Allocating tasks when some may get canceled or ...

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Allocating tasks when some may get canceled or additional ones may arrive∗ Azar Abizada† and Siwei Chen‡ September 16, 2011

Abstract We look for a systematic way of assigning a set of objects, e.g. houses, tasks, to a group of individuals having preferences over these objects. We consider situations in which there might be more objects to be assigned than it was planned, or there might be fewer objects. We model this kind of unpredictable changes in two new properties, composition up and composition down. We characterize the family of rules that satisfy either of these properties, together with non-wastefulness, neutrality, strategyproofness, and non-bossiness. They are the sequential priority rules. JEL classification: C78, D70. Keywords: composition up, composition down, sequential priority

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Introduction Consider a manager who has to allocate a certain number of tasks to the workers in his

team. Each worker can be assigned at most one task. Tasks diﬀer. Each worker has strict preferences over the tasks. Also, each worker has the option of not working. We look for rules assigning tasks to workers that satisfy certain properties. ∗

We would like to thank William Thomson for his guidance, and invaluable comments. Department of Economics, University of Rochester. [email protected] ‡ Department of Economics, University of Rochester. [email protected] †

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In economics jargon, a task 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 have been analyzed for this problem (Svensson, 1999; Ergin, 2000; Ehlers and Klaus, 2004; Bu, 2011). Here, we consider the case where the set of objects is not ﬁxed. In practice, for various reasons, it happens that tasks are canceled, or that new tasks arise. How should we handle such events? In general, when circumstances in which a group of agents ﬁnd themselves change, there might be several ways to handle the change. The document which binds the agents as a group may specify a rule to solve the decision problems they face, but which one of several perspectives may be taken in applying the rule when circumstances change is not always written down. Indeed, there often are several perspectives. These perspectives may aﬀect the welfare of diﬀerent agents diﬀerently: hurt some and beneﬁt others. However, they may well be equally legitimate. In order to avoid possible conﬂicts about how to proceed, a natural robustness principle on the rule is that no matter which perspective is taken, it should produce the same outcome. Think of the following scenario: after a decision has been made about some ﬁrst situation and payoﬀs have been distributed, a second situation occurs in which opportunities have expanded. The ﬁrst position that one may take is to declare the ﬁrst situation irrelevant. Then, we simply ignore 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. Our robustness principle says that each agent should receive the same ﬁnal payoﬀ in the two procedures. This principle 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). The axiom that has been examined in the context of the adjudication of conﬂicting claims (ONeill, 1983), under the name of composition (Young, 1987), can also be seen as an expression of our robustness principle. In that context, a shrinking of opportunities has been considered too, and a dual principle (“composition down”) formulated, which can also be understood as an expression of the robustness principle. For each class of problems under consideration, the robustness principle needs to be

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adapted in order to best take account of the structure of the class. The conceptual innovation in our paper is a recursive construction. We propose two ways of applying the composition idea to our model. The ﬁrst requirement is related to the possibility that opportunities expand, that is, additional tasks become available. Two possible ways of proceeding are (i) to ignore the choice made initially and to reapply the rule to the new problem, the one in which opportunities have expanded; (ii) to use the initial choice as point of departure, and to simply take care of the distribution of the beneﬁts made possible by the augmentation in resources that has taken place, let us call it the “secondary” distribution. The ﬁnal welfare eventually experienced by agents would be obtained by augmenting their initial welfare. We require the ﬁnal welfare of each worker to be the same in the two procedures. Here is how the secondary distribution is deﬁned. We need to specify who is involved and what resources are available to them. We proceed inductively. For resources, we simply start from the set of tasks newly made available. As for the workers, if a worker prefers his initial assignment to any of the new tasks, there is no reason to include him in the secondary distribution. Therefore, we only include each worker who prefers at least one of the new tasks to his initial assignment. The task initially assigned to any such worker is released, and the set of tasks to be assigned in the secondary distribution is augmented by this task. Then, we identify any worker who initially had not been included in the secondary distribution because he was not interested in any of the newly available tasks, but who prefers at least one of the tasks just released to his initial assignment. Any such worker is now included in the secondary distribution. This may cause some additional tasks to be released, and so on. We proceed until none of the workers who had not been included in the secondary distribution prefers any of the available tasks in this distribution to his initial assignment. The second requirement pertains to the opposite change in opportunities, that is, when some tasks are canceled. Two possible ways of proceeding are (i) once again, to ignore the choice made initially and to reapply the rule to the new problem, the one in which opportunities have shrunk; (ii) to ignore the choice made initially and reapply the rule to the problem in which opportunities have shrunk and only the initial assignees are considered, since they are the ones who came ready to perform their initially assigned tasks. Both of these procedures provide a plausible way of handling the change in the number of tasks. However, as before, they may aﬀect the welfare of diﬀerent workers diﬀerently; hurt some and beneﬁt others. In order to avoid possible conﬂicts between the workers, we require

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the ﬁnal welfare of each worker to be the same.1 In addition to our composition properties, we are also interested in the following properties: no worker should prefer an unassigned task to his assignment (non-wastefulness); the chosen assignment should not depend on the names of tasks (neutrality); no worker should ever beneﬁt by misrepresenting his preferences (strategy-proofness); and if a worker’s preferences change, and his assignment remains the same, then each other worker’s assignment should remain the same as well (non-bossiness). We characterize the family of allocation rules satisfying either of the composition properties together with the properties just stated. It is the family of so-called “sequential priority rules”. To deﬁne such a rule, we ﬁrst pick an order on the set of workers. Then, for each problem, the sequential priority rule associated with that order works as follows; let the workers arrive one at a time according to the order. The rule assigns to each worker his most preferred task among the remaining ones. 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 sequential priority rules. In Section 5 we provide our main results.

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Model There is a ﬁnite and ﬁxed set workers N = {1, 2, . . . , n}. There is a ﬁnite set of potential

tasks, T = {t1 , t2 , . . . , tm }. Each worker has the option of not working. We denote this option ∅. Each worker can be assigned at most one task. Each worker i ∈ N has a strict preference relation Ri over the potential tasks and the option of not working, that is T ∪ {∅}. 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 ≡ (T, R) ∈ 2T × RN . Let Π be the set of all problems. An assignment for π ≡ (T, R) is a mapping µ : N −→ T ∪ {∅} such that each t ∈ T is assigned to at most one worker, that is |{i ∈ N : µi = t}| ≤ 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 1

Since preferences of the workers are strict, this requirement is equivalent to requiring the outcomes to

be the same.

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φ: Π →

∪

A(π) such that for each π ∈ Π, φ(π) ∈ A(π).

π∈Π

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Axioms Let (T, R) ∈ Π. Let µ ∈ A(T, R). Task t is acceptable to worker i with preferences Ri if t Pi ∅. Let Di (T, R) ≡

{t ∈ T : t Pi ∅} be the set of acceptable tasks to worker i at (T, R). Let Chi (T, R) ∈ T ∪ {∅} be the most preferred task for worker i at (T, R), that is for each t ∈ T ∪ {∅}, Chi (T, R) Ri t. We call a worker who is assigned a task at µ an assignee, and a worker who is not assigned any task a non-assignee. Let M (µ) ≡ {i ∈ N : µi ∈ T } 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 T is a one-to-one mapping f : T ∪ {∅} → T ∪ {∅} such that f (∅) = ∅. Let F be the set of all permutations on T . For each R ∈ RN , let Rf ∈ R be deﬁned as follows: for each i ∈ N and each pair t, t′ ∈ T ∪ {∅}, f (t) Pif f (t′ ) if and only if t Pi t′ . Let R∅ ≡ {Ri ∈ R : for each t ∈ T , ∅ Pi t} be the set of preference relations where ∅ is top-ranked. Next, we introduce the axioms. Let φ be a rule. • No worker should prefer an unassigned task or the option of not working to his assignment: Non-wastefulness: For each (T, R) ∈ Π and each i ∈ N , if µ ≡ φ(T, R), then for each ∪ t ∈ (T \ µi ) ∪ {∅}, µi Ri t. i∈N

• The chosen assignment should be independent of the names of tasks: Neutrality: For each (T, R) ∈ Π and each f ∈ F , we have φ(f (T ), Rf ) = f (φ(T, R)). • No worker should ever beneﬁt by misrepresenting his preference: 5

Strategy-proofness: For each (T, R) ∈ Π, each i ∈ N , and each Ri′ ∈ R, we have φi (T, R) Ri φi (T, (Ri′ , R−i )). • If a worker’s preferences change, and his assignment remains the same, then each other worker’s assignment should remain the same as well (Satterthwaite and Sonnenschein, 1981): Non-bossiness: For each (T, R) ∈ Π, each i ∈ N , and each Ri′ ∈ R, if φi (T, (Ri′ , R−i )) = φi (T, R), then φ(T, (Ri′ , R−i )) = φ(T, R). • A weaker requirement is: if a non-assignee’s preferences change, and he remains a nonassignee, then each other worker’s assignment should remain the same as well: Weak non-bossiness: For each (T, R) ∈ Π, each i ∈ U (φ(T, R)), and each Ri′ ∈ R, if φi (T, (Ri′ , R−i )) = φi (T, R), then φ(T, (Ri′ , R−i )) = φ(T, R). • If more tasks are available, then each worker should be made at least as well oﬀ as before (Chun and Thomson, 1984, and Roemer, 1986): Resource monotonicity: For each (T, R) ∈ Π, each T ′ ⊃ T , and each i ∈ N , we have φi (T ′ , R) Ri φi (T, R). • We introduce two composition properties related to possible changes in the number of tasks. Let an assignment be chosen for a problem. Let some additional tasks unexpectedly arrive. Two perspectives can be taken in handling the change. One possibility is simply to cancel the initial assignment and assign the available tasks (including the additional ones). Another possibility is as follows. The manager declares all the newly arrived tasks available. Some worker may prefer one of these tasks to his initial assignment. Let the manager consider any such worker for the available tasks. The tasks that are initially assigned to these workers are released and become available too. Then, some other worker may prefer one of

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the released tasks to his initial assignment. Let the manager include any such worker for consideration in assigning the available tasks. Here too, the tasks initially assigned to these workers are released and become available too. This process continues until no additional worker prefers one of the tasks in the augmented set of tasks at that point to his initial assignment. In the end, a group of workers has been identiﬁed, each of whom prefers one of the tasks that has been made available in succession to his initial assignment. The manager deals with the problem of assigning the available tasks among these workers. For the workers outside this group, the manager maintains their assignments. Both of these procedures provide a plausible way of handling the change in the number of tasks. However, they may aﬀect the welfare of diﬀerent workers diﬀerently; hurt some and beneﬁt others. In order to avoid possible conﬂicts between the workers, we require the outcomes to be the same. To introduce the requirement formally, we need some notation. For each π = (T, R) ∈ Π, each µ ∈ A(T, R), and each T ′ ⊃ T , let ′ ∆N 1 (π, µ, T ′ ) ≡ {i ∈ N : there t Pi µi }. {∪ is t ∈ T \ T, } ∆T 1 (π, µ, T ′ ) ≡ (T ′ \ T ) ∪ i∈∆N 1 (π,µ,T ′ ) µi . 1 ′ ∆N 2 (π, µ, T ′ ) ≡ {i ∈ N : there {∪ is t ∈ ∆T (π,}µ, T ), t Pi µi }. ∆T 2 (π, µ, T ′ ) ≡ (T ′ \ T ) ∪ i∈∆N 2 (π,µ,T ′ ) µi .

······ t−1 ′ ∆N t (π, µ, T ′ ) ≡ {i ∈ N : there {∪ is t ∈ ∆T (π, } µ, T ), t Pi µi }. ∆T t (π, µ, T ′ ) ≡ (T ′ \ T ) ∪ i∈∆N t (π,µ,T ′ ) µi .

······ Let s be such that ∆N s+1 (π, µ, T ′ ) = ∆N s (π, µ, T ′ ). Let ∆N (π, µ, T ′ ) ≡ ∆N s (π, µ, T ′ ) and ∆T (π, µ, T ′ ) ≡ ∆T s (π, µ, T ′ ). Composition up2 : For each (T, R) ∈ Π and each T ′ ⊃ T , if N ′′ ≡ ∆N ((T, R), φ(T, R), T ′ ), T ′′ ≡ ∆T ((T, R), φ(T, R), T ′ ), and for each j ∈ / N ′′ , Rj′ ∈ R∅ , then φ−N ′′ (T ′ , R) = φ−N ′′ (T, R), ′ φN ′′ (T ′ , R) = φN ′′ (T ′′ , (RN ′′ , R−N ′′ )).

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The terminology we use is proposed by Thomson (2003)

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• The next requirement pertains to the opposite change in the number of tasks. Let an assignment be chosen for a problem, and suppose that some of the tasks are unexpectedly canceled. Two perspectives can be taken in handling the change. Once again, one possibility is simply to cancel the initial assignment and assign the remaining tasks among all the workers. Another is to cancel the initial assignment and assign the remaining tasks only among the initial assignees. Here too, both of these procedures provide a plausible way of handling the change in the number of tasks. But again, they may aﬀect diﬀerent workers diﬀerently; hurt some and beneﬁt others. In order to avoid possible conﬂicts between the workers, we require the outcomes to be the same. Composition down: For each (T, R) ∈ Π and each T ′ ⊂ T , if µ ≡ φ(T, R) and for each j ∈ U (µ), Rj′ ∈ R∅ , then φ(T ′ , R) = φ(T ′ , (RU′ (µ) , RM (µ) )).

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Rules A priority order is a linear order ≻ on N . Let Γ be the set of all priority orders. We

associate a sequential priority rule to each linear order over the workers. For each problem, we let the workers arrive one at a time according to the order, and assign to him his most preferred task among the available ones. Formally, the sequential priority rule associated with ≻ ∈ Γ, SP ≻ , is deﬁned as follows: W.l.o.g. assume N = {k1 , k2 , ... , kn } and k1 ≻ k2 ≻ ... ≻ kn . For each (T, R) ∈ Π, SPk≻1 (T, R) = Chk1 (T, R), SPk≻2 (T, R) = Chk2 (T \ SPk≻1 (T, R), R), ······

∪

j−1

SPk≻j (T, R)

= Chkj (T \

SPk≻i (T, R), R),

i=1

······ SPk≻n (T, R) = Chkn (T \

n−1 ∪ i=1

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SPk≻i (T, R), R).

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Results It is easy to verify that the sequential priority rules are non-wasteful, neutral, strategy-

proof, and non-bossy.3 The sequential priority rules are also resource-monotonic.4 Next, we show that they satisfy composition up and composition down. Proposition 1 The sequential priority rules satisfy composition up. Proof. Let ≻ ∈ Γ. Suppose by contradiction that SP ≻ violates composition up: there are π = (T, R) ∈ Π, T ′ ⊃ T , R′ ∈ (R∅ )N , and i ∈ N such that if µ ≡ SP ≻ (T, R), ′ µ′ ≡ SP ≻ (T ′ , R), N ′′ ≡ ∆N (π, µ, T ′ ), T ′′ ≡ ∆T (π, µ, T ′ ), and µ′′ ≡ SP ≻ (T ′′ , (RN ′′ , R−N ′′ ))

then µ′i ̸= µ′′1 . Case 1: i ∈ / 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 SP ≻ , there is j ∈ N such that j ≻ i, µj = µ′i , and µ′j Pj µj . Again, by the deﬁnition of SP ≻ , 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 (T, R). By the deﬁnitions of N ′′ and T ′′ , for each t ∈ T ′′ , Chm (T, R) Pm t. Since T ′ \ T ⊆ T ′′ , then Chm (T ′ , R) = Chm (T, R). Since for each s ∈ N , µ′s Rs µs , then µ′m = Ch(T, 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 ′′ = µ−N ′′ . By the deﬁnition of SP ≻ , we have SPN≻′′ (T ′ \µ′ (−N ′′ ), (RN ′′ , R−N ′′ )) = µN ′ ∪ ∪ ∪ ∪ Since i∈N \N ′′ µ′i = i∈N \N ′′ µi , we have T ′ \ i∈N \N ′′ µ′i = T ′ \ i∈N \N ′′ µi = T ′′ . This im∪ ′ ≻ ′ ′ ′ ′ plies that µ′′N ′ = SP ≻ (T ′′ , (RN ′′ , R−N ′′ )) = SP (T \ i∈N \N ′′ µi , (RN ′′ , R−N ′′ )) = µN ′′ . This

contradicts our assumption that µ′i ̸= µ′′i .

Proposition 2 The sequential priority rules satisfy composition down. Proof. Let ≻ ∈ Γ. Let (T, R) ∈ Π and T ′ ⊂ T . 3 4

See Svensson (1999). See Ehlers and Klaus (2004).

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By resource monotonicity, for each i ∈ N , SPi≻ (T, R) Ri SPi≻ (T ′ , R). By the deﬁnition of SP ≻ , for each i ∈ N , SPi≻ (T ′ , R) Ri ∅. Thus, for each i ∈ U (SP ≻ (T, R)), we have SPi≻ (T ′ , R) = SPi≻ (T, R) = ∅. Let R′ ∈ (R∅ )N . Let i ∈ U (SP ≻ (T, R)). Let worker i’s preferences change from Ri to Ri′ . By the deﬁnition of SP ≻ , we have SPi≻ (T ′ , (Ri′ , R−i )) = ∅ = SPi≻ (T ′ , R). By weak non-bossiness, SP ≻ (T ′ , (Ri′ , R−i )) = SP ≻ (T ′ , R). Let j ∈ U (SP ≻ (T, R))\{i}. Let worker j’s preferences change from Rj to Rj′ . By the deﬁnition of SP ≻ , we have SPj≻ (T ′ , (Ri′ , Rj′ , R−ij )) = ∅ = SPj≻ (T ′ , R). By weak non-bossiness, SP ≻ (T ′ , (Ri′ , Rj′ , R−ij )) = SP ≻ (T ′ , R). Repeating the argument, we obtain SP ≻ (T ′ , (RU′ (SP ≻ (T,R)) , RM (SP ≻ (T,R)) )) = SP ≻ (T ′ , R).

Our ﬁrst main result is a characterization of the sequential priority rules.

Theorem 1 The sequential priority rules are the only non-wasteful, neutral, strategy-proof, weak non-bossy rules to satisfy composition up. Proof. By Proposition 1, the sequential priority rules satisfy 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 = SP ≻ . Let R ∈ RN be such that for each pair i, i′ ∈ N , Ri = Ri′ . Let t ∈ Di (T , R). By non-wastefulness, there is k ∈ N such that φk (t, R) = t. R1 = · · · = Rn .. . t .. . ∅ .. . Claim 1: For each R′ ∈ RN , if t ∈ Dk (T , R′ ), then φk (t, R′ ) = t. Proof of Claim 1: Let R′ ∈ RN and t ∈ Dk (T , R′ ). Let i ∈ N \ {k}. Let worker i’s preferences change from Ri to Ri′ . Since t Pi ∅, by strategy-proofness, φi (t, (Ri′ , R−i )) ̸= t. Thus, φi (t, (Ri′ , R−i )) = ∅. By weak non-bossiness, φ(t, (Ri′ , R−i )) = φ(t, R). 10

Let j ∈ N \ {k, i}. Let worker j’s preferences change from Rj to Rj′ . Since t Pj ∅, by strategy-proofness, φj (t, (Ri′ , Rj′ , R−ij )) ̸= t. Thus, φj (t, (Ri′ , Rj′ , R−ij )) = ∅. By weak non′ bossiness, φ(t, (Ri′ , Rj′ , R−ij )) = φ(t, R). Repeating the argument, we obtain φ(t, (Rk , R−k )) =

φ(t, R). Let worker k ′ s preferences change from Rk to Rk′ . If φk (t, R′ ) = ∅, then when worker k’s ′ true preferences are Rk′ and he faces R−k , he is better oﬀ by misrepresenting his preferences

to be Rk . This is a violation of strategy-proofness. Thus, φk (t, R′ ) = t. Claim 2: For each R′ ∈ RN and each t′ ∈ Dk (T , R′ ), we have φk (t′ , R′ ) = t′ . Proof of Claim 2: Let R′ ∈ RN and t′ ∈ Dk (T , R′ ). Let f ∈ F be such that f (t) = t′ , f (t′ ) = t, and for each t′′ ∈ T \ {t, t′ }, f (t′′ ) = t′′ . Since t′ ∈ Dk (T , R′ ) and f (t′ ) = t, then t ∈ Dk (T , (R′ )f ). By Claim 1, φk (t, (R′ )f ) = t. By neutrality, f (φk (t′ , R′ )) = φk (f (t′ ), (R′ )f ) = t. Thus, φk (t′ , R′ ) = t′ . Claim 3: For each (T, R′ ) ∈ Π, we have φk (T, R′ ) = Chk (T, R′ ). Proof of Claim 3: Let (T, R′ ) ∈ Π. There are two cases. Case 1: Chk (T, R′ ) = ∅. By non-wastefulness, φk (T, R′ ) Rk′ ∅. By the deﬁnition of Chk (T, R′ ), we have ∅ Rk′ φk (T, R′ ). Thus, φk (T, R′ ) = ∅ = Chk (T, R′ ). Case 2: Chk (T, R′ ) ∈ T . Let t′ ≡ Chk (T, R′ ). By Claim 2, φk (t′ , R′ ) = t′ . Let the set of available tasks change from {t′ } to T . Let N ′ ≡ ∆N ((t′ , R′ ), φ(t′ , R′ ), T ). Since φk (t′ , R′ ) = Chk (T, R′ ), then k∈ / N ′ . By composition up, φk (T, R′ ) = φk (t′ , R′ ) = t′ = Chk (T, R′ ). We have shown that for each (T, R) ∈ Π, worker k receives his most preferred task. Thus, worker k has the highest priority. We denote him d1 . Let Rd′′1 ∈ R∅ . Let R−d1 and t be the same as before.

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Rd′′1 ∅ .. .

R1 = · · · Rd1 −1 = Rd1 +1 · · · Rk .. . t .. . ∅ .. .

By Claim 3, φd1 (t, (Rd′′1 , R−d1 )) = ∅. By non-wastefulness, there is l ∈ N \ {d1 } such that φl (t, (Rd′′1 , R−d1 )) = t. Claim 4: For each R′ ∈ RN , if Rd′ 1 ∈ R∅ and t ∈ Dl (T , R′ ), then φl (t, R′ ) = t. Proof of Claim 4: Let R′ ∈ RN be such that Rd′ 1 ∈ R∅ . Let t ∈ Dl (T , R′ ). Let i ∈ N \ {d1 , l}.

Let worker i’s preferences change from Ri to Ri′ .

Since t Pi ∅, by

strategy-proofness, φi (t, (Rd′′1 , Ri′ , R−d1 i )) ̸= t. Thus, φi (t, (Rd′′1 , Ri′ , R−d1 i )) = ∅. By weak non-bossiness, φ(t, (Rd′′1 , Ri′ , R−d1 i )) = φ(t, (Rd′′1 , R−d1 )). Let j ∈ N \ {d1 , l, i}. Let worker j’s preferences change from Rj to Rj′ . Since t Pj ∅, by strategy-proofness, φj (t, (Rd′′1 , Ri′ , Rj′ , R−d1 ij )) ̸= t. Thus, φj (t, (Rd′′1 , Ri′ , Rj′ , R−d1 ij )) = ∅. By weak non-bossiness, φ(t, (Rd′′1 , Ri′ , Rj′ , R−d1 ij )) = φ(t, (Rd′′1 , R−d1 )). Repeating the argument, ′ we obtain φ(t, (Rd′′1 , Rl , R−d )) = φ(t, (Rd′′1 , R−d1 )). 1l ′ Let worker d1 ’s preferences change from Rd′′1 to Rd′ 1 . By non-wastefullness, φd1 (t, (Rl , R−l )) = ∅. ′ By weak non-bossiness, φ(t, (Rl , R−l )) = φ(t, (Rd′′1 , R−d1 )).

Let worker l′ s preferences change from Rl to Rl′ . If φl (t, R′ ) = ∅, then when worker l’s ′ true preferences are Rl′ and he faces R−l , he is better oﬀ by misrepresenting his preferences

to be Rl . This is a violation of strategy-proofness. Thus, φl (t, R′ ) = t. Claim 5: For each R′ ∈ RN and each t′ ∈ Dl (T , R′ ), if Rd′ 1 ∈ R∅ , then φl (t′ , R′ ) = t′ . Proof of Claim 5: Let R′ ∈ RN be such that Rd′ 1 ∈ R∅ . Let t′ ∈ Dl (T , R′ ). Let f ∈ F be such that f (t) = t′ , f (t′ ) = t, and for each t′′ ∈ T \ {t, t′ }, f (t′′ ) = t′′ . Since f (∅) = ∅, then (Rd′ 1 )f ∈ R∅ . Since t′ ∈ Dl (T , R′ ) and f (t′ ) = t, then t ∈ Dl (T , (R′ )f ). By Claim 4, φl (t, (R′ )f ) = t. By neutrality, f (φl (t′ , R′ )) = φl (f (t′ ), (R′ )f ) = t. Thus, φl (t′ , R′ ) = t′ . Claim 6: For each (T, R′ ) ∈ Π, we have φl (T, R′ ) = Chl (T \ Chd1 (T, R′ ), R′ ). 12

Proof of Claim 6: Let (T, R′ ) ∈ Π. Let t′ ≡ Chd1 (T, R′ ). By Claim 3, φd1 (T, R′ ) = t′ . There are two cases. Case 1: Chl (T \ t′ , R′ ) = ∅. By non-wastefulness, φl (T, R′ ) Rl′ ∅. Since φd1 (T, R′ ) = t′ , Chl (T \ t′ , R′ ) Rl′ φl (T, R′ ). Thus, φl (T, R′ ) = ∅ = Chl (T \ t′ , R′ ). Case 2: Chl (T \ t′ , R′ ) ∈ T . Let t′′ ≡ Chl (T \ t′ , R′ ). By Claim 3, φd1 (t′ t′′ , R′ ) = t′ . Let Rd′′′1 ∈ R∅ . By Claim 5, ′ φl (t′′ , (Rd′′′1 , R−d )) = t′′ . By composition up, φl (t′ t′′ , R′ ) = t′′ . Let the set of available tasks 1

change from {t′ , t′′ } to T . Let N ′ ≡ ∆N ((t′ t′′ , R′ ), φ(t′ t′′ , R′ ), T ). Since φd1 (t′ t′′ , R′ ) = Chd1 (T, R′ ), then d1 ∈ / N ′ . Since φl (t′ t′′ , R′ ) = Chl (T \ t′ , R′ ), then l ∈ / N ′ . By composition up, φl (T, R′ ) = φl (t′ t′′ , R′ ) = t′′ = Chl (T \ t′ , R′ ). We have shown that for each (T, R) ∈ Π, after d1 receives his most preferred task, worker l receives his most preferred task among the remaining ones. Thus, worker l has the second highest priority. We denote him d2 . Let Rd′′2 ∈ R∅ . Let Rd′′1 , R−d1 d2 , and t be the same as before. ······ Repeating the argument, we obtain that for each preference proﬁle, worker d1 receives his most preferred task, then worker d2 receives his most preferred task among the remaining ones · · · · · · . Let ≻ ∈ Γ be such that d1 ≻ d2 ≻ d3 ≻ . . . ≻ dn . Then, φ = SP ≻ . Our second main result is also a characterization of the sequential priority rules. It differs from Theorem 1 in two ways. First, we replace composition up with composition down. Second, we impose non-bossiness instead of weak non-bossiness. Theorem 2 The sequential priority rules are the only non-wasteful, neutral, strategy-proof, non-bossy rules to satisfy composition down. Proof. In Proposition 2, we verify that the sequential priority 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 = SP ≻ . 13

Let R ∈ RN be such that for each pair i, i′ ∈ N , Ri = Ri′ . Let t ∈ Di (T , R). By non-wastefulness, there is k ∈ N such that φk (t, R) = t. Claim 1: For each R′ ∈ RN , if t ∈ Dk (T , R′ ), then φk (t, R′ ) = t. Claim 2: For each R′ ∈ RN and each t′ ∈ Dk (T , R′ ), we have φk (t′ , R′ ) = t′ . Because composition up is not used, and non-bossiness is stronger than weak non-bossiness, the proofs of these two claims are the same as in Theorem 1. Claim 3: For each (T, R′ ) ∈ Π, we have φk (T, R′ ) = Chk (T, R′ ). Proof of Claim 3: Let (T, R′ ) ∈ Π. There are two cases. Case 1: Chk (T, R′ ) = ∅. By non-wastefulness, φk (T, R′ ) Rk′ ∅. By the deﬁnition of Chk (T, R′ ), we have ∅ Rk′ φk (T, R′ ). Thus, φk (T, R′ ) = ∅ = Chk (T, R′ ). Case 2: Chk (T, R′ ) ∈ T . Let t′ ≡ Chk (T, R′ ). Let µ′ ≡ φ(T, R′ ). Let R′′ ∈ RN be such that for each i ∈ N , Ri′′ ∈ R∅ . Suppose k ∈ U (µ′ ). Let the set of available tasks change from T to {t′ }. ′ ′ ′ ′ By composition down, φ(t′ , R′ ) = φ(t′ , (RU′′ (µ) , RM (µ) )). By Claim 2, φk (t , R ) = t . By ′ ′ ′ Case 1, φk (t′ , (RU′′ (µ′ ) , RM (µ′ ) )) = ∅. This contradicts our previous conclusion that φ(t , R ) = ′ ′ φ(t′ , (RU′′ (µ′ ) , RM (µ′ ) )). Thus, k ∈ M (µ ). ′ Let Rk∗ ∈ R be such that for each t′′ ∈ T \ {t′ }, t′ Pk∗ ∅ Pk∗ t′′ . Let µ∗ ≡ φ(T, (Rk∗ , R−k )).

By a similar argument as above, k ∈ M (µ∗ ). Thus, µ∗k = t′ . By strategy-proofness, µ′k Rk′ µ∗k . Since t′ ≡ Chk (T, R′ ), µ′k = t′ . We have shown that for each (T, R) ∈ Π, worker k receives his most preferred task. Thus, worker k has the highest priority. We denote him d1 . Let Rd′′1 ∈ R∅ . Let R−d1 and t be the same as before. By Claim 3, φd1 (t, (Rd′′1 , R−d1 )) = ∅. By non-wastefulness, there is l ∈ N \ {d1 } such that φl (t, (Rd′′1 , R−d1 )) = t. Claim 4: For each R′ ∈ RN , if Rd′ 1 ∈ R∅ and t ∈ Dl (T , R′ ), then φl (t, R′ ) = t. Claim 5: For each R′ ∈ RN and each t′ ∈ Dl (T , R′ ), if Rd′ 1 ∈ R∅ , then φl (t′ , R′ ) = t′ . Again, because composition up is not used, and non-bossiness is stronger than weak nonbossiness, the proofs of these two claims are the same as in Theorem 1.

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Claim 6: For each (T, R′ ) ∈ Π, we have φl (T, R′ ) = Chl (T \ Chd1 (T, R′ ), R′ ). Proof of Claim 6: Let (T, R′ ) ∈ Π. Let t′ ≡ Chd1 (T, R′ ). By Claim 3, φd1 (T, R′ ) = t′ . There are two cases. Case 1: Chl (T \ t′ , R′ ) = ∅. By non-wastefulness, φl (T, R′ ) Rl′ ∅. Since φd1 (T, R′ ) = t′ , Chl (T \ t′ , R′ ) Rl′ φl (T, R′ ). Thus, φl (T, R′ ) = ∅ = Chl (T \ t′ , R′ ). Case 2: Chl (T \ t′ , R′ ) ∈ T . Let t′′ ≡ Chl (T \ t′ , R′ ). Let Rd∗1 ∈ R be such that for each t′′′ ∈ T \ {t′ }, t′ Pd∗1 ∅ Pd∗1 t′′′ . ′ Let µ′ ≡ φ(T, (Rd∗1 , R−d )). By Claim 3, µ′d1 = t′ = φd1 (T, R′ ). By non-bossiness, µ′ = 1

φ(T, R′ ). Suppose l ∈ U (µ′ ). Let the set of available tasks change from T to t′′ . Let R′′ ∈ RN be such that for each i ∈ N , Ri′′ ∈ R∅ .

′ By composition down, φl (t′′ , (Rd∗1 , R−d )) = 1

′ ′′ ∗ ′ φl (t′′ , (Rd∗1 , RU′′ (µ′ ) , RM (µ′ )\d1 )). By Claim 3, φd1 (t , (Rd1 , R−d1 )) = ∅. Let worker d1 ’s pref′ erences change from Rd∗1 to Rd′′1 ∈ R∅ . By Claim 3, φd1 (t′′ , (Rd′′1 , R−d1 )) = ∅. By non′ ′ ′ bossiness, φ(t′′ , (Rd′′1 , R−d1 )) = φ(t′′ , (Rd∗1 , R−d )). By Claim 5, φl (t′′ , (Rd′′1 , R−d )) = t′′ . Thus, 1 1 ′ ′ φl (t′′ , (Rd∗1 , R−d )) = t′′ . Since l ∈ U (µ′ ), by non-wastefulness, φl (t′′ , (Rd∗1 , RU′′ (µ′ ) , RM (µ′ )\d1 )) = ∅. 1 ′ ′ This contradicts our previous conclusion that φl (t′′ , (Rd∗1 , R−d )) = φl (t′′ , (Rd∗1 , RU′′ (µ′ ) , RM (µ′ )\d1 )). 1

Thus, l ∈ M (µ′ ). ′ Let Rl∗ ∈ R be such that for each t′′′ ∈ T , t′′ Pl∗ ∅ Pl∗ t′′′ . Let µ∗ ≡ φ(T, (Rd∗1 , Rl∗ , R−d )). 1l

By a similar argument as above, l ∈ M (µ∗ ). Thus, µ∗l = t′′ . By strategy-proofness, µ′l Rl′ µ∗l . Since µ′d1 = t′ and t′′ ≡ Chl (T \ t′ , R′ ), t′′ Rl′ µ′l . Thus, µ′l = t′′ . Since µ′ = φ(T, R′ ), φl (T, R′ ) = t′′ = Chl (T \ t′ , R′ ). We have shown that for each (T, R) ∈ Π, after d1 receives his most preferred task, worker l receives his most preferred task among the remaining ones. Thus, worker l has the second highest priority. We denote him d2 . Let Rd′′2 ∈ R∅ . Let Rd′′1 , R−d1 d2 , and t be the same as before. ······ Repeating the argument, we obtain that for each preference proﬁle, worker d1 receives his most preferred task, then worker d2 receives his most preferred task among the remaining 15

ones · · · · · · . Let ≻ ∈ Γ be such that d1 ≻ d2 ≻ d3 ≻ . . . ≻ dn . Then, φ = SP ≻ .

• On the independence of the axioms in Theorem 1 and Theorem 2 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. For those that require more work, we give examples to show violations of the axioms. (1) Let φ1 be the rule deﬁned as follows: For each (T, R) ∈ Π and each i ∈ N , φ1i (T, R) = ∅.

(2) Let φ2 be the rule deﬁned as follows: Let ≻, ≻′ ∈ Γ be such that 1 ≻ 2 ≻ 3 ≻ 4 ≻ ... ≻ n and 2 ≻′ 1 ≻′ 3 ≻′ 4 ≻′ ... ≻′ n. Let t ∈ T . For each (T, R) ∈ Π,  SP ≻′ (T, R) if Ch2 (T, R) = t, 2 φ (T, R) = SP ≻ (T, R) otherwise. (3) Let φ3 be the rule deﬁned as follows: Let ≻, ≻′ ∈ Γ be such that 1 ≻ 2 ≻ 3 ≻ 4 ≻ ... ≻ n and 2 ≻′ 1 ≻′ 3 ≻′ 4 ≻′ ... ≻′ n. Let t ∈ T . For each (T, R) ∈ Π,  SP ≻′ (T, R) if R2 = R1 , 3 φ (T, R) = SP ≻ (T, R) otherwise. Example 1 The rule φ3 is not strategy-proof. Let T ≡ {t1 }. Let R ∈ RN be such that t1 P1 ∅, t1 P2 ∅, and R2 ̸= R1 . By the deﬁnition of φ3 , we have φ3 (T, R) = SP ≻ (T, R). Thus, φ31 (T, R) = t1 , and for each i ̸= 1, φ3i (T, R) = ∅. Let R2′ ≡ R1 . Suppose that when facing R−2 , worker 2 with true preference R2 mis′

represents his preference to be R2′ . Then, φ3 (T, (R2′ , R−2 )) = SP ≻ (T, (R2′ , R−2 )). Thus, 16

φ32 (T, (R2′ , R−2 )) = t1 , and for each i ̸= 2, φ3i (T, (R2′ , R−2 )) = ∅. Thus, φ32 (T, (R2′ , R−2 )) P2 φ32 (T, R). This is in violation of strategy-proofness. (4) Let φ4 be the rule deﬁned as follows: Let ≻, ≻′ ∈ Γ be such that 1 ≻ 2 ≻ 3 ≻ 4 ≻ ... ≻ n and 2 ≻′ 1 ≻′ 3 ≻′ 4 ≻′ ... ≻′ n. For each (T, R) ∈ Π,  SP ≻′ (T, R) if Rn = R2 , 4 φ (T, R) = SP ≻ (T, R) otherwise. Example 2 The rule φ4 is not weak non-bossy. Let T ≡ {t1 }. Let R ∈ RN be such that t1 P1 ∅, t1 P2 ∅, and Rn ̸= R2 . By the deﬁnition of φ4 , we have φ4 (T, R) = SP ≻ (T, R). Thus, φ41 (T, R) = t1 , and for each i ̸= 1, φ4i (T, R) = ∅. Let Rn′ ≡ R2 . Suppose that when facing R−n , worker n reports his preference to be Rn′ . Then, ′

φ4 (T, (Rn′ , R−n )) = SP ≻ (T, (Rn′ , R−n )). Thus, φ42 (T, (Rn′ , R−n )) = t1 , and for each i ̸= 2, φ4i (T, (Rn′ , R−n )) = ∅. Thus, φ4n (T, (Rn′ , R−n )) = φ4n (T, R) = ∅, φ41 (T, (Rn′ , R−n )) ̸= φ41 (T, R), and φ42 (T, (Rn′ , R−n )) ̸= φ42 (T, R). This is in violation of weak non-bossiness. (5) Let φ5 be the rule deﬁned as follows: Let ≻, ≻′ ∈ Γ be distinct. For each (T, R) ∈ Π,  SP ≻ (T, R) if |T | = 1, φ5 (T, R) = SP ≻′ (T, R) if |T | ≥ 2. Example 3 The rule φ5 does not satisfy composition up. Let T ≡ {t1 }. Let R ∈ RN be such that t1 P1 t2 P1 ∅, t1 P2 t2 P2 ∅, and for each i ∈ / {1, 2}, t2 Pi ∅. By the deﬁnition of φ5 , we have φ5 (T, R) = SP ≻ (T, R). Thus, φ51 (T, R) = t1 , and for each i ̸= 1, φ5i (T, R) = ∅. Let T ′ ≡ {t1 , t2 }. Then, N ′′ ≡ ∆N ((T, R), φ5 (T, R), T ′ ) = N \ {1} and T ′′ ≡ ∆T ((T, R), ′

φ5 (T, R), T ′ )) = {t2 }. By the deﬁnition of φ5 , we have φ5 (T ′ , R) = SP ≻ (T ′ , R). Thus, / {1, 2}, φ5i (T ′ , R) = ∅. Thus, φ51 (T ′ , R) ̸= φ51 (T ′ , R) = t2 , φ52 (T ′ , R) = t1 , and for each i ∈ φ51 (T, R). This is in violation of composition up.

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Example 4 The rule φ5 does not satisfy composition down. Let T ≡ {t1 , t2 }. Let R ∈ RN be such that t1 P1 ∅ P1 t2 , t1 P2 t2 P2 ∅, and for each i∈ / {1, 2}, t2 Pi ∅. ′

By the deﬁnition of φ5 , we have φ5 (T, R) = SP ≻ (T, R). Thus, φ52 (T, R) = t1 , φ53 (T, R) = t2 , and for each i ∈ / {2, 3}, φ5i (T, R) = ∅. Let T ′ ≡ {t1 }. By the deﬁnition of φ5 , we have φ5 (T ′ , R) = SP ≻ (T ′ , R). Thus, φ51 (T ′ , R) = t1 , and for each i ̸= 1, φ5i (T ′ , R) = ∅. Let R′ ∈ RN be such that for each i ∈ N , Ri′ ∈ R∅ . Sim′ ′ )). )) = SP ≻ (T ′ , (R2 , R3 , R−{2,3} ilarly, by the deﬁnition of φ5 , we have φ5 (T ′ , (R2 , R3 , R−{2,3} ′ ′ )) = t1 , and for each i ̸= 2, φ5i (T ′ , (R2 , R3 , R−{2,3} )) = ∅. Thus, Thus, φ52 (T ′ , (R2 , R3 , R−{2,3} ′ φ5 (T ′ , R) = φ5 (T ′ , (R2 , R3 , R−{2,3} )). This is in violation of composition down.

(1) The rule φ1 satisﬁes all the axioms of Theorems 1 and 2 except for non-wastefulness. (2) The rule φ2 satisﬁes all the axioms of Theorems 1 and 2 except for neutrality. (3) The rule φ3 satisﬁes all the axioms of Theorems 1 and 2 except for strategy-proofness. (4) The rule φ4 satisﬁes all the axioms of Theorem 1 except for weak non-bossiness. (4′ ) The rule φ4 satisﬁes all the axioms of Theorem 2 except for non-bossiness. (5) The rule φ5 satisﬁes all the axioms of Theorem 1 except for composition up. (5′ ) The rule φ5 satisﬁes all the axioms of Theorem 2 except for composition down.

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