House allocation when availability of houses may change unexpectedly

Mathematical Social Sciences 81 (2016) 29–37

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House allocation when availability of houses may change unexpectedly✩ Azar Abizada a,∗ , Siwei Chen b a

School of Business, ADA University, 11 Ahmadbay Aghaoglu St., Baku AZ1008, Azerbaijan

b

Lingnan College, Sun Yat-Sen University, Guangzhou, 510275, China

article

info

Article history: Received 11 February 2014 Received in revised form 20 February 2016 Accepted 1 March 2016 Available online 19 March 2016

abstract We study the problem of allocating a set of objects, e.g. houses, tasks, offices to a group of people having preferences over these objects. For various reasons, there may be more or fewer objects than initially planned and allocated. How should such unexpected changes be handled? One way is to declare the initial decision irrelevant and reallocate all available objects. Alternatively, one can use the initial decision as starting point in allocating the new objects. Since both perspectives seem equally reasonable, a natural robustness principle on the rule is that it should produce the same outcome no matter which one is taken. We define two robustness properties based on this idea, pertaining to more objects and fewer objects, respectively. We characterize the family of rules that satisfy mild efficiency, fairness and incentives requirements, together with either one of our robustness properties. They are the family of serial dictatorship rules. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Many universities offer on-campus housing for their faculty members. Each year, a university has to allocate the available houses among the new faculty members. Each faculty member can receive at most one house. The faculty members have the outside option of living off campus. The university provides the new faculty members with the complete list of houses (vacant or not). Each faculty member then submits strict preferences over the houses and the outside option. We look for rules allocating houses to faculty members. Some other applications of our model are assigning vacant offices among new employees, assigning on campus apartments among new graduate students, and so on. A house is an indivisible good, or ‘‘object’’. The problem of allocating objects among people is first studied by Hylland and Zeckhauser (1979). In the basic model, the set of objects is fixed, and each person receives at most one object. Various properties of efficiency, fairness, and robustness under strategic behavior

✩ We would like to thank William Thomson for his guidance, and invaluable comments. We also would like to thank two anonymous referees and the participants of GAMES 2012 and SED 2013 for their feedbacks. ∗ Corresponding author. E-mail addresses: [email protected] (A. Abizada), [email protected] (S. Chen).

http://dx.doi.org/10.1016/j.mathsocsci.2016.03.002 0165-4896/© 2016 Elsevier B.V. All rights reserved.

have been analyzed in this context (Hylland and Zeckhauser, 1979; Svensson, 1999; Ergin, 2000; Ehlers and Klaus, 2004; Bu, 2012). We consider situations where the set of objects is not fixed. Throughout the paper, we will use the example of on-campus housing for faculty members. Although in real life, the assignment procedure may not be exactly the same as described here, we choose this (partially artificial) example for two reasons: (i) it does capture important features of real-life problems, (ii) it helps us explain the key notions clearly and intuitively. Suppose that an assignment of available houses among incoming faculty members has been made. Closer to the arrival of the new faculty members, it may happen that some houses become unavailable: some current faculty members who were supposed to terminate their leases choose to extend their stay. Thus, their current houses become unavailable for this year. It is also possible that some additional houses become available: some current faculty members decide to terminate their leases earlier (either they move off campus or for some other reason). Thus, their houses become vacant. When one of these unexpected changes happens, how should university handle it? In general, when the circumstances in which a group of people find themselves change, there might be several perspectives that can be taken and the document binding the people as a group may not always specify which one should be chosen. Yet, outcomes may differ depending on which perspective is taken. When outcomes differ, no matter what action is chosen, someone maybe unhappy

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A. Abizada, S. Chen / Mathematical Social Sciences 81 (2016) 29–37

that the other action is not taken. Therefore, the choice of which perspective should be taken may need to have a good justification. To avoid such difficulties that one can face, a natural robustness principle on the rule is that when there are more than one equally legitimate way of handling a change, all should produce the same outcome. To understand what we consider as legitimate perspectives, consider the following scenario: after a decision has been made about some first situation, a second situation occurs in which opportunities have changed. The first position that we may take is to declare the choice made in the first situation irrelevant. Then, we simply cancel the initial payoffs and handle the second situation on its own. Alternatively, we can use the initial payoffs as starting point in dealing with the second situation. Both procedures seem equally reasonable. For that reason, we require that each person should receive the same payoff independent of which procedure is followed. When opportunities expand, this idea has been studied under the name of ‘‘step-by-step negotiations’’ for bargaining (Kalai, 1977), and it is also reminiscent of the ‘‘path independence’’ axiom for choice functions (Plott, 1973). ‘‘Lower composition’’ (Moulin, 2000) and ‘‘composition’’ (Young, 1988) for adjudication of conflicting claims, ‘‘lower composition’’ (Moulin and Stong, 2002) for queueing problem, ‘‘agenda independence’’ (Herrero and Villar, 1998, 2000) and ‘‘composition up’’ (Abizada and Chen, 2014) for rationing problem when preferences are single-peaked, and ‘‘composition’’ (Moreno-Ternero and Roemer, 2012) for resource allocation with utilities, can all be viewed as expressions of this robustness principle in different contexts. In the contexts of adjudication of conflicting claims and queueing problem, a shrinking of opportunities has been considered too. Axioms pertaining to this situation have been formulated (Moulin, 1987, 2000; Moulin and Stong, 2002; Abizada and Chen, 2014), which can also be understood as expressions of the general principle. This is the first paper that applies this general robustness principle to the objects assignment problem. We formulate two main robustness requirements, pertaining to either contraction or expansion of opportunities. Suppose that a rule has been selected. Consider a problem and apply the rule to this problem, thereby obtaining an assignment. We first consider the possibility that opportunities shrink; in our application, this means that some current faculty members who were supposed to terminate their leases choose to extend them. Thus, their current houses become unavailable. One possible way to deal with this change is simply to cancel the initial assignment and to apply the rule to assign the remaining houses among everyone. Another possibility is to cancel the initial assignment and to apply the rule to assign the remaining houses among the initial assignees only; after all, they are the ones who were promised houses. Both of these procedures seem equally legitimate in handling the change in the number of available houses. However, some people may prefer the first procedure to the second, and the others the second to the first. In such cases, no matter which of the two procedures the university follows, someone would complain and justifying particular choice may be difficult. To avoid such difficulties, we require that no matter which perspective the university follows, the welfare of each person should be the same.1 The second robustness requirement pertains to the opposite changes in opportunities; in our application it means that some current faculty members decide to terminate their leases earlier. Thus, additional houses become available. Once again, one possible way to handle such a change is to cancel the initial assignment and

1 Since preferences of the people are strict, this requirement is equivalent to requiring the outcomes to be the same.

to apply the rule to assign all available houses (including the newly available ones). Another procedure is simply to let initial assignees to keep their assigned houses and to apply the rule to assign the newly available houses among the initially unassigned people. What we do in the second procedure is to ignore the people who are fully satisfied in assignment of the newly available resources. This is in line with the usual procedure followed in the literature in describing this robustness principle. However, there are some critical differences between our model and the models studied earlier. In all of the previous models the allocated resource was homogeneous. Therefore, when resources suddenly increase, people who were initially (fully) satisfied, were not interested in the newly available resource, as it was the same resource. Differently, in our model resources are objects that differ from one another and when resources suddenly increase, new objects which differ from the already available ones are introduced. Therefore, even if a person is (fully) satisfied, i.e. he was initially assigned an object, he may still be interested in obtaining one of the newly available objects instead of his initial assignment. Therefore we need to modify the (second) procedure described above to consider interested initial assignees as well. Otherwise, this procedure may result in inefficient outcome.2 We propose the following alternative procedure (instead of the second procedure described above) to handle the increase in the available houses, the university informs all the new faculty members of the newly available houses, and asks whether any of them would be interested in any of these houses. Some faculty members may prefer one (perhaps several) of the new houses to their initial assignments. The university considers all such faculty members for the assignment of new houses and declares their initial assignments released and thus available.3 The university then asks the other faculty members whether any of them would be interested in any of the houses just released. Some of them may indeed prefer one (perhaps several) of these houses to their initial assignments. The university adds these faculty members to the group that is considered for the available houses and declares their initial assignments released and thus available. This process continues until no one prefers any of the available houses to his initial assignment. The university maintains the initial assignments for the faculty members who are satisfied with their initial assignments, and applies the rule to assign available houses among the faculty members who are interested in them. Once again, this procedure and the one described earlier (canceling initial decision and reassigning) provide equally legitimate way of handling the change in the number of available houses. In order to avoid possible complaints, we require that no matter which of these two procedures the university follows, the welfare of each person should be the same.4 In addition to our robustness requirements, we are also interested in the following desirable requirements: a mild efficiency requirement that no person should prefer an unassigned house to his assignment (non-wastefulness); a mild fairness requirement that the chosen assignment should not depend on the names of

2 It is possible that some of the initial assignees prefer some of the newly available houses to their initial assignments. The formulation in which the initial assignees keep their assignments ignores the initial assignees’ preferences when additional houses become available. Thus, it is inefficient. 3 The houses that were initially assigned to these faculty members are declared available since each faculty member can be assigned at most one house. Moreover, some other faculty members may prefer one of these houses to their assignments. By declaring them available, we could recover possible efficiency losses. 4 As one can note, both of our robustness requirement is a weaker notion of ‘‘consistency’’ requirement which was studied for various resource allocation models (Thomson, 1990, 1994; Tadenuma and Thomson, 1991), including our indivisible object allocation model (Ergin, 2000; Sönmez and Ünver, 2010).

A. Abizada, S. Chen / Mathematical Social Sciences 81 (2016) 29–37

houses (neutrality); a strong incentive compatibility requirement that no person should ever benefit by misrepresenting his preferences (strategy-proofness); finally, a mild technical robustness requirement that if a person’s preferences change and his assignment remains the same, then each other person’s assignment should remain the same as well (non-bossiness). We characterize the family of allocation rules satisfying either one of the robustness requirement together with the requirements just stated. It is the well known family of ‘‘serial dictatorship’’ rules. To define such a rule, we first pick an order on the set of people. The serial dictatorship rule associated with that order works as follows: we let the people arrive one at a time in the selected order; we assign the first person his most preferred house; assign the second person his most preferred house among the remaining ones; and so on, until each person has been considered. Svensson (1999) characterizes the serial dictatorship rules using a subset of the requirements that we impose (neutrality, non-bossiness, and strategy-proofness). However, his result does not imply ours, due to the differences in the models studied in these two papers: our model is more general in that (i) we do not require the number of people and the number of houses to be the same and (ii) we allow people to have an outside option, i.e. the option of not receiving any house. Ergin (2000) also characterizes the serial dictatorship rules using a strong ‘‘consistency’’ requirement and the notion of ‘‘neutrality’’. The rest of the paper is organized as follows. In Section 2 we define the model. In Section 3 we define axioms. In Section 4 we define the family of serial dictatorship rules and present our main results. In Section 5 we have concluding remarks and discussions. 2. Model There is a finite set of potential people N = {1, 2, . . . , n}. Let N ⊂ N be the set of people present. There is a finite set of potential houses H = {h1 , h2 , . . . , hm }. Let H ⊂ H be the set of available houses. Each person has the outside option of living off campus. We denote this option ∅. Each person can be assigned at most one house. Each person i ∈ N has a strict preference relation Ri over the potential houses and the option of living off campus, that is, over the set H ∪ {∅}. Let R be the set of all preference relations. Let R ≡ (Ri )i∈N be the preference profile. Let RN be the set of all preference profiles. A problem is a list π ≡ (N , H , R) ∈ 2N × 2H × RN . Let 5 be the set of all problems. An assignment for π ≡ (N , H , R ) is a mapping µ : N −→ H ∪ {∅} such that each h ∈ H is assigned to at most one person, that is, |{i ∈ N : µi = h}| ≤ 1. Let A(π) be the set of all assignments for π . A rule associates with each problem an assignment for it.  Formally, it is a mapping ϕ : Π → π∈Π A(π ) such that for each π ∈ Π , ϕ(π ) ∈ A(π ). 3. Axioms Before defining the requirements that we are interested in, we need to introduce several notions. Let (N , H , R) ∈ Π . Let µ ∈ A(N , H , R). House h is acceptable to person i with preferences Ri if hPi ∅. Let Di (H , R ) ≡ {h ∈ H : hPi ∅} be the set of acceptable houses for person i. Let Chi (H , R ) ∈ H ∪ {∅} be the most preferred house for person i, i.e. for each h ∈ H ∪ {∅}, Chi (H , R)Ri h. We call a person who is assigned a house at µ an assignee, and a person who is not assigned any house a non-assignee. Let M (µ) ≡ {i ∈ N : µi ∈ H } be the set of assignees at µ and U (µ) ≡ N \ M (µ) = {i ∈ N : µi = ∅} be the set of non-assignees at µ. A permutation on H is a oneto-one mapping f : H ∪{∅} → H ∪{∅} such that f (∅) = ∅. Let F be the set of allpermutations on H . For each f ∈ F and each H ⊆ H , let f (H ) ≡ h∈H {f (h)}. For each R ∈ RN , let R f ∈ R be defined as

31 f

follows: for each i ∈ N and each pair h, h′ ∈ H ∪ {∅}, f (h)Pi f (h′ ) if and only if hPi h′ . Also, we abuse notation by using µj instead of {µj }, to simplify the expression. We will do so throughout the paper whenever it does not cause any confusion. Next, we introduce the axioms. Let ϕ be a rule. Our first requirement is  that no person should prefer an unassigned house in H \ j∈N µj or his outside option to his assignment. This is a very mild efficiency requirement.5 Non-wastefulness: For each (N , H , R)  ∈ Π and each i ∈ N, if µ ≡ ϕ(N , H , R), then for each h ∈ (H \ j∈N µj ) ∪ {∅}, µi Ri h. Our next requirement is that the chosen assignment should be independent of the names of houses. Neutrality: For each (N , H , R) ∈ Π and each f ∈ F , we have ϕ(N , f (H ), Rf ) = f (ϕ(N , H , R)). Next we introduce a strong incentive compatibility requirement. No person should ever benefit by misrepresenting his preference. Strategy-proofness: For each (N , H , R) ∈ Π , each i ∈ N, and each R′i ∈ R, we have ϕi (N , H , R)Ri ϕi (N , H , (R′i , R−i )). Next we introduce two technical robustness requirements. First, if a person’s preferences change, and his assignment remains the same, then assignment of other people should remain the same as well (Satterthwaite and Sonnenschein, 1981). Non-bossiness: For each (N , H , R) ∈ Π , each i ∈ N, and each R′i ∈ R, if ϕi (N , H , (R′i , R−i )) = ϕi (N , H , R), then ϕ(N , H , (R′i , R−i )) = ϕ(N , H , R). A weaker requirement is that if the preferences of a nonassignee in U (ϕ(N , H , R)) change, and he remains unassigned, then assignment of other people should remain the same as well. Weak non-bossiness: For each (N , H , R) ∈ Π , each i ∈ U (ϕ(N , H , R)), and each R′i ∈ R, if ϕi (N , H , (R′i , R−i )) = ϕi (N , H , R), then ϕ(N , H , (R′i , R−i )) = ϕ(N , H , R). If more houses are available, then each person should be made at least as well off as before (Roemer, 1986; Chun and Thomson, 1988). This is a solidarity requirement. Resource monotonicity: For each (N , H , R) ∈ Π , each H ′ ⊃ H, and each i ∈ N, we have ϕi (N , H ′ , R)Ri ϕi (N , H , R). Suppose that, after an assignment is made, some people take their assigned houses and leave. Forthe reduced problem of assigning the remaining houses, H \ j̸∈N ′ ϕj (H , R), among the remaining people, N ′ , we should obtain the same allocation as before. This consistency idea has been formulated and studied for various models (Thomson, 1990, 1994; Tadenuma and Thomson, 1991; Ergin, 2000; Sönmez and Ünver, 2010). Consistency: For each (N , H , R) ∈ Π and each N ′ ⊆ N, we have for each i ∈ N ′ , ϕi (N ′ , H \ j̸∈N ′ ϕj (N , H , R), RN ′ ) = ϕi (N , H , R). Next, we introduce two composition properties related to possible changes in the set of houses. These are our central concepts. Let an assignment be chosen for a problem. We first consider the possibility that some houses become unavailable. One possible way to deal with this change is simply to cancel the initial assignment and to apply the rule to assign the remaining houses among everyone. Another possibility is to cancel the initial assignment and to apply the rule to assign the remaining houses among the initial assignees only; after all, they are the ones who were promised houses. Both of these procedures provide a

5 A widely used notion for property rights is individual rationality, which says that no person should be worse than his initial situation (i.e. outside option of not taking part in assignment). Formally, Individual Rationality: For each (N , H , R) ∈ Π and each i ∈ N, if µ ≡ ϕ(N , H , R), then µi Ri ∅. Our notion of non-wastefulness implies individual rationality, and thus is stronger requirement.

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A. Abizada, S. Chen / Mathematical Social Sciences 81 (2016) 29–37

plausible way of handling the change in the set of houses. However, some people may prefer the first procedure to the second, and the others may prefer the second to the first. Therefore, no matter which procedure university follows, some people may complain that the university should have considered the other action. In order to avoid possible complaints, we require the assignments of both procedures to be the same.6 Composition down7 : For each (N , H , R) ∈ Π and each H ′ ⊂ H, if µ ≡ ϕ(N , H , R), then

∀i ∈ U (µ),

ϕi (N , H ′ , R) = ∅,

∀j ∈ M (µ),

ϕj (N , H ′ , R) = ϕj (M (µ), H ′ , RM (µ) ).

and

Remark 1. It can be easily verified that composition down is weaker than consistency. For consistency, we consider all possible scenarios of a group of people leaving with their assignments. For composition down, we restrict our attention to the situations where only people who are not assigned any house leave. The next requirement pertains to the opposite change in the set of houses. Let an assignment be chosen for a problem, and let some additional houses become available: it may happen that some current faculty members decide to terminate their leases earlier. Two perspectives can be taken in handling the change. Once again, one possibility is to cancel the initial assignment and to apply the rule to assign the available houses, including the newly available ones among everyone. Another possibility is simply to let initial assignees to keep their assigned houses and to apply the rule to assign the newly available houses among the initial non-assignees. As before, in order to avoid possible complaints, we require the assignments of both procedures to be the same. Composition up8 : For each (N , H , R) ∈ Π and each H ′ ⊃ H, if µ ≡ ϕ(N , H , R), then

∀i ∈ M (µ),

ϕi (N , H ′ , R) = ϕi (N , H , R),

∀j ∈ U (µ),

ϕj (N , H , R) = ϕj (U (µ), H \ H , RU (µ) ). ′

and

Let an assignment µ be chosen for a problem (N , H , R). Let some additional houses, H ′ \ H, become available. Here too, two perspectives can be taken in handling the change. One possibility is simply to cancel the initial assignment and assign all available houses, H ′ , including the new ones. The other perspective that we propose is as follows: the university informs all the faculty members of the newly available houses, and asks whether any of them would be interested in any of these houses. Some faculty members, 1N 1 (π , µ, H ′ ), may indeed prefer one (perhaps several) of the new houses to their initial assignments. The university considers all such faculty members for the assignment of the new houses and declares the houses 1H 1 (π , µ, H ′ ) initially assigned to them available. The university then asks the other faculty members whether any of them would be interested in any of the houses just released. Some of them, 1N 2 (π , µ, H ′ ), may prefer one (perhaps several) of these houses to their initial assignments. The university adds these faculty members to the group that is considered for the assignment of the available houses and declares the houses, 1H 2 (π , µ, H ′ ), initially assigned to them available. (That is, the set of available houses is augmented by the houses 1H 2 (π , µ, H ′ ) that have just been freed.) This continues until no faculty member prefers any of the available houses to his initial assignment. For those faculty members, who prefer their initial assignments, N \ 1N s (π , µ, H ′ ), the university keeps their initial assignments. Then available houses, 1H s (π , µ, H ′ ), are assigned among the remaining faculty members, 1N s (π , µ, H ′ ). Once again, both of these procedures provide a plausible way of handling the change. In order to avoid possible complaints, we require the assignments of both procedures to be the same. For each π = (N , H , R) ∈ Π , each µ ∈ A(N , H , R), and each H ′ ⊃ H, let

1H 0 (π , µ, H ′ ) ≡ H ′ \ H . 1N 1 (π , µ, H ′ ) ≡ {i ∈ N : there is h ∈ 1H 0 (π , µ, H ′ ), hPi µi }.  1H 1 (π , µ, H ′ ) ≡ µi . i∈1N 1 (π ,µ,H ′ )

′

Our second robustness requirement, composition up, is a very strong requirement. As we will show in the next proposition it is incompatible with even a very weak notion of efficiency. Therefore, it is not a so desirable requirement. We consider it only because the formulation matches similar properties studied in other problems where they are appealing.

1N (π , µ, H ) ≡ {i ∈ N \ 1N 1 (π , µ, H ′ ) : there is h ∈ 1H 1 (π , µ, H ′ ), hPi µi }.  1H 2 (π , µ, H ′ ) ≡ µi . 2

′

i∈1N 2 (π ,µ,H ′ )

···  1N (π , µ, H ) ≡ t

′

i∈N\

Proposition 1. No rule satisfies non-wastefulness and composition up. Proof. Let ϕ be non-wasteful and satisfy composition-up. Let N = {1} and H = {h1 , h2 }. Preferences are as follows: h1 R1 h2 . Let H = {h2 }. Let µ ≡ ϕ(N , H , R). Then, by non-wastefulness, µ1 = h2 . Let house h1 become available, i.e. H ′ = {h1 , h2 }. By composition up, ϕ1 (N , H ′ , R) = h2 . Since h1 R1 h2 , ϕ1 (N , H ′ , R) = h2 contradicts non-wastefulness.  In general, for each class of problems, each principle needs to be adapted in order to best take account of the structure of the class. In the light of Proposition 1, we need to explore alternative expressions of the composition idea when new houses become available.

6 In fact, the requirement is that two procedures should provide the same welfare to each person. But since preferences of the people are strict, this requirement is equivalent to requiring the assignments to be the same. 7 The terminology that we use is proposed by Thomson (2003). 8 The terminology that we use is proposed by Thomson (2003).



1N r (π , µ, H ′ ) :

r ≤ t −1

 there is h ∈ 1H

1H t (π , µ, H ′ ) ≡

··· .

t −1

(π , µ, H ), hPi µi . ′



µi .

i∈1N t (π ,µ,H ′ )

Let s be such that 1N s (π , µ, H ′ ) = ∅. Since there are finitely many people, such s exists. By definition, sets 1N 0 (·), . . . , 1N s (·). s Also note that sets 1H 0 (·),  . . . , 1tH (·) are ′disjoint. ′ ′  Let 1Nt (π , µ, H′ ) ≡ t ≤s 1N (π , µ, H ) and 1H (π , µ, H ) ≡ t ≤s 1H (π , µ, H ). Iterative composition up: For each π ≡ (N , H , R) ∈ Π with µ = ϕ(π ) and each H ′ ⊃ H, if N ′′ ≡ 1N (π , µ, H ′ ) and H ′′ ≡ 1H (π , µ, H ′ ), then

∀i ∈ N \ N ′′ , ϕi (N , H ′ , R) = ϕi (N , H , R),

and

∀j ∈ N ′′ , ϕj (N , H ′ , R) = ϕj (N ′′ , H ′′ , RN ′′ ). To make the notion clear, we present an example.

A. Abizada, S. Chen / Mathematical Social Sciences 81 (2016) 29–37

Example 1 (Illustration of Iterative Composition Up). Let N {1, 2, 3} and H = {h1 , h2 , h3 , h4 }. Preferences are as follows R1 h1 h2 h3 h4

R2 h2 h3 h1 h4

=

R3 h4 h2 h1 h3

33

time according to the order, and assign to each person his most preferred house among the available ones. Formally, the serial dictatorship rule10 associated with ≻∈ Γ , SD≻ , is defined as follows: For each (N , H , R) ∈ Π , let N = {k1 , k2 , . . . , kl } with k1 ≻ k2 ≻ . . . ≻ kl . SD≻ k1 (N , H , R) ≡ Chk1 (H , R), ≻ SD≻ k2 (N , H , R) ≡ Chk2 H \ SDk1 (N , H , R), R ,





··· Let H = {h2 , h3 , h4 }. Let ϕ be a rule. Let µ ≡ ϕ(N , H , R). Let µ1 = h2 , µ2 = h3 , and µ3 = h4 . Let house h1 become available, i.e. H ′ = {h1 , h2 , h3 , h4 }. Two perspectives can be taken. (i) The first perspective is to cancel the initial assignment and assign the new set of houses H ′ = {h1 , h2 , h3 , h4 }. We apply rule ϕ to (N , H ′ , R). Let µ′ ≡ ϕ(N , H ′ , R). (ii) The second perspective is to take assignment µ as starting point. Then, we proceed in the following iterative steps to identify H ′′ in the definition of iterative composition up. Let 1H 0 (π , µ, H ′ ) = {h1 }. Step 1: First, we identify the set of people who prefer the newly introduced house h1 to their initial assignments (i.e. 1N 1 (π , µ, H ′ )). Since h1 R1 h2 = µ1 , µ2 = h3 R2 h1 , and µ3 = h4 R3 h1 , we consider person 1 in assignment of house h1 . Person 1’s initial assignment, µ1 = h2 , is declared available (i.e. 1H 1 (π , µ, H ′ ) in the definition is {h2 }). Step 2: Next we identify the set of people who prefer the newly available house h2 to their initial assignments (i.e. 1N 2 (π , µ, H ′ )). Since, h2 R2 h3 = µ1 and µ3 = h4 R3 h2 , we consider person 2 in addition to person 1 from Step 1, in assignment of houses in {h1 , h2 }. Person 2’s initial assignment, µ2 = h3 , is declared available (i.e. 1H 2 (π , µ, H ′ ) in the definition is {h3 }). Since person 3 prefers his assignment to all the available houses {h1 , h2 , h3 }, we stop. We define H ′′ ≡ {h1 , h2 , h3 } and N ′′ ≡ {1, 2}. For person 3, we let him keep his initial assignment. We then apply the rule ϕ to (N ′′ , H ′′ , (R1 , R2 )). Let µ′′N ′′ ≡ ϕ(N ′′ , H ′′ , (R1 , R2 )) and µ′′3 ≡ ϕ3 (N , H , R). The requirement is µ′ = µ′′ . Next, we discuss the logical relations between iterative composition up and two standard requirements: consistency and resource monotonicity. Remark 2. It can be easily verified that iterative composition up is also weaker than consistency. For consistency, we consider all possible scenarios of a group of people leaving with their assignments. For iterative composition up, if we take (N , H ′ , R) as the initial problem, then we only consider the reduced problems where for different H ⊂ H ′ , people in N \ 1N ((N , H , R), ϕ(N , H , R), H ′ ) leave with their assignments.

 SDkj (N , H , R) ≡ Chkj ≻

H\

j −1 

 SDki (N , H , R), R , ≻

i=1

···  SDkn (N , H , R) ≡ Chkl ≻

H\

l −1 

 SDki (N , H , R), R . ≻

i =1

It is easy to verify that the serial dictatorship rules are non-wasteful, neutral, strategy-proof, and non-bossy.11 The serial dictatorship rules are also resource-monotonic,12 and consistent.13 The next two results state that they satisfy iterative composition up and composition down. Proposition 2. The serial dictatorship rules satisfy iterative composition up. The proof of Proposition 2, together with the proofs of all the other results are provided in the Appendix. Proposition 3. The serial dictatorship rules satisfy composition down. Next is our first main result, which is a characterization of the serial dictatorship rules by imposing iterative composition up and some standard properties. Theorem 1. A rule ϕ satisfies non-wastefulness, neutrality, strategyproofness, weak non-bossiness, and iterative composition up if and only if there exists a linear order ≻∈ Γ such that ϕ = SD≻ . Next is our second main result. It differs from Theorem 1 in two ways. First, we replace iterative composition up with composition down. Second, we impose non-bossiness instead of weak nonbossiness. Theorem 2. A rule ϕ satisfies non-wastefulness, neutrality, strategyproofness, non-bossiness, and composition down if and only if there exists a linear order ≻∈ Γ such that ϕ = SD≻ .

• On the independence of the axioms in Theorems 1 and 2

Remark 3. Iterative composition up and resource monotonicity are logically unrelated. The proof of this claim is presented in the Appendix.

We define five rules that we will use to establish the independence of the axioms in our theorems. For some rules, it is straightforward to see whether they satisfy the axioms in our theorems or not. Whenever it is not so straightforward, we give examples to show violations. Let ϕ 1 be the rule defined as follows: For each (N , H , R) ∈ Π and each i ∈ N,

4. Results

ϕi1 (N , H , R) = ∅.

Our main results are characterizations of the following family of rules. A priority order is a linear order9 ≻ on N . Let 0 be the set of all priority orders. We associate a serial dictatorship rule to each such order. For each problem, we let the people arrive one at a

9 A linear order is a complete, transitive and antisymmetric binary relation.

10 This family has been introduced and widely studied in the literature under the name ‘‘serial dictatorship’’. 11 See Svensson (1999). 12 See Ehlers and Klaus (2004). 13 Ergin (2000) characterizes the serial dictatorship rules by weak efficiency, consistency, and neutrality. The key difference between his model and ours is that he assumes that the number of people and the number of houses are the same. Thus, his result has no implication in our setting.

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A. Abizada, S. Chen / Mathematical Social Sciences 81 (2016) 29–37

Let ϕ 2 be the rule defined as follows: Let ≻, ≻′ ∈ Γ be such that 1 ≻ 2 ≻ 3 ≻ 4 ≻ . . . ≻ n and 2 ≻′ 1 ≻′ 3 ≻′ 4 ≻′ . . . ≻′ n. Let h ∈ H . For each (N , H , R) ∈ Π ,



′

≻ ϕ (N , H , R) = SD≻ (N , H , R) if Ch2 (H , R) = h, SD (N , H , R) otherwise. 2

Let ϕ 3 be the rule defined as follows: Let ≻, ≻′ ∈ Γ be such that 1 ≻ 2 ≻ 3 ≻ 4 ≻ . . . ≻ n and 2 ≻′ 1 ≻′ 3 ≻′ 4 ≻′ . . . ≻′ n. Let h ∈ H . For each (N , H , R) ∈ Π ,

 ≻′ ϕ 3 (N , H , R) = SD≻ (N , H , R) if R2 = R1 , SD (N , H , R) otherwise. Example 2. The rule ϕ 3 is not strategy-proof. Let N = {1, 2}. Let H ≡ {h1 }. Let R ∈ RN be such that h1 P1 ∅, h1 P2 ∅, and R2 ̸= R1 . By the definition of ϕ 3 , we have ϕ 3 (N , H , R) = SD≻ (N , H , R). Thus, ϕ13 (N , H , R) = h1 and ϕ23 (N , H , R) = ∅. Let R′2 ≡ R1 . Suppose that when facing R−2 , person 2 with true preference R2 misrepresents his preference to be ′ R′2 . Then, ϕ 3 (N , H , (R′2 , R−2 )) = SD≻ (N , H , (R′2 , R−2 )). Thus, 3 3 ′ ϕ2 (N , H , (R2 , R−2 )) = h1 . Thus, ϕ2 (N , H , (R′2 , R−2 ))P2 ϕ23 (N , H , R). This is in violation of strategy-proofness. Let ϕ 4 be the rule defined as follows: Let ≻, ≻′ ∈ Γ be 1 ≻ 2 ≻ 3 ≻ . . . ≻ n − 1 ≻ n and 1 ≻′ 2 ≻′ 3 ≻′ . . . ≻ n ≻′ n − 1. For each (N , H , R) ∈ Π ,

 ≻′ ϕ 4 (N , H , R) = SD≻ (N , H , R) if R1 = R2 , SD (N , H , R) otherwise. Example 3. The rule ϕ 4 is not weakly non-bossy. Let H ≡ {h1 }. Let R ∈ RN be such that for each i ∈ N \{n − 1, n}, we have ∅Pi h1 , and h1 Pn−1 ∅, h1 Pn ∅, and R1 ̸= R2 . By the definition of ϕ 4 , we have ϕ 4 (N , H , R) = SD≻ (N , H , R). Thus, ϕn4−1 (N , H , R) = h1 , and for each i ̸= n−1, ϕi4 (N , H , R) = ∅. Let R′1 ≡ R2 . Suppose that when facing R−1 , person 1 reports his preference to be R′1 . Then, ϕ 4 (N , H , (R′1 , R−1 )) = ′

SD≻ (N , H , (R′1 , R−1 )). Thus, ϕn4 (N , H , (R′1 , R−1 )) = h1 , and for each i ̸= n, ϕi4 (N , H , (R′1 , R−1 )) = ∅. Thus, ϕ14 (N , H , (R′1 , R−1 )) = ϕ14 (N , H , R) = ∅, ϕn4 (N , H , (R′1 , R−1 )) ̸= ϕn4 (N , H , R), and ϕn4−1 (N , H , (R′1 , R−1 )) ̸= ϕn4−1 (N , H , R). This is in violation of weak non-bossiness. Let ϕ 5 be the rule defined as follows: Let ≻, ≻′ ∈ Γ be such that 1 ≻ 2 ≻ 3 ≻ 4 ≻ . . . ≻ n and 2 ≻′ 1 ≻′ 3 ≻′ 4 ≻′ . . . ≻′ n. Let h ∈ H . For each (N , H , R) ∈ Π ,

ϕ 5 (N , H , R) =



SD≻ (N , H , R) if |H | = 1, ′

SD≻ (N , H , R)

if |H | ≥ 2.

Example 4. The rule ϕ 5 does not satisfy iterative composition up. Let N = {1, 2}. Let H ≡ {h1 }. Let R ∈ RN be such that h1 P1 h2 P1 ∅ and h1 P2 h2 P2 ∅. Let π ≡ (N , H , R). By the definition of ϕ 5 , we have ϕ 5 (π ) = SD≻ (π ). Thus, 5 ϕ1 (π) = h1 and ϕ25 (π ) = ∅. Let H ′ ≡ {h1 , h2 }. Then, N ′′ ≡ 1N (π , ϕ 5 (π ), H ′ ) = {2} and ′′ H ≡ 1H (π , ϕ 5 (π ), H ′ ) = {h2 }. By the definition of ϕ 5 , we ′ have ϕ 5 (N , H ′ , R) = SD≻ (N , H ′ , R). Thus, ϕ15 (N , H ′ , R) = h2 , and ϕ25 (N , H ′ , R) = h1 . Thus, ϕ15 (N , H ′ , R) ̸= h1 = ϕ15 (π ). This is in violation of iterative composition up. Example 5. The rule ϕ 5 does not satisfy composition down.

Let N = {1, 2}. Let H ≡ {h1 , h2 }. Let R ∈ RN be such that h1 P1 ∅P1 h2 and h1 P2 h2 P2 ∅. ′ By the definition of ϕ 5 , we have ϕ 5 (N , H , R) = SD≻ (N , H , R). 5 5 Thus, ϕ2 (N , H , R) = h1 and ϕ1 (N , H , R) = ∅. Thus, M (ϕ 5 (N , H , R)) = {2}. Let H ′ ≡ {h1 }. By the definition of ϕ 5 , we have ϕ 5 (N , H ′ , R) = ≻ SD (N , H ′ , R). Thus, ϕ15 (N , H ′ , R) = h1 and ϕ25 (N , H ′ , R) = ∅. By the definition, we have ϕ25 ({2}, {h1 }, R2 ) = h1 . Thus, ϕ25 (N , H ′ , R) ̸= ϕ25 ({2}, {h1 }, R2 ). This is in violation of composition down. (1) The rule ϕ 1 satisfies all the axioms of Theorems 1 and 2 except for non-wastefulness. (2) The rule ϕ 2 satisfies all the axioms of Theorems 1 and 2 except for neutrality. (3) The rule ϕ 3 satisfies all the axioms of Theorems 1 and 2 except for strategy-proofness. (4) The rule ϕ 4 satisfies all the axioms of Theorem 1 except for weak non-bossiness. (4′ ) The rule ϕ 4 satisfies all the axioms of Theorem 2 except for non-bossiness. (5) The rule ϕ 5 satisfies all the axioms of Theorem 1 except for iterative composition up. (5′ ) The rule ϕ 5 satisfies all the axioms of Theorem 2 except for composition down. 5. Concluding remarks and discussions We study the problem of allocating a set of objects among a group of people having preferences over these objects. We define two robustness properties pertaining to unexpected changes in set of objects, more objects and fewer objects, respectively. We characterize the family of rules that satisfy mild efficiency, fairness and incentives requirements, together with either one of our robustness properties (Theorems 1 and 2). While defining robustness requirements we restricted ourselves to two perspectives that can be taken in handling the unexpected changes in the set of available houses. First and the most natural way is to cancel the initial assignment and re-assign the current set of houses. It has been used in almost all the previous models for which these requirements were studied. However, for the second perspective, which is taking the initial assignment as starting point and making assignment from thereon, there might be some alternatives procedures as well. For instance, when more houses become available, some possible alternative procedures are (i) assign the new houses among the non-assignees; if there are some houses left, assign them among the initial assignees, or (ii) assign the initially available houses among the non-assignees, and assign the new houses among the initial assignees. Similarly, when some houses become unavailable, some possible alternatives are (i) assign the remaining houses among the non-assignees, or (ii) assign the remaining houses among the initial assignees whose houses are still available. Obviously, any of these alternatives is less reasonable than the ones proposed in our requirements. Moreover, it is easy to see that a requirement associated with any of these alternatives would lead to incentive problems. It would be interesting to also take the planner (housing office) into consideration: if the planner is better off under a specific procedure, then having the same outcome under both procedures may not be the most crucial point of the analysis. However, it is beyond the scope of this paper, and will be left for future research. Appendix. Omitted proofs Proof of Remark 3. We first show that a resource monotonic rule does not necessarily satisfy iterative composition up.

A. Abizada, S. Chen / Mathematical Social Sciences 81 (2016) 29–37

Let ϕ 6 be the rule defined as follows: Let ≻∈ Γ be such that 1 ≻ 2 ≻ 3 ≻ . . . ≻ n. For each (N , H , R) ∈ Π and each i ∈ N,

ϕi6 (N , H , R) =



∅ if |H | = 1, SD≻ i (N , H , R) if |H | ≥ 2.

Clearly, ϕ 6 is resource monotonic. Let N = {1, 2, 3} and H = {h1 , h2 }. Let R ∈ RN be such that h1 P1 h2 P1 ∅, h1 P2 h2 P2 h3 P2 ∅, and h3 P3 h2 P3 ∅. Let π ≡ (N , H , R). By the definition of ϕ 6 , we have ϕ16 (π ) = h1 , ϕ26 (π ) = h2 , and 6 ϕ3 (π) = ∅. Let H ′ = {h1 , h2 , h3 }. Then, 1N (π , ϕ 6 (π ), H ′ ) = {3}, and 1H (π , ϕ 6 (π ), H ′ ) = {h3 }. Let N ′′ ≡ {3} and H ′′ ≡ {h3 }. By the definition of ϕ 6 , we have ϕ36 (N ′′ , H ′′ , R3 ) = ∅. Again, by the definition of ϕ 6 , we have ϕ 6 (N , H ′ , R) = SD≻ (N , H ′ , R). Thus, ϕ16 (N , H ′ , R) = h1 , ϕ26 (N , H ′ , R) = h2 , and ϕ36 (N , H ′ , R) = h3 . Then, ϕ36 (N , H ′ , R) ̸= ϕ36 (N ′′ , H ′′ , R3 ). This is in violation of iterative composition up. We next show that a rule satisfying iterative composition up is not necessarily resource monotonic. Let ϕ 7 be the rule defined as follows: Let ≻∈ Γ be such that 1 ≻ 2 ≻ 3 ≻ . . . ≻ n. For each (N , H , R) ∈ Π and each i ∈ N, ϕ 7 assigns the person with the highest priority his least preferred house among H ∪ {∅}. The rule then assigns the person with the second highest priority his least preferred house among the remaining ones, and so on. We show that ϕ 7 satisfies iterative composition up. Consider the following 2 cases: (1) Person i is assigned a house initially. Then, he would like to participate when new houses become available, since his assignment is below ∅, which is always available. (2) Person i is assigned ∅ initially. If one of the initially available houses is unassigned, then person i prefers this house to ∅, and would like to participate. If all the initially available houses are assigned, by Case (1), all the assignees would like to participate when new houses become available. Then, the houses assigned to them become available. Then, person i would like to participate. All the people would participate, which makes the two assignment problems of the two perspectives in the definition of iterative composition up identical. Thus, the assignments are the same. Let N = {1, 2} and H = {h1 }. Let R ∈ RN be such that h1 P1 ∅P1 h2 and h2 P2 ∅P2 h1 . By the definition of ϕ 7 , we have ϕ17 (N , H , R) = ∅ and 7 ϕ2 (N , H , R) = h1 . Let H ′ = {h1 , h2 }. Then, ϕ17 (N , H ′ , R) = h2 and ϕ27 (N , H ′ , R) = h1 . This is in violation of resource monotonicity.  Proof of Proposition 2. Let ≻∈ Γ . Suppose by contradiction that SD≻ violates iterative composition up: there are π = (N , H , R) ∈ Π , H ′ ⊃ H, and i ∈ N such that if µ ≡ SD≻ (N , H , R), µ′ ≡ SD≻ (N , H ′ , R), N ′′ ≡ 1N (π , µ, H ′ ), H ′′ ≡ 1H (π , µ, H ′ ), and µ′′ ≡ SD≻ (N ′′ , H ′′ , RN ′′ ), then either i ∈ N \ N ′′ and µ′i ̸= µi , or i ∈ N ′′ and µ′i ̸= µ′′i . Case 1: i ∈ N \ N ′′ and µ′i ̸= µi . By resource monotonicity, for each s ∈ N, µ′s Rs µs . Since µ′i ̸= µi , then µ′i Pi µi . By the definition of SD≻ , there is j ∈ N such that j ≻ i, µj = µ′i , and µ′j Pj µj . Again, by the definition of SD≻ , there is k ∈ N such that k ≻ j, µk = µ′j , and µ′k Pk µk . Since N is finite, repeating the argument, we obtain that there are l, m ∈ N such that m ≻ l, µ′l Pl µl , and µ′l = µm = Chm (H , R). If m ∈ N ′′ , then i, j, k, l ∈ N ′′ , which contradicts the assumption that i ∈ N \ N ′′ . Thus, m ∈ N \ N ′′ . By the definitions of N ′′ and H ′′ , for each h ∈ H ′′ , Chm (H , R)Pm h. Since H ′ \ H ⊆ H ′′ ,

35

then Chm (H ′ , R) = Chm (H , R). Since for each s ∈ N, µ′s Rs µs , then µ′m = Ch(H , R) = µm . This contradicts our previous conclusion that µ′l = µm . Case 2: i ∈ N ′′ and µ′i ̸= µ′′i . By Case 1, µ′N \N ′′ = µN \N ′′ . Since the people in N \ N ′ are as-

signed the same houses in (N , H ′ , R) as before, their presence does ≻ not affect the  assignments of the others. (N , H ′ , R) = N ′′   Thus, SD ≻ ′ ′′ ′ ′′ SD (N , H \ i∈N \N ′′ µi , RN ). Since i∈N \N ′′ µi = i∈N \N ′′ µi ,   we have H ′ \ i∈N \N ′′ µ′i = H ′ \ i∈N \N ′′ µi = H ′′ . Therefore, µ′′ = SD≻ (N ′′ , H ′′ , RN ′′ ) = SD≻ (N , H ′ , R) = µ′N ′′ . This contradicts N ′′ our assumption that µ′i ̸= µ′′i .  Proof of Proposition 3. Let ≻∈ Γ . Let (N , H , R) ∈ Π and H ′ ⊂ H. Let µ ≡ SD≻ (N , H , R) and µ′ ≡ SD≻ (N , H ′ , R). By resource monotonicity, for each i ∈ N, µi Ri µ′i . By the definition of SD≻ , for each i ∈ N, µ′i Ri ∅. Thus, for each i ∈ U (µ), we have µ′i = µi = ∅. Since all the people in U (µ) are assigned ∅ in (N , H ′ , R), by the definition of SD≻ , their presence does not affect the assignments ′ of the others. Thus, for each i ∈ M (µ), SD≻ = i (N , H , R) ′ SD≻ ( M (µ), H , R ) .  M (µ) i Proof of Theorem 1. By Proposition 2, the serial dictatorship rules satisfy iterative composition up. It is straightforward to verify that they satisfy the other properties. Conversely, let ϕ be a rule satisfying these properties. We show that there is ≻∈ Γ such that ϕ = SD≻ . We will identify the order in the following steps. Step 1: Identifying the person with the highest priority. Pick h ∈ H . Let R ∈ RN be such that for each pair i, i′ ∈ N , Ri = Ri′ , and hPi ∅. R1 = · · · = Rn

.. . h

.. . ∅ .. . By non-wastefulness, there is k ∈ N such that ϕk (N , h, R) = h. Claim 1. For each R′ ∈ RN , if h ∈ Dk (H , R′ ), then ϕk (N , h, R′ ) = h. Proof of Claim 1. Let R′ ∈ RN and h ∈ Dk (H , R′ ). Let i ∈ N \ {k}. Let i’s preferences change from Ri to R′i . Since hPi ∅, by strategyproofness, ϕi (N , {h}, (R′i , R−i )) ̸= h. Thus, ϕi (N , {h}, (R′i , R−i )) = ∅. By weak non-bossiness, ϕ(N , {h}, (R′i , R−i )) = ϕ(N , {h}, R). Let j ∈ N \ {k, i}. Let j’s preferences change from Rj to R′j . Since hPj ∅, by strategy-proofness, ϕj (N , {h}, (R′i , R′j , R−ij )) ̸= h. Thus, ϕj (N , {h}, (R′i , R′j , R−ij )) = ∅. By weak non-bossiness, ϕ(N , {h}, (R′i , R′j , R−ij )) = ϕ(N , {h}, R). Repeating the argument, we obtain ϕ(N , {h}, (Rk , R′−k )) = ϕ(N , {h}, R). Let k′ s preferences change from Rk to R′k . If ϕk (N , {h}, R′ ) = ∅, then when k’s true preferences are R′k and he faces R′−k , he is better off by misrepresenting his preferences to be Rk . This is in violation of strategy-proofness. Thus, ϕk (N , {h}, R′ ) = h.  Claim 2. For each R′ ∈ RN and each h′ ∈ Dk (H , R′ ), we have

ϕk (N , {h′ }, R′ ) = h′ .

Proof of Claim 2. Let R′ ∈ RN and h′ ∈ Dk (H , R′ ). Let f ∈ F be such that f (h) = h′ , f (h′ ) = h, and for each h′′ ∈ H \ {h, h′ }, f (h′′ ) = h′′ . Since h′ ∈ Dk (H , R′ ) and f (h′ ) = h, then h ∈ Dk (H , (R′ )f ). By Claim 1, ϕk (N , {h}, (R′ )f ) = h. By neutrality, f (ϕk (N , {h′ }, R′ )) = ϕk (N , {f (h′ )}, (R′ )f ) = h. Thus, ϕk (N , {h′ }, R′ ) = h′ . 

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A. Abizada, S. Chen / Mathematical Social Sciences 81 (2016) 29–37

Claim 3. For each (N , H , R′ ) ∈ Π , we have ϕk = Chk (H , R′ ). Proof of Claim 3. Let (N , H , R′ ) ∈ Π . There are two cases. Case 1: Chk (H , R′ ) = ∅. By non-wastefulness, ϕk (N , H , R′ )R′k ∅. By the definition of Chk (H , R′ ), we have ∅R′k ϕk (N , H , R′ ). Thus, ϕk (N , H , R′ ) = ∅ = Chk (H , R′ ). Case 2: Chk (H , R′ ) ∈ H. Let h′ ≡ Chk (H , R′ ). By the previous result, ϕk (N , {h′ }, R′ ) = ′ h . Let the set of available houses change from {h′ } to H. Let N ′ ≡ 1N ((N , {h′ }, R′ ), ϕ(N , {h′ }, R′ ), H ). Since ϕk (N , {h′ }, R′ ) = Chk (H , R′ ), then k ̸∈ N ′ . By iterative composition up, ϕk (N , H , R′ ) = ϕk (N , {h′ }, R′ ) = h′ = Chk (H , R′ ).  Pick h ∈ H. Let R ∈ RN be such that for each i, i′ ∈ N , Ri = Ri′ , and there exists h′ ∈ H \ {h}, hPi ∅Pi h′ . Let R¯ ∈ RN be such that for each i ∈ N , Chi (H , R¯ ) = ∅. Claim 4. For each N ⊆ N , if k ∈ N, then ϕk (N , {h}, RN ) = h. Proof of Claim 4. By non-wastefulness, some person in N is assigned h. Suppose by contradiction that there is j ∈ N \ {i} such that ϕj (N , {h}, RN ) = h. By non-wastefulness, for each i ∈ N , ϕi (N , {h′ }, (RN , R¯ N \N )) = ∅. Let the set of available houses change from {h′ } to {h, h′ }. Let π ≡ (N , {h}, (RN , R¯ N \N )). Let N ′ ≡ 1N (π , ϕ(π ), {h, h′ }) and H ′ ≡ 1H (π , ϕ(π ), {h, h′ }). Then, N ′ = N and H ′ = {h}. By iterative composition up, ϕj (N , {h, h′ }, (RN , R¯ N \N )) = ϕj (N ′ , H ′ , RN ′ ) = ϕj (N , {h}, RN ) = h. This contradicts Claim 2. Thus, ϕk (N , {h}, RN ) = h.  Claim 5. For each (N , H , R′ ) ∈ Π , if k ∈ N, then ϕk (N , H , R′ ) = Chk (H , R′ ). The proof is the same as the proofs for Claims 1–3, replacing N by N. We have shown that for each problem, person k is always assigned his most preferred house. Thus, person k has the highest priority. We denote him as d1 . Step 2: Identifying the person with the second highest priority. Let RN \{d1 } and h be the same as before. R1 = · · · Rd1 −1 = Rd1 +1 · · · Rk

.. . h

.. . ∅ .. . By non-wastefulness, there is l ∈ N \ {d1 } such that ϕl (N \

{d1 }, {h}, RN \{d1 } ) = h.

By a similar argument as the one in Step 1, we obtain that for each (N , H , R ′ ) ∈ 5, if d1 ̸∈ N and l ∈ N , then ϕl (N , H , R ′ ) = Chl (H , R ′ ). Claim 6. For each (N , H , R′ ) ∈ Π , if d1 , l ∈ N and h ≡ Chd1 (H , R′ ), then ϕl (N , H , R′ ) = Chl (H \ {h}, R′ ). Proof of Claim 6. Let (N , H , R′ ) ∈ Π with d1 , l ∈ N. Let h ≡ Chd1 (H , R′ ). By Claim 5, ϕd1 (N , H , R′ ) = h. There are two cases. Case 1: Chl (H \ {h}, R′ ) = ∅. Since ϕd1 (N , H , R′ ) = h, then Chl (H \ {h}, R′ )R′l ϕl (N , H , R′ ). By non-wastefulness, ϕl (N , H , R′ )R′l ∅. Thus, ϕl (N , H , R′ ) = ∅ = Chl (H \ {h}, R′ ). Case 2: Chl (H \ {h}, R′ ) ∈ H.

Let h′ ≡ Chl (H \ {h}, R′ ). By Claim 5, ϕd1 (N , {h, h′ }, R′ ) = h. By the above result in bold, ϕl (N \ {d1 }, {h′ }, R′N \{d1 } ) = h′ . By iterative composition up, ϕl (N , {h, h′ }, R′ ) = ϕl (N \ {d1 }, {h′ }, R′N \{d1 } ) = h′ . Let the set of available houses change from {h, h′ } to H. Let N ′ ≡ 1N ((N , {h, h′ }, R′ ), ϕ(N , {h, h′ }, R′ ), H ). Since ϕd1 (N , {h, h′ }, R′ ) = Chd1 (H , R′ ), then d1 ̸∈ N ′ . Since ϕl (N , {h, h′ }, R′ ) = Chl (H \ {h}, R′ ), then l ̸∈ N ′ . By iterative composition up, ϕl (N , H , R′ ) = ϕl (N , {h, h′ }, R′ ) = h′ = Chl (H \ {h}, R′ ).  We have shown that for each problem, person d1 is assigned his most preferred house. Person l is then assigned his most preferred house among the remaining ones. Thus, person l has the second highest priority. We denote him as d2 . Repeating the argument, we obtain ≻∈ Γ with d1 ≻ d2 ≻ d3 ≻ . . . ≻ dn , such that for each problem, person d1 is assigned his most preferred house, person d2 is then assigned his most preferred house among the remaining ones, and so on. Therefore, ϕ = SD≻ .  Proof of Theorem 2. By Proposition 3, the serial dictatorship rules satisfy composition down. It is straightforward to verify that they satisfy the other properties. Conversely, let ϕ be a rule satisfying these properties. We show that there is ≻∈ Γ such that ϕ = SD≻ . We will identify the order in the following steps. Step 1: Identifying the person with the highest priority. Pick h ∈ H . Let R ∈ RN be such that for each pair i, i′ ∈ N, Ri = Ri′ , and hPi ∅. By non-wastefulness, there is k ∈ N such that ϕk (N , {h}, R) = h. Claim 1. For each R′ ∈ RN , if h ∈ Dk (H , R′ ), then ϕk (N , {h}, R′ ) = h. Claim 2. For each R′ ∈ RN and each h′ ∈ Dk (H , R′ ), we have

ϕk (N , {h′ }, R′ ) = h′ .

Since the proofs of Claims 1 and 2 in Theorem 1 only consider the axioms in common14 with Theorem 2, they can be applied here as well. Claim 3. For each (N , H , R′ ) Chk (H , R′ ).

∈ Π , we have ϕk (N , H , R′ ) =

Proof of Claim 3. Let (N , H , R′ ) ∈ Π . There are two cases. Case 1: Chk (H , R′ ) = ∅. By non-wastefulness, ϕk (N , H , R′ )R′k ∅. By the definition of Chk (H , R′ ), we have ∅R′k ϕk (N , H , R′ ). Thus, ϕk (N , H , R′ ) = ∅ = Chk (H , R′ ). Case 2: Chk (H , R′ ) ∈ H. Let h′ ≡ Chk (H , R′ ). Let µ′ ≡ ϕ(N , H , R′ ). Suppose that k ∈ U (µ′ ). Let the set of available houses change from H to {h′ }. By composition down, ϕk (N , {h′ }, R′ ) = ϕk (N , H , R′ ) = ∅. This contradicts Claim 2. Thus, k ∈ M (µ′ ). Let R∗k ∈ R be such that for each h ∈ H \ {h′ }, h′ Pk∗ ∅Pk∗ h. Let µ∗ ≡ ϕ(N , H , (R∗k , R′−k )). By a similar argument as above, k ∈ M (µ∗ ). Thus, µ∗k = h′ . By strategy-proofness, µ′k R′k µ∗k . Since µ∗k = h′ = Chk (H , R′ ), then µ′k = h′ . Pick h ∈ H. Let R ∈ RN be such that for each i, i′ ∈ N , Ri = Ri′ , and there exists h′ ∈ H \ {h}, hPi h′ Pi ∅. Let R¯ ∈ RN be such that for each i ∈ N , Chi (H , R¯ ) = ∅. Claim 4. For each N ⊆ N , if k ∈ N, then ϕk (N , {h}, RN ) = h.

14 Note that, non-bossiness is stronger than weak non-bossiness.

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Proof of Claim 4. By non-wastefulness, some person in N is assigned h. Suppose by contradiction that there is j ∈ N \ {i} such that ϕj (N , {h}, RN ) = h. Let i ∈ N \ {j, k}. Let i’s preferences change from Ri to R¯ i . By non-wastefulness, ϕi (N , {h}, (R¯ i , RN \{i} )) = ∅. By non-bossiness, ϕ(N , {h}, (R¯ i , RN \{i} )) = ϕ(N , {h}, RN ). Repeating the argument, we obtain ϕ(N , {h}, (R¯ N \{j,k} , Rj , Rk )) = ϕ(N , {h}, RN ). Let us consider (N , {h, h′ }, (R¯ N \{j,k} , Rj , Rk )). Let µ ≡ ϕ(N , {h, h′ }, (R¯ N \{j,k} , Rj , Rk )). By non-wastefulness, M (µ) = {j, k}. Let the set of available houses change from {h, h′ } to {h}. By composition down, ϕk ({j, k}, {h}, (Rj , Rk )) = ϕk (N , {h}, (R¯ N \{j,k} , Rj , Rk )) = ∅. Let us consider (N , {h, h′ }, (R¯ N \{j,k} , Rj , Rk )). Let µ′ ≡ ϕ(N , {h, h′ }, (R¯ N \{j,k} , Rj , Rk )). By non-wastefulness, M (µ′ ) = {j, k}. Let the set of available houses change from {h, h′ } to {h}. By composition down, ϕk ({j, k}, {h}, (Rj , Rk )) = ϕk (N , {h}, (R¯ N \{j,k} , Rj , Rk )). By Claim 2, ϕk (N , {h}, (R¯ N \{j,k} , Rj , Rk )) = h. Thus, ϕk ({j, k}, {h}, (Rj , Rk )) = h. This contradicts the above result ϕk ({j, k}, {h}, (Rj , Rk )) = ∅. Thus, ϕk (N , {h}, RN ) = h. Claim 5. For each (N , H , R′ ) ∈ Π , if k ∈ N, then ϕk (N , H , R′ ) = Chk (H , R′ ). The proof is the same as the proofs for Claims 1–3, by replacing

N by N. We have shown that for each problem, person k is always assigned his most preferred house. Thus, person k has the highest priority. We denote him d1 . Step 2: Identifying the person with the second highest priority. Let RN \{d1 } and h be the same as before. By non-wastefulness, there is l ∈ N \ {d1 } such that ϕl (N \ {d1 }, h, RN \{d1 } ) = h. By a similar argument as the one in Step 1, we obtain that for each (N , H , R ′ ) ∈ 5, if d1 ̸∈ N and l ∈ N , then ϕl (N , H , R ′ ) = Chl (H , R ′ ). Claim 6. For each (N , H , R′ ) ∈ Π , if d1 , l ∈ N and h ≡ Chd1 (H , R′ ), then ϕl (N , H , R′ ) = Chl (H \ {h}, R′ ). Proof of Claim 6. Let (N , H , R′ ) ∈ Π with d1 , l ∈ N. Let h ≡ Chd1 (H , R′ ). By Claim 5, ϕd1 (N , H , R′ ) = h. There are two cases. Case 1: Chl (H \ {h}, R′ ) = ∅. Since ϕd1 (N , H , R′ ) = h, then Chl (H \ {h}, R′ )R′l ϕl (N , H , R′ ). By non-wastefulness, ϕl (N , H , R′ )R′l ∅. Thus, ϕl (N , H , R′ ) = ∅ = Chl (H \ {h}, R′ ). Case 2: Chl (H \ {h}, R′ ) ∈ H. Let h′ ≡ Chl (H \ {h}, R′ ). Let R∗d1 ∈ R be such that for each ′′ h ̸= h, hPd∗1 ∅Pd∗1 h′′ . Let µ ≡ ϕ(N , H , R′ ) and µ′ ≡ ϕ(N , H , (R∗d1 , R′N \{d1 } )). By Claim 5, µ′d1 = µd1 = h. By non-bossiness, µ′ = µ. Suppose l ∈ U (µ′ ). Let the set of available houses change from H to H \ {h}. Let µ′′ ≡ ϕ(N , H \ {h}, (R∗d1 , R′N \{d1 } )). By nonwastefulness, µ′′d1 = ∅. Thus, d1 ∈ U (µ′′ ). By composition down, µ′′l = ϕl (N \ {d1 }, H \ {h}, RN \{d1 } ). By the above result in bold, ϕl (N \ {d1 }, H \ {h}, RN \{d1 } ) = h′ . Thus, µ′′l = h′ . Since l ∈ U (µ′ ),

37

by composition down, µ′′l = ∅. This contradicts the previous result µ′′l = h′ . Thus, l ∈ M (µ′ ). Let R∗l ∈ R for each h′′ ̸= h′ , h′ Pl∗ ∅Pl∗ h′′ . Let µ∗ ≡ ϕ(N , H , (R∗d1 , R∗l , R′N \{d1 ,l} )). By a similar argument as above, l ∈ M (µ∗ ). Thus, µ∗l = h′ . By strategy-proofness, µ′l R′l µ∗l . Since µ′d1 = h and Chl (H \ {h}, R′ ) = h′ , then h′ R′l µ′l . Thus, µ′l = µ∗l = h′ . Since µ = µ′ , then µl = h′ = Chl (H \ {h}, R′ ).  We have shown that for each problem, person d1 is assigned his most preferred house. Person l is then assigned his most preferred house among the remaining ones. Thus, person l has the second highest priority. We denote him d2 . Repeating the argument, we obtain ≻∈ Γ with d1 ≻ d2 ≻ d3 ≻ . . . ≻ dn , such that for each problem, person d1 is assigned his most preferred house, person d2 is then assigned his most preferred house among the remaining ones, and so on. Therefore, ϕ = SD≻ .  References Abizada, A., Chen, S., 2014. A characterization of the uniform rule based on new robustness properties. Math. Social Sci. 71, 80–85. Bu, N., 2012. Characterizations of serial dictatorships in the assignment of object types, mimeo. Chun, Y., Thomson, W., 1988. Monotonicity properties of Bargainin solutions when applied to economics. Math. Social Sci. 15, 11–27. Ehlers, L., Klaus, B., 2004. Resource-monotonicity for house allocation problems. Internat. J. Game Theory 32, 545–560. Ergin, H., 2000. Consistency in house allocation problems. J. Math. Econom. 34 (1), 77–97. Herrero, C., Villar, A., 1998. Agenda independence in allocation problems with single-peaked preferences, mimeo. Herrero, C., Villar, A., 2000. An alternative characterization of the equal-distance rule for allocation problems with single-peaked preferences. Econom. Lett. 66, 311–317. Hylland, A., Zeckhauser, R., 1979. The efficient allocation of individuals to positions. J. Polit. Econ. 87, 293–314. Kalai, E., 1977. Proportional solutions to Bargaining situations: interpersonal utility comparisons. Econometrica 45, 1623–1630. Moreno-Ternero, J., Roemer, J., 2012. A common ground for resource and welfare egalitarianism. Games Econom. Behav. 75, 832–841. Moulin, H., 1987. Equal or proportional division of a surplus, and other mehods. Internat. J. Game Theory 16, 161–186. Moulin, H., 2000. Priority rules and other asymmetric rationing methods. Econometrica 68, 643–684. Moulin, H., Stong, R., 2002. Fair queuing and other probabilistic allocation methods. Math. Oper. Res. 27, 1–30. Plott, C.R., 1973. Path independence, rationality and social choice. Econometrica 41, 1075–1091. Roemer, J.E., 1986. The mismarriage of the Bargaining theory and distributive justice. Ethics 97, 88–110. Satterthwaite, M.A., Sonnenschein, H., 1981. Strategy-proof allocation mechanisms at differentiable points. Rev. Econom. Stud. 48, 587–597. Sönmez, T., Ünver, U., 2010. House allocation with existing tenants: A Characterization. Games Econom. Behav. 69, 425–445. Svensson, L.-G., 1999. Strategy-proof allocation of indivisible goods. Soc. Choice Welf. 16 (4), 557–567. Tadenuma, K., Thomson, W., 1991. No-envy and consistency in economies with indivisible goods. Econometrica 59, 1755–1767. Thomson, W., 1990. The consistency principle. In: Ichiishi, T., Neyman, A., Tauman, Y. (Eds.), Game Theory and Applications. Academic Press, New York, pp. 187–215. Thomson, W., 1994. Consistent solutions to the problem of fair division when preferences are single-peaked. J. Econom. Theory 63, 219–245. Thomson, W., 2003. Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey. Math. Social Sci. 45, 249–297. Young, H.P., 1988. Distributive justice in taxation. J. Econom. Theory 44, 321–335.