Outside Options in Neutral Allocation of Discrete Resources

Transcription

1 Outside Options in Neutral Allocation of Discrete Resources Marek Pycia M. Utku Ünver This Draft: December 2016, First Draft: May 2007 Abstract Serial dictatorships have emerged as natural direct mechanisms in the literature on the allocation of indivisible goods without transfers. Svensson (Social Choice and Welfare, 1999) previously showed that serial dictatorships are the only neutral and group strategy-proof mechanisms when agents have no outside options and individual rationality constraints do not exist. In this paper, we take into account individual rationality constraints in detail. Our main result establishes that the class of group strategy-proof, object-neutral, non-wasteful, and individual rational mechanisms is equal to a class of mechanisms we call binary serial dictatorships under voluntary participation. This class is also Pareto e cient. Ünver gratefully acknowledges the research support of National Science Foundation. We thank Fuhito Kojima and Vikram Manjunath for discussions and an associate editor and two referees whose comments significantly shortened and improved the paper. Address: University of California at Los Angeles, Department of Economics, 9371 Bunche Hall Los Angeles, CA 90095, USA; pycia@ucla.edu Address: Boston College, Department of Economics, 140 Commonwealth Ave., Chestnut Hill, MA, 02467, USA; unver@bc.edu 1

2 1 Introduction Serial dictatorships have often emerged as natural direct mechanisms in the literature on the allocation of indivisible goods without transfers and with single-unit demands (cf. Hylland and Zeckhauser, 1979 model). Each serial dictatorship mechanism allocates goods by ordering agents, and then letting the first agent choose her most preferred house, thereafter letting the second agent choose her most preferred house among those still available, etc (see Svensson, 1994). There are many simple serial dictatorship mechanisms: one per each possible ordering of agents. The characterizations of serial dictatorships such as Svensson (1999) have not included agents outside options and the resulting individual rationality constraints. Svensson (1999) showed that serial dictatorship is the only neutral and group strategy-proof mechanism when agents have no outside options. A mechanism is neutral or object-neutral if its outcome does not depend on the names of objects. A mechanism is group strategy-proof if there is no group of agents that can misstate their preferences and obtain a weakly better house, and such that at least one agent in the group gets a strictly better house. We allow for the outside options: agents can remain unmatched if she chooses to, i.e., participation is voluntary. In this setting, individual rationality ensures voluntary participation: no agent is assigned a house worse than her outside option. We also use a weak e ciency property: A mechanism is non-wasteful if there is no unassigned house that an agent prefers to be matched with rather than her assignment. Our main result establishes that the class of group strategy-proof, object-neutral, nonwasteful and individually rational mechanisms is equal to a new class of mechanisms we call binary serial dictatorships. 1 This class of mechanisms extends and generalizes serial dictatorship to the setting with outside options. Each mechanism from this class starts by assigning a selected agent her most preferred house, or if the agent prefers her outside option to all houses by making the agent unmatched. As in serial dictatorship, a second agent is then selected and obtains her most preferred house, or remains unmatched. In contrast to the simple serial dictatorship, the identity of the second agent can depend on whether the first agent is matched with a house, or whether she is left unmatched. The mechanism then repeats the procedure, selecting a third agent whose identity depends on whether the first and second agent were matched with houses, or left unmatched, etc. We obtain two corollaries of our main characterization. Since sequential dictatorships are Pareto e result with Pareto e cient, we replace non-wastefulness and individual rationality in our characterization ciency to obtain a trivial corollary. We also obtain a characterization of 1 A mechanism is object-neutral in environments with outside options if its outcome does not depend on the names of objects (houses); permuting the names of the outside option and an object may a ect the mechanism s outcome. 2

3 serial dictatorships in a subdomain of our preference domain. If the outside option is always ranked last by all agents (as in Pápai, 2000) then a mechanism is group strategy-proof, objectneutral, and non-wasteful if and only if it is a serial dictatorship. 2 In the last section of the paper, we discuss the di erence between object- and outcome-neutrality in environments with outside options. In the setting with single-unit demands, in addition to Svensson (1999), serial dictatorships were studied by others: Svensson (1994) introduced serial dictatorships as a strategy-proof mechanism, Ergin (2000) characterized this class through a variable population and resource axiom, consistency, in addition to neutrality. Abdulkadiroğlu and Sönmez (1998) showed that, given a fixed preference profile, each Pareto e serial dictatorship. 3 Sönmez and cient outcome can be obtained by running a Ünver (2010) studied neutrality and strategy-proofness, together with additional axioms, and allow agents to have property rights over some of the goods; they do not allow outside options (cf. Abdulkadiroğlu and Sönmez, 1999). These papers also did not allow agents to take outside options. The present paper is one of the first papers that analyzes outside option and voluntary participation in allocation of indivisible goods without transfers. Following its initial draft, others have examined outside options in related environments. Bu (2014) used neutrality and additional axioms to characterize sequential dictatorships. Erdil (2014) shows in a domain without transfers that non-wasteful and strategy-proof deterministic mechanisms are not dominated by strategy-proof deterministic mechanisms. Pycia and Ünver (2016b) characterized Arrovian e cient and strategy-proof mechanisms as a class very close to sequential dictatorships that di er only in the last step of its algorithm when there could be two agents owning two goods and they can trade those; Pycia and Troyan (2016) shows that a similar class of sequential-dictatorship-like mechanisms characterizes strong obvious strategy-proofness and Pareto e ciency. In school-choice domain, Kesten and Kurino (2016) show that with outside options there is no mechanism that Pareto-dominates the student-optimal stable school-choice mechanism. They also study maximal subdomains of preferences where such result no longer holds. In a more general setting with or without transfers, Alva and Manjunath (2016) recently show that if a pair of individual rational and strategy-proof mechanisms are participation equivalent (i.e., if at every problem every agent either receives her outside option under both mechanisms 2 We also provide a simple and direct proof extending to the environment with outside options Papai s (2001) insight that group strategy-proofness is equivalent to individual strategy-proofness and non-bossiness. 3 Abdulkadiroğlu and Sönmez also show that randomizing over serial dictatorships is equivalent to randomizing over Gale s top trading cycles (cf. Shapley and Scarf 1974, Ma 1994). For further studies of random serial dictatorships see Sönmez and Ünver (2005), Pathak and Sethuraman (2010), Che and Kojima (2010), Carroll (2010), Liu and Pycia (2011), and Pycia and Troyan (2016). 3

4 or assigned a real outcome under both) then they should be welfare equivalent. 4 While we show that Svensson s insight is basically robust to allowing agents to take outside options, there are many other standard mechanism design problems in which whether agents have the ability to take an outside option crucially a ects the standard results. For instance, in the setting with monetary transfers and quasi-linear utilities, the strategy-proof and e cient mechanisms of Groves (1973) lack, in general, individual rationality (see Green and La ont, 1977, and Holmstrom, 1979). Myerson and Satterthwaite (1983) impossibility of ex-post Pareto e cient and Bayesian incentive compatible bilateral trade crucially depends on individual rationality. The Coasian dynamics of Gul, Sonneschein, and Wilson (1986) hinges on the inability of buyers to take an outside option (see Board and Pycia, 2014). 2 House Allocation Problem with Outside Options Let I be a finite set of agents. Let H be a finite set of indivisible goods that we refer to as houses (following the terminology of Shapley and Scarf, 1974). Each agent i has a strict preference relation over H and her outside option denoted by ;. The strict preference relation is denoted by i. Let i be the induced weak preference relation from i, 5 that is for any x, y 2 H [ {;}, x i y () x = y or x i y. We denote the preference relation of agent i by the induced weak preference relation i. Let R be the set of preference relations. Let =( i ) i2i 2 R I be a preference profile. Each agent has not only right to hold on to her own house, but also have rights on the vacant houses, which are social endowments. Triple hi,h, i is a house allocation problem with outside options. An outcome of a problem is a matching. We define a submatching first. A submatching is an assignment that assigns a subset of agents a house or the remaining unmatched option, and no two agents the same house. Formally, for any given J I, a submatching is a oneto-one function : J! H [ {;} such that for any i, j 2 J, (i) = (j) ) i = j. We will occasionally use the set interpretation of functions to denote the submatching as well, 4 For other characterizations involving strategy-proofness in the house allocation domain see, for example, Pápai (2000), Ehlers (2002), Ehlers, Klaus, and Pápai (2002), Bogomolnaia, Deb, and Ehlers (2005), Kesten (2009), Bu (2014), Velez (2014), and Pycia and Ünver (2016a). See Sönmez and Ünver (2011) for a recent survey of the literature. In the setting with multi-unit demand, Pápai (2001), Ehlers and Klaus (2003), and Hatfield (2009) characterized sequential dictatorships not allowing for outside options. 5 The weak preference relation is a linear order on H, i.e. a binary relation on H that is antisymmetric, transitive, complete, and reflexive. 4

5 i.e., = {(i, (i))} i2j. Let S be the set of submatchings, which includes also the empty submatching?. We denote the set of agents over which the submatching is defined as I = J; moreover, let H be the houses matched in the submatching : H = (I ) \{;}. A matching is a submatching such that I = I. Let M denote the set of matchings. Let M = S\Mdenote the set of submatchings that are not matchings.. A mechanism is a systematic procedure that assigns a matching for each problem. Throughout the paper, we fix I and H, and thus, we denote a problem through the preference profile. Therefore, formally a mechanism is a function ' : R I! M. 3 The Axioms In this section, we introduce our axioms. Amatchingisindividually rational, if no agent receives a house worse than the remaining unmatched option. Formally, a matching µ 2 M is individually rational if for all i 2 I, µ(i) i ;. A mechanism is individually rational, if it always finds an individually rational matching. Amatchingisnon-wasteful, if no agent receives an option that is worse than a house that is unassigned. Formally, a matching µ 2 M is non-wasteful for all i 2 I, µ(i) i h for all h 2 H \ µ(i). Non-wastefulness would imply individual rationality if ; were considered as a house will I copies, which is an equivalent theoretical treatment of outside option with ours. Thus, one can think of individual rationality as a special instance of non-wastefulness. AmatchingisPareto e cient, if there is no matching that makes everybody weakly better o, and at least one agent strictly better o. That is, a matching µ 2 M is Pareto e cient if there exists no matching 2 M such that for all i 2 I, (i) i µ(i), and for some i 2 I, (i) i µ(i). A mechanism is Pareto e cient, if it always finds a Pareto e cient matching. Individual rationality, non-wastefulness and Pareto e ciency are related concepts. Lemma 1 If a matching is Pareto e cient then it is individually rational and non-wasteful. Proof of Lemma 1. Let µ be an individually irrational or wasteful matching. Then there exists some agent i 2 I, with g i µ(i) suchthateitherg = ; or g 2 H is not assigned to any agent. Consider the following matching : for all j 2 I\{i}, (j) = µ(j) and (i) = g. Clearly Pareto-dominates µ; hence, µ is not Pareto e cient. A mechanism is non-bossy if whenever an agent misreports her preferences and cannot change her house assigned by the mechanism, then she cannot change the matching assigned 5

6 by the mechanism, either. Formally, a mechanism ' is non-bossy if for all 2 R I,i2I, and 0 i2 R, '[ 0 i, i ](i) ='[ ](i) ) '[ 0 i, i ]='[ ]. A mechanism is strategy-proof if an agent cannot receive a better house by misreporting her preferences. Formally, a mechanism ' is strategy-proof if for all 2 R I, for all i 2 I, and 0 i2 R, '[ ](i) i '[ 0 i, i ](i). A mechanism is group strategy-proof if there is no group of agents that can misstate their preferences so that they all obtain a weakly better house and at least one agent in the group gets a strictly better house. Formally, a mechanism ' is group strategy-proof if there are no 2 R I, J I, and 0 J 2 R J such that '[ 0 J, J ](i) i '[ ](i) 8i 2 J, and '[ 0 J, J ](j) j '[ ](j) 9j 2 J. A mechanism is (Maskin) monotonic if whenever the preferences of agents change in a way such that the lower contour set at the assigned house under the original preferences is a subset of the lower contour set at the same house under the new preferences, then the matching assigned by the mechanism does not change. Formally, a mechanism ' is monotonic if for all, 0 2 R I and i 2 I, {h 2 H : '[ ](i) i h} {h 2 H : '[ ](i) 0 i h} ) '[ 0 ]='[ ]. Axioms of strategy-proofness, non-bossiness, group strategy-proofness and monotonicity are very related concepts, and the following lemmata show their relationships: Lemma 2 (Pápai, 2000) A mechanism is group strategy-proof if and only if it is strategyproof and non-bossy. Lemma 3 (Takemiya, 2001) A mechanism is monotonic if and only if it is group strategyproof. These lemmata were previously proven in a domain without outside options but the proofs carry over to our setting. For independent interest, we provide a simple alternative proof of Lemma 2 here: Proof of Lemma 2. One direction is obvious. To prove the other direction, suppose ' is non-bossy and strategy-proof, and, contrary to the claim, suppose it is not group-strategy 6

7 proof. Then, there exists a coalition J I such that for some and 0 J,wehave'[ 0 J, J ](j) j '[ ](j) forallj 2 J and at least one agent has strict preference. Let 0 =[ 0 J, J]. Let µ = '[ ]andµ 0 = '[ 0 J, J]. Consider J such that j ranks µ 0 (j) firstand µ(j) secondforallj 2 J such that µ(j) 6= µ 0 (j) and j ranks µ(j) firstforallj 2 J such that µ 0 (j) =µ(j). For any j 2 J, strategy-proofness implies that '[ j, j ](j) j '[ ](j) =µ(j), and, hence, '[ j, j ](j) 2 {µ(j),µ 0 (j)}; the strategy-proofness also implies that µ(j) ='[ ](j) j '[ j, j ](j), and, hence, '[ j, j ](j) =µ(j) ='[ ](j). By nonbossiness, '[ j, j ]='[ ]. Proceeding in this way, we can replace j with j in one at atimeforeachj 2 J and we end up obtaining '[ J, J]='[ K, K]=µ for all K J. Proceeding similarly, we can show that '[ J, J]=µ 0. Indeed, for any j 2 J, the strategy-proofness implies that '[ j, 0 j](j) j '[ 0 ](j) = µ 0 (j), and, hence, '[ j, j ](j) =µ 0 (j). By non-bossiness, '[ j, 0 j]='[ 0 ]. We can thus modify each 0 j without changing the relative ranking of µ(j) andµ 0 (j) bypushingallotheroptionsbelowµ(j) forall agents in J, one agent at a time, and we still have µ 0 as the outcome of ' at each step. We have thus shown that '[ 0 J, J](j) =µ 0 = µ = '[ ](j), a contradiction proving that ' is group strategy-proof. Next, we introduce an object neutrality concept. In order to introduce this concept, we need to define some other concepts: A relabeling is a function : H [ {;}! H [ {;} is a one-to-one and onto function with (;) =;. That is, under a relabeling, the names of houses are exchanged. Let be the set of relabeling functions. For example, under relabeling 2, for house h 2 H, (h) ishouseh s new name. For any 2 R I,and 2, therelabeled preference profile 2 R I is such that for any i 2 I, x i y () 1 (x) i 1 (y) 8x, y 2 H [ {;}. That is, under the relabeled preference profile, the original names of the houses are replaced by their new names. A mechanism is object-neutral if renaming of houses results with everybody receiving the house which is the renamed version of her old assignment. Formally a mechanism ' is object-neutral if for any 2 R I and 2, '[ ](i) = ('[ ](i)) 8i 2 I. 4 Binary Serial Dictatorships In this section, we introduce our proposed mechanisms that characterize the axioms group strategy-proofness, object neutrality, individual rationality, and non-wastefulness. First, we 7

8 need to introduce some new concepts. We start with the definition of a standard sequential dictatorship. A sequential order is a function f : M! I such that f( ) 2 I \ I for all 2 M. Asequential dictatorship is a mechanism f, which is induced by a sequential order f, and its outcome is found by the following iterative algorithm given a preference profile : Step 1: Agent i 1 = f(?) isassignedherfavoriteoptioninh [{;}; let this option be denoted as h 1.. Step t: Let t 1 = {(i 1,h 1 ), (i 2,h 2 ),...,(i t 1,h t 1 )}. Agent i t = f( t 1 )isassigned her favorite option in [H \{h 1,h 2,...,h t as h t. 1 }][{;}; let this option be denoted Sequential dictatorships are strategy-proof, non-bossy, and Pareto e not object-neutral in general. cient. But they are We need to restrict the set of sequential dictatorships considerably to obtain an objectneutral mechanism. A binary serial order is a sequential order f such that for all, 0 2 M satisfying I = I 0 and (i) =; () 0 (i) =; for all i 2 I,wehavef( )=f( 0 ). We refer to a sequential dictatorship induced by a binary serial order as a binary serial dictatorship. Since all it matters is whether the previous agents are assigned the outside option or a house in determining the next agent with the priority to choose, we can substantially simplify the way we denote a binary serial order: Let B = {J {0, 1}} J(I. An element 2 B is referred to as a binary submatching and can also be denoted in the functional form as well, i.e., (i) =b whenever (i, b) 2. Moreover, let (i, 0) refer to i is assigned the outside option and (i, 1) refer to i is assigned a house. Let I = J be the set of agents matched under. Redefine a binary serial order as a function f : B! I such that for all 2 B, f( ) 2 I \ I. Moreover, we refer to 2 S as consistent with 2 B if I = I and (i) =I{ (i) 2 H} for all i 2 I. 6 Given a sequential order f, arelevant submatching is a submatching = {(i s,h s )} t s=1 for some t 0suchthati s = f({(i 1,h 1 ),...,(i s 1,h s 1 )}) foralls<t. Let S f be the set of all relevant submatchings for f. These definitions hold for binary submatchings, too. In particular, let B f be the set of relevant binary submatchings for a binary serial order f. We also define another subclass of sequential dictatorships. A serial dictatorship is a sequential dictatorship f such that f( )=f( 0 )forall, 0 2 M such that I = I 0. We refer to such a sequential order f as a linear order. 6 I{apple} =1ifapple is a true statement, and I{apple} = 0 otherwise. 8

9 5 The Main Characterization Our main theorem of the paper is stated as follows: Theorem 1 A mechanism is group strategy-proof, object-neutral, non-wasteful, and individually rational if and only if it is a binary serial dictatorship. We prove Theorem 1 using two propositions: Proposition 1 Any binary serial dictatorship is group strategy-proof, object-neutral, nonwasteful, individually rational, and Pareto e cient. Proof of Proposition 1. Let f be a binary serial dictatorship. Since f is a hierarchical exchange mechanism as introduced by Pápai (2000), by her main theorem (although she does not have outside options in her model explicitly, her proof can almost without generality modified so that), f is group strategy-proof and Pareto e cient. Since the definition of f does not depend on the names of houses assigned, but it depends on whether an agent remains unassigned or receives a house, f is object-neutral. Its Pareto e ciency implies non-wastefulness and individual rationality. Proposition 2 Every group strategy-proof, object-neutral, non-wasteful, and individually rational mechanism ' induces a binary serial order f such that f = '. Proof of Proposition 2. Let ' be a group strategy-proof, object-neutral, non-wasteful, and individually rational mechanism. By Lemma 1, ' is individually rational. By Lemma 2, ' is strategy-proof and non-bossy. By Lemma 3, ' is monotonic. Consider a preference profile 2 R I with = {(i s,g s )} t s=1 2 S, where g 1,...,g t 2 H [{;}, and with x 2 (H \{g 1,g 2,...,g t }) [ {;} such that 1. for all i s 2 I, is ranks the houses {g 1,g 2,...,g s 1 } \ H in order of their indexes, then g s, and then other options. 2. for all i 2 I \ I, i ranks houses {g 1,g 2,...,g t } \ H in order of indexes, then x, and then other options. Let R,x be the domain of such preference profiles. For some binary serial order f that will be constructed below, we will show that for any 2 R I, '[ ] = f [ ]. We use induction in our construction and and main part of our proof: 9

10 Initial Step: Pick h 2 H. Consider a profile?,h 2 R?,h. There exists some agent i = ' 1 [?,h ](h) bynon-wastefulness. Setf(?) =i. By monotonicity for all 2 R I such that agent i ranks h first, we have '[ ](i) =h. By object neutrality, i receives her first choice in every profile if it is a house. By individual rationality she receives the outside option whenever she ranks it first. Inductive Assumption: Fix t 2 {2, 3,..., I 1}. For all 2 B with I = t 0 <t, assume that we constructed f( ); and for all = {(i s,g s )} t0 s=1 2 S, which is consistent with and satisfying {(i s 0,g s 0)} s s 0 =1 2 Sf for all s apple t 0, we proved v[ ](i s )=g s = f [ ] for all s apple t 0 for all 2 R,x for all x 2 (H \{g 1,g 2,...,g t 0}) [ {;}. We prove that the inductive assumption also holds for t 0 = t: Let 2 B be such that I = t. If 62 B f then arbitrarily set f( ). Assume otherwise, i.e., 2 B f. Thus, for all 2 S that is a consistent submatching with,wehave = {(i s,g s )} t s=1 such that {(i s,g s )} t0 s=1 2 S f for all t 0 apple t. Let 2 R,h for some house h. By the inductive assumption, for all i s 2 I, v[ ](i s )=g s. By non-wastefulness, there exists some i 2 I \ I such that v[ ](i) =h. Set f( )=i. For any 0 = [ {(i, g)} and 0 2 R 0,x for any option x, g 2 (H \{g 1,...,g t }) [ {;}, wehavev[ 0 ](i) =g if g 2 H, by monotonicity and object neutrality of ', and if g = ;, byindividualrationalityof'. By monotonicity and the inductive assumption, '[ 0 ](i s )=g s for all s apple t. Moreover, by construction of f [ 0 ](i s )=g s for all s apple t and f [ 0 ](i) =g, concluding the proof of the induction. f,wehave To finish the proof, take an arbitrary 2 R I. For any 0 2 R f [ ],;, by the induction above, we have '[ 0 ]= f [ 0 ]. Since is a monotonic transformation of 0 for both ' and f, by monotonicity we have '[ ] ='[ 0 ]= f [ 0 ]= f [ ], concluding the proof. Our main theorem also has an immediate corollary: Corollary 1 A mechanism is group strategy-proof, object-neutral, and Pareto e only if it is a binary serial dictatorship. cient if and The result follows from Theorem 1, Proposition 1, and Lemma 1. 10

11 6 Independence of the Axioms In this section, we relax each axiom one at a time, and show that there exists a mechanism which is not a binary serial dictatorship and yet satisfies the remaining axioms. We consider group strategy-proofness as two axioms by Lemma 1: strategy-proofness and non-bossiness. Example 1 A mechanism that is non-strategy-proof, non-bossy, object-neutral, non-wasteful, and individually rational: Take a binary serial order. Run the associated binary serial dictatorship with the following modification: Reverse the preference order of each agent for all houses that she ranked higher than the outside option and keep the relative order of other options the same. Example 2 A mechanism that is strategy-proof, bossy, neutral, non-wasteful, and individually rational: Let f,f 0 be two binary serial orders such that the initial dictator is the same agent: i = f(?) =f 0 (?), but otherwise the orders do not match in general, i.e., f 6= f 0. Let ' be a mechanism such that ( f [ ] if 8h 2 H, h i ; '[ ] = f 0 [ ] otherwise, i.e., the binary serial order that will be used in the binary serial dictatorship is determined by the preferences of the initial dictator (but not necessarily by her assigned option), depending on whether she prefers all houses to the outside option or not. Example 3 A mechanism that is strategy-proof, non-bossy, non-object neutral, nonwasteful, and individually rational: A top-trading-cycles mechanism (a la Pápai, 2000) that gives the ownership rights of objects to at least two di erent agents at the beginning. Example 4 A mechanism that is strategy-proof, non-bossy, object-neutral, wasteful, and individually rational: A mechanism that leaves every agent always unmatched. Example 5 A mechanism that is strategy-proof, non-bossy, object-neutral, non-wasteful, and individually irrational: Take a binary serial order. Run the associated binary serial dictatorship with the following modification: During her turn each agent is assigned the best available house according to her preferences if there are still available houses (even if the outside option is preferred to that house) and the option otherwise. 11

12 7 Serial Dictatorships with Outside Options We also obtain a variant of Svensson s serial dictatorship characterization result. Svensson s (1999) characterization critically hinges on the the fact that there are no outside options. A natural way to relax this assumption without imposing voluntary participation and individual rationality constraints is as follows: Consider a preference domain where the outside option is always ranked at the bottom of preferences. This is the preference domain invoked in Pàpai (2000) and in the following literature. Let this restricted set of preferences be denoted by ˆR. Proposition 3 A mechanism defined over ˆR I is group strategy-proof, neutral, and nonwasteful if and only if it is a serial dictatorship. The proof of one direction follows from Proposition 1 and the proof of the other direction is identical to the proof of Proposition 2 verbatim using the restricted domain. 7 Here, unlike Svensson s characterization, non-wastefulness is not a redundant axiom. A mechanism that leaves all agents unmatched satisfies all axioms but non-wastefulness. 8 Dichotomy of Neutrality and Outside Options: Scarcity vs Abundance Before concluding, we would like to discuss the role of object neutrality in our main characterization. Our object neutrality is an axiom specific to houses but not to the outside option. 8 This is the natural definition of neutrality in our domain. This can be contrasted with outcome neutrality, i.e., the outcome of the mechanism is independent of the name of the option each agent is assigned. These two neutrality conditions coincide with the neutrality axiom in Svensson (1999) s domain. On the other hand, in our domain outcome neutrality does not have a natural translation, as outside option can be assigned to anybody who wants it while the houses are scarce and each can be assigned to only one agent. One may think that sequential dictatorship structure in our characterization is not unexpected given Svensson s (1999) characterization of serial dictatorships without outside options. So one may expect that given that we have two types of goods, houses and outside option, and 7 Since the preference domain of this result is included in Pycia and Ünver (2016a), this proposition can alternatively be proven using their characterization. However, Theorem 1 cannot. 8 See Sönmez and Ünver (2010) and Bu (2014) for characterizations with other weakenings of neutrality in similar domains. 12

13 we only have neutrality among houses, a binary serial dictatorship is the expected mechanism in the characterization. However, consider another weakening of neutrality such that there is no outside option but there are two classes of houses, where neutrality strictly applies within each class but not across classes. We have two types of goods, again. But now, we can have top trading cycles mechanisms (TTC) (cf. Pápai, 2000, also known as hierarchical exchange) where in each round two di erent agents, but not a single one, are endowed by the mechanism the property rights of these two classes of houses, respectively. Thus, each of these two agents would get her top available choice if it is one of her houses, and they would trade with each other if they like each other s houses the most. However whenever there is a conflict in their top choices, the property right holder for that conflicted house would win, not necessarily the same agent. If there were three such classes of houses, then we can even have trading cycles mechanisms with brokers (TC) (cf. Pycia and Ünver, 2016a). Thus, subclasses of sequential dictatorships cannot characterize group strategy-proof, non-wasteful, and individually rational mechanisms under all weakenings of neutrality. A similar richer class arises when outside option can be attained by less than I agents at a time using our object neutrality axiom. Thus, a weakening of neutrality seems to induce a meaningful restriction other than sequential dictatorships only when the goods are scarce. An outside option can be interpreted as a good with multiple copies such that the copies are abundant and su cient for all agents. Therefore, the agent, who owns the property right of this outside option good in the TTC or TC sense, does not have any bargaining power in trade as this good is abundant. Hence, all scarce goods need to be owned by a single agent, alattcsense,tosatisfyourobjectneutrality,whiletheownershipstatusoftheoutside option does not matter. Moreover, the dichotomy of our object neutrality gives us choice who will own all houses next: if the current agent is assigned a house we can choose one agent, and if the current agent is assigned the outside option, we can choose another agent. Indeed, we can even have a richer class of sequential dictatorship problem domains with such properties. Suppose we have multiple types of outside options that can each be attained by any agent, and agents value such outside options di erently. In this case, a variant of our characterization would continue to hold: In this case, instead of a binary serial dictatorship, we will obtain a (k + 1) sequential dictatorship where k is the number of di erent outside options, and recursively define its sequential order as follows: f( [ {(i, x)} can have k +1 di erent values according to one of the k +1di erent types of options x can be, i.e., one of the k outside options or a house, where is a relevant submatching of the sequential order f and i = f( ). Our proof can be extended to this case easily. 13

14 References [1] Atila Abdulkadiroğlu and Tayfun Sönmez (1998) Random serial dictatorship and the core from random endowments in house allocation problems, Econometrica, 66: [2] Atila Abdulkadiroğlu and Tayfun Sönmez (1999) House allocation with existing tenants, Journal of Economic Theory, 88: [3] Nanyang Bu (2014) Characterizations of the sequential priority rules in the assignment of object types, Social Choice and Welfare, 43: [4] Samson Alva and Vikram Manjunath (2016) Dominant strategy mechanism design with outside options, working paper. [5] Simon Board and Marek Pycia (2014) Outside Options and the Failure of the Coase Conjecture American Economic Review 104: 656?671. [6] Anna Bogomolnaia, Rajat Deb, and Lars Ehlers (2005) Strategy-proof assignment on the full preference domain, Journal of Economic Theory, 123: [7] Anna Bogomolnaia and Herve Moulin (2001) A New Solution to the Random Assignment Problem, Journal of Economic Theory, 100, 295?328. [8] Carroll, G. (2010): A General Equivalence Theorem for Allocation of Indivisible Objects, Working paper. [9] Che, Y.-K. and F. Kojima (2010) Asymptotic Equivalence of Random Priority and Probabilistic Serial Mechanisms, Econometrica, 78, 1625?1672. [10] Lars Ehlers (2002) Coalitional strategy-proof house allocation, Journal of Economic Theory, 105: [11] Lars Ehlers and Bettina Klaus (2003) Coalitional strategy-proof and resource-monotonic solutions for multiple assignment problems, Social Choice and Welfare, 21: [12] Lars Ehlers, Bettina Klaus, and Szilvia Pápai (2002) Strategy-proofness and populationmonotonicity in house allocation problems, Journal of Mathematical Economics, 38: [13] Aytek Erdil (2014) Strategy-proof stochastic assignment, Journal of Economic Theory, 151:

15 [14] Haluk Ergin (2000) Consistency in house allocation problems, Journal of Mathematical Economics, 34: [15] Faruk Gul, Hugo Sonnenschein, and Robert Wilson (1986) Foundations of Dynamic Monopoly and the Coase Conjecture. Journal of Economic Theory 39: 155?90. [16] Jerry R. Green and Jean-Jacques La ont (1977) Characterization of Satisfactory Mechanisms for the Revelation of Preferences for Public Goods, Econometrica, 45: [17] Theodore Groves (1973) Incentives in Teams, Econometrica, 41: [18] John W. Hatfield (2009) Strategy-proof and nonbossy quota allocations, Social Choice and Welfare, 33: [19] Bengt Holmstrom (1979) Moral Hazard and Observability, The Bell Journal of Economics, 10: [20] Aanund Hylland and Richard Zeckhauser (1979) The e cient allocation of individuals to positions, Journal of Political Economy, 87: [21] Onur Kesten (2009) Coalitional strategy-proofness and resource monotonicity for house allocation problems, International Journal of Game Theory, 38: [22] Onur Kesten and Morimitsu Kurino (2016) Do outside options matter in matching? A new perspective on the trade-o s in student assignment, working paper. [23] Qingmin Liu and Marek Pycia (2011) Ordinal E ciency, Fairness, and Incentives in Large Markets, Working Paper. [24] Jinpeng Ma (1994) Strategy-proofness and strict core in a market with indivisibilities. International Journal of Game Theory, 23: [25] Roger B. Myerson and Mark A. Satterthwaite (1983) E cient mechanisms for bilateral trading, Journal of Economic Theory, 29: [26] Szilvia Pápai (2000) Strategyproof assignment by hierarchical exchange. Econometrica, 68: [27] Szilvia Pápai (2001) Strategyproof and nonbossy multiple assignments, Journal of Public Economic Theory, 3: [28] Pathak, P. A. and J. Sethuraman (2010) Lotteries in Student Assignment: An Equivalence Result, Theoretical Economics, forthcoming. 15

16 [29] Marek Pycia and Peter Troyan (2016) Obvious Dominance and Random Priority, working paper. [30] Marek Pycia and M. Utku Ünver (2016) Incentive Compatible Allocation and Exchange of Indivisible Resources, Theoretical Economics, forthcoming. [31] Marek Pycia and M. Utku resources, working paper. Ünver (2016) Arrovian e ciency in allocation of discrete [32] Lloyd Shapley and Herbert Scarf (1974) On cores and indivisibility, Journal of Mathematical Economics, 1: [33] Tayfun Sönmez and M. Utku Ünver (2005) House Allocation with Existing Tenants: An Equivalence Games and Economic Behavior 52: [34] Tayfun Sönmez and M. Utku Ünver (2010) House allocation with existing tenants: A characterization, Games and Economic Behavior, 69: [35] Tayfun Sönmez and M. Utku Ünver (2011) Matching, allocation, and exchange of discrete resources, in J. Benhabib, A. Bisin, and M. Jackson (eds.), Handbook of Social Economics, Vol. 1A. The Netherlands: North-Holland, [36] Koji Takemiya (2001) Coalition strategy-proofness and monotonicity in Shapley-Scarf housing markets, Mathematical Social Sciences, 41: [37] Lars-Gunnar Svensson (1994) Queue allocation of indivisible goods, Social Choice and Welfare, 11: [38] Lars-Gunnar Svensson (1999) Strategy-proof allocation of indivisible goods, Social Choice and Welfare, 16: [39] Rodrigo A. Velez (2014) Consistent strategy-proof assignment by hierarchical exchange, Economic Theory, 56: