The proof also works when we further weaken the alternative condition by requiring S to be a trading cycle in the TTC allocation and a to coincide with the TTC allocation on S. In our next result, we characterize the set of TTC allocations obtained by varying the endowment allocation e for a given house allocation problem N, H, R. It is known that, without externalities, the set of TTC allocations is equal to the set of Pareto efficient allocations (see Lemma 1 of Abdulkadiroğlu and Sönmez, 1998). However, we can expect that this result does not generalize to the case of egocentric preferences for the following two reasons. First, TTC allocations depend only on preferences R over own-allotments, whereas Pareto efficiency requires the knowledge of preferences R over allocations. Second, any TTC allocation is stable as shown in Theorem 1, while the set of stable allocations can be strictly smaller than that of Pareto efficient allocations as shown in Example 1. The following result shows that we can generate any stable allocation as a TTC allocation for some endowment allocation. Proposition 1. Consider a house allocation problem N, H, R with egocentric preferences. An allocation a A is stable in N, H, R if and only if it is the TTC allocation of N, H, R, e for some endowment allocation e A. Proof. See the Appendix. From Proposition 1, we can see that TTC allocations are more closely related to stable allocations than to Pareto efficient allocations. As mentioned following Example 1, 15

16 the concept of Pareto efficiency is too weak when there are externalities. In the proof of Proposition 1, in order to show that any stable allocation a is a TTC allocation for some endowment allocation, we set the endowment allocation as a itself. Abdulkadiroğlu and Sönmez (1998) take exactly the same approach to prove that any Pareto efficient allocation is in the core for some endowment allocation when there are no externalities. Remark 3. When there are externalities, it is not obvious how to define competitive equilibrium, as an agent s most preferred house in a budget set may depend on others choices. However, with egocentric preferences, one s favorite house is well-defined for any budget set, and so we can apply the usual concept of competitive equilibrium. Then competitive equilibria of a housing market N, H, R, e with egocentric preferences coincide with those of the related housing market N, H, R, e without externalities. Thus, a competitive allocation of a housing market N, H, R, e with egocentric preferences is unique and given by the TTC allocation (see Theorem 2 of Roth and Postlewaite, 1977). 4 TTC Mechanism and Strategy-Proofness In this section, we fix N, H, and e, and consider a scenario where each agent i reports his preferences R i and an allocation is chosen based on agents reports. Again, we assume that each agent s preferences belong to the class of egocentric preferences, and thus he is required to report an egocentric preference relation. Let R be the set of all egocentric preference relations on A. A mechanism is a rule that selects an allocation for each egocentric preference profile. That is, a mechanism can be represented by a mapping φ : R N A. We can consider a mechanism that selects the TTC allocation for each preference profile. We call it the TTC mechanism and denote it by φ T. Note that the TTC mechanism utilizes only preferences over own-allotments, and this feature can be regarded as an informational advantage of the TTC mechanism in that it suffices for agents to report their preferences over own-allotments rather than those over allocations. We can also consider a mechanism that selects an allocation in the ω-core (resp. the α-core) for each preference profile and call it an ω-core mechanism (resp. an α-core mechanism). Without externalities, a core mechanism is unique, coinciding with the TTC mechanism. However, with egocentric preferences, there can be multiple allocations in the ω-core, and thus there can be an ω-core mechanism and an α-core mechanism different from the TTC mechanism. For any coalition T N, let R T = (R i ) i T R T and R T = (R i ) i/ T R N\T denote the preference profiles of agents in T and those not in T, respectively. When T consists of a single agent i, we write R i instead of R {i}. A mechanism φ is strategy-proof if for any R R N, for any i N, and for any R i R, φ(r)r iφ(r i, R i). That is, when a mechanism is strategy-proof, no agent can gain by misreporting his preferences. A mechanism φ is 16

17 coalitionally strategy-proof if for any R R N and for any nonempty T N, there exists no R T RT such that φ(r T, R T )R i φ(r) for all i T and φ(r T, R T )P i φ(r) for some i T. In other words, when a mechanism is coalitionally strategy-proof, no coalition of agents can make at least one member better off and none worse off by jointly misreporting their preferences. Obviously, if a mechanism is coalitionally strategy-proof, it is strategyproof. It has been shown that, when there are no externalities, the TTC mechanism is strategyproof (Roth, 1982) and coalitionally strategy-proof (Moulin, 1995, Lemma 3.3). 13 In the following proposition, we extend these existing results to the case where the domain of preferences consists of egocentric preferences. Proposition 2. The TTC mechanism φ T is coalitionally strategy-proof. Proof. See the Appendix. Roth (1982) proves that no agent can obtain a more preferred house by misreporting his preferences in the TTC mechanism. Similarly, Moulin (1995, Lemma 3.3) shows that, when a group of agents misreports their preferences in the TTC mechanism, no member can obtain a more preferred house without making another member receive a less preferred house. Thus, there is no profitable manipulation by an individual agent or a group of agents in which a misreporting agent receives a better house. Without externalities, this is sufficient for (coalitional) strategy-proofness. However, when there are externalities, there is another kind of potentially profitable manipulation in which misreports by an agent or a group of agents influence others allotments while misreporting agents receive the same allotments. In the proof of Proposition 2, we show that this second kind of manipulation cannot be profitable. When a group of agents misreports their preferences in the TTC mechanism in a way that they maintain the same allotments, the allotments of the remaining agents are not affected either. 14 We say that an allocation a A is individually rational in the housing market N, H, R, e if a(i) R i e(i) for all i N. Note that individual rationality can be checked only with preferences over own-allotments and that any allocation in the α-core is individually rational Bird (1984) proves that the TTC mechanism is coalitionally strategy-proof in a weaker sense that there is no joint misreport that makes every member better off, but in a more general setting where indifferences between distinct houses are allowed. Roth (1982) also considers a setting that allows indifferences. 14 A mechanism is called nonbossy if no agent can influence others allotments without changing his own allotment (Satterthwaite and Sonnenschein, 1981). Miyagawa (2002) argues that the TTC mechanism is nonbossy. It is easy to see that, if a mechanism is nonbossy, no group of agents can influence the outsiders allotments without changing the members allotments. 15 The preference ranking of houses below one s endowment house does not play a role when we apply the TTC algorithm and when we check whether an allocation in the α-core is stable. Hence, Theorems 1 and 2 as well as the results in this section remain valid as long as each agent s preference ranking of allocations satisfies the egocentricity assumption weakly above his endowment house. 17