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A characterization of the uniform rule based on new robustness properties
Accepted Manuscript A characterization of the uniform rule based on new robustness properties Azar Abizada, Siwei Chen PII: DOI: Reference:
S0165-4896(14)00045-6 http://dx.doi.org/10.1016/j.mathsocsci.2014.05.003 MATSOC 1740
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Mathematical Social Sciences
Received date: 13 June 2013 Revised date: 17 March 2014 Accepted date: 16 May 2014 Please cite this article as: Abizada, A., Chen, S., A characterization of the uniform rule based on new robustness properties. Mathematical Social Sciences (2014), http://dx.doi.org/10.1016/j.mathsocsci.2014.05.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Two characterizations of the uniform rule based on new robustness properties∗ Azar Abizada† and Siwei Chen‡ March 17, 2014
Abstract We study the problem of allocating a divisible good among a group of people. Each person’s preferences are single-peaked. We consider situations in which there might be more of the resource to be assigned than was planned, or there might be less of the resource. Two robustness properties are formulated, which we call one-sided composition up and one-sided composition down. We show that only one rule satisfies irrelevance of null agents, the equal-division lower bound, and our robustness properties. This rule is the uniform rule. JEL classification: C71; D63. Keywords: one-sided composition up, one-sided composition down, irrelevance of null agents, uniform rule
1
Introduction
Consider a department chair having to assign teaching duties to faculty. Each faculty member has single-peaked preferences over the workload he receives: he has an ideal workload (i.e. peak of his preferences) and his welfare decreases as his workload moves away from this ideal amount. We look for rules allocating the total workload that satisfy certain desirable properties. We would like to thank William Thomson for his guidance, and invaluable comments. We are grateful to two anonymous referees for their useful comments. † School of Business, ADA University, 11 Ahmadbay Aghaoglu St., Baku AZ1008, Azerbaijan. Email: [email protected]. ‡ Lingnan College, Sun Yat-sen University, Guangzhou, 510275, China. Email: [email protected]. ∗
1
Various properties of efficiency, fairness, and robustness under strategic behavior have been analyzed for the problem where agents have single-peaked preferences (Sprumont, 1991; Ching, 1992, 1994, 2010; Thomson, 1994a, 1994b, 1995, 1997, 2011). Here, we consider the case where the total workload is not fixed. Once teaching duties have been assigned, imagine that too many students have registered for some course. In such a case, the chair has to increase the number of sections for the course. Alternatively, it may be that too few students have registered for some course. Then, some sections have to be canceled. How should we handle these changes? In general, when the circumstances in which a group of agents find themselves change, there might be several ways to handle the change. Each of these ways may well be equally legitimate. Yet, outcomes may differ depending on which perspective is taken. In such cases, no matter what action is chosen, someone maybe unhappy that the other action is not taken. Therefore, the choice of which perspective should be taken may need to have 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, no matter which one is taken, it should produce the same outcome. When opportunities expand, this principle has been investigated under the name of “stepby-step negotiations” (Kalai, 1977) for bargaining, and it is also reminiscent of the “path independence” axiom for choice functions (Plott, 1973). “Lower composition” (Moulin, 2000) and “composition” (Young, 1988) for the adjudication of conflicting claims, “lower composition” (Moulin and Stong, 2002) for queuing problems, “composition” (Moreno-Ternero and Roemer, 2012) for resource allocation with utilities, and “iterative composition up” (Abizada and Chen, 2013) for allocation of the indivisible goods, can all be viewed as expressions of this robustness principle in different contexts. In the context of the adjudication of conflicting claims, queuing problem, and allocation of the indivisible goods, a shrinking of opportunities has been considered too. Axioms pertaining to such situation have been formulated (Moulin, 1987, 2000; Moulin and Stong, 2002; Abizada and Chen, 2013), which can also be understood as expressions of the general principle. For each class of problems, a general principle needs to be adapted in order to best take into account the structure of the class. We propose two new properties that are expressions of the robustness idea to our model. Return to the example. Our first property is related to the possibility that the total workload increases, that is, when new sections have to be offered. Two possible ways of proceeding are (i) to ignore the choice made initially, and to reapply the rule to the new problem, the one with the larger workload; (ii) to use the initial choice as point of departure. For each faculty member we keep his initial workload and
2
“revise his preferences by that amount”. We allocate the additional workload using these revised preferences. The final workload assigned to each faculty member is obtained by augmenting his initial workload by the additional workload computed in this manner. Each of these procedures provides a plausible way of handling the change in the total workload. However, they may affect the welfare of different faculty members differently. If that is so, depending on which procedure is chosen, some faculty members may object that the other one was not chosen. In order to avoid such situations, we require the final assignments to be the same, independently of which procedure is followed. The second property pertains to the opposite change in the total workload, that is, when some courses have to be 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 with the smaller workload; (ii) to use the initial choice as point of departure. Similarly, since each faculty member keeps his initial workload, we should revise his preferences in the new situation. We allocate the cancelled courses by treating them as an “endowment” (changing the sign from negative to positive), using the revised preferences. The final workload assigned to each faculty member is obtained by subtracting the amount computed in this manner from his initial workload. Again, we require the final assignments to be the same irrespective of the perspective that is followed. We are interested in efficiency, which states that there should be no other assignment at which at least one faculty member is better off and no faculty member is worse off. We consider a new property called irrelevance of null agents, which states that whenever there is excess demand, the assignment should not be affected by the departure of faculty members who ideally prefer not to teach. This property can be viewed as a mild mixture of efficiency and “consistency”1 (Thomson, 1994b; Herrero and Villar, 2000). We also consider two standard punctual fairness properties: first, no faculty member should prefer someone else’s assignment to his own (no envy); second, each faculty member should find his or her assignment at least as desirable as an equal share of the total workload (the equal-division lower bound ). Herrero and Villar (2000) is the first paper to study the robustness principle for this model. They propose a robustness property that pertains to situations in which the endowment increases, under the name “agenda-independence”. However, as we will show, this property is incompatible with efficiency and either one of our fairness properties2 . As disWe will illustrate the logical relations among these properties in Section 3 where they are formally defined. 2 They characterize the “equal distance” rule using efficiency, agenda independence, and some other properties. 1
3
cussed above, we also define a similar robustness property which pertains to situations in which the endowment decreases. Again, this property is incompatible with efficiency and either one of our fairness properties. These incompatibilities occur when initially there is too much (the sum of peaks is smaller than than the endowment) and the endowment increases, or when initially there is not enough (the sum of peaks is larger than the endowment) and the endowment decreases. We propose two more reasonable robustness properties, which are conditional upon the endowment remaining on the same side of the sum of the peaks. This kind of conditioning is introduced in this model in Thomson (1994a). The first property applies to the case of “there is still not enough after an increase”, and the second property applies to the case of “there is still too much after a decrease”. These are the central notions of our paper. Our main result is that there is a unique rule satisfying irrelevance of null agents, the equal-division lower bound, and the two robustness properties. It is the rule known as the “uniform rule”, which is the most central rule in the literature. We also find that the uniform rule is the only rule satisfying a weaker version of irrelevance of null agents, no envy, peak only, and the two robustness properties. The uniform rule has already come out of a number of axiomatic studies of this class of problems. Our study confirms its importance. The remainder of the paper is organized as follows. In Section 2 we define the model. In Section 3 we state the axioms and define the rules. In Section 4 we provide our main result and some discussions.
2
Model
There is a finite set of potential agents N = {1, 2, · · · , n}.3 Let N ⊆ N . Each agent i ∈ N has a continuous preference relation Ri defined over R. Let Pi be the strict preference relation associated with Ri , and Ii the indifference relation. The preference relation Ri is single-peaked: there is a number p(Ri ) ∈ R such that for each pair xi , yi ∈ R, if yi < xi ≤ p(Ri ) or p(Ri ) ≤ xi < yi , then xi Pi yi . The number p(Ri ) is the peak of Ri .4 The number p(R ¯ i ) ≡ max{p(Ri ), 0} is the most preferred non-negative amount at Ri . Let R be the set of all such preference relations. Let R ≡ (Ri )i∈N ∈ R N be the preference Most of our analysis is for a fixed population. However, one mild property involves variable populations. Here we allow negative peaks. Consider the example in the introduction. For each faculty member, there is a minimum of 10 hours teaching per week required by the university. This minimum requirement is the origin in our model. Some faculty member may prefer to teach only 8 hours per week. In our model, it means that his peak is at -2. 3
4
4
profile of N . There is an endowment M ∈ R+ of an infinitely divisible commodity that has to be fully distributed among the agents. A problem with agent set N is a pair (R, M) ∈ R N × R+ . Let ΠN be the set of problems with agent set N. An assignment for (R, M ) is a list x = (xi )i∈N ∈ RN + such that P i∈N xi = M. Let X(R, M ) be the set of assignments for (R, M). A rule associates with S S each problem an assignment for it. Formally, it is a mapping ϕ : N ⊆N ΠN → N ⊆N RN + N such that for each N ⊆ N and each (R, M) ∈ Π , ϕ(R, M) ∈ X(R, M).
3
Rules and axioms
We now define various rules that have played an important role in the literature. They will also help us understand the strength of our axioms and establish their independence in our characterizations. First is the rule that assigns each agent an equal share of the endowment. Equal-division rule, EDiv : For each N ⊆ N , each (R, M) ∈ ΠN , and each i ∈ N EDivi (R, M) =
M . |N|
The second rule is the expression for this model of an idea that has played a central role in the theory of fairness, that of proportionality. It assigns each agent an equal proportion of his most preferred non-negative amount. Proportional rule, P rop : For each N ⊆ N , each (R, M) ∈ ΠN , and each i ∈ N P ropi (R, M) =
P
p(Ri ) M p(Rj )
j∈N
M
if
P
j∈N
p(Rj ) > 0,
otherwise.
|N |
The next rule evaluates sacrifices made by agents at an allocation by distances to peaks and it seeks to equate sacrifices, adjustment being made if needed, to respect account our non-negativity constraint.
5
Equal distance rule, EDist : For each N ⊆ N , each (R, M) ∈ ΠN , and each i ∈ N EDisti (R, M) = max{0, p(Ri ) + λ} where λ ∈ R solves
P
j∈N
EDistj (R, M) = M.
The last rule is arguably the most central rule in the literature. It gives agents an equal opportunity by choosing a closed interval, and assigning each one his most preferred point in the interval. When there is not enough, the left endpoint of the interval is 0. When there is too much, the right endpoint of the interval is M. Uniform rule, U : For each N ⊆ N , each (R, M) ∈ ΠN , and each i ∈ N, Ui (R, M) = where λ ∈ R+ solves
P
j∈N
min{p(R ), λ} i
if
P
j∈N
p(Rj ) ≥ M,
max{p(Ri ), λ} otherwise.
Uj (R, M) = M.5
Next, we introduce the axioms, starting with punctual fairness notions. Let ϕ be a rule. • In discussing fairness, equal division is frequently thought of as a natural reference point. equal-division is typically not efficient, but if we think of equal division as specifying ownership rights, an appealing fairness property is that each agent should find his assignment at least as desirable as an equal share of the endowment.6 It is satisfied by the equal division rule and the uniform rule, but is violated by the other two rules that we defined. Equal-division lower bound: For each N ⊆ N , each (R, M) ∈ ΠN , and each i ∈ N, M . ϕi (R, M) Ri |N | • Another way of assessing the fairness of an assignment is to let each agent compare his assignment to everybody else’s assignment. We require that each agent should find his assignment at least as desirable as each other agent’s assignment (Foley, 1967). It too is satisfied by the equal division rule and the uniform rule, but is violated by the other two rules that we defined. It is defined in B´enassy (1982). The first axiomatic characterization of it is provided by Sprumont (1991). This property has been studied for various models (Moulin, 1991; Maniquet, 1996), including our model (Thomson, 1994b, 1997; S¨ onmez, 1994) 5
6
6
No envy: For each N ⊆ N , each (R, M) ∈ ΠN , and each pair i, j ∈ N, ϕi (R, M) Ri ϕj (R, M).7 • The standard property here is that there should be no assignment at which each agent is at least as well off, and some agent is better off. It is violated by the equal division rule, but is satisfied by the other three. Efficiency: For each N ⊆ N and each (R, M) ∈ ΠN , there is no x ∈ X(R, M) at which for each i ∈ N, xi Ri ϕi (R, M), and for some j ∈ N, xj Pj ϕj (R, M). • The next property, which is variable population requirement, is that when there is excess demand, removing the agents with peaks at zero should not affect the other agents’ assignments (Ching, 2010). It is violated by the equal division rule, but is satisfied by the other three. Irrelevance of null agents: For each N ⊆ N and each (R, M) ∈ ΠN with N 0 ≡ {i ∈ P N : p(Ri ) = 0}, if j∈N p(Rj ) ≥ M, then for each j ∈ N \ N 0 , ϕj (R, M) = ϕj (RN \N 0 , M). • A weaker property is that when there is excess demand, each agent with a peak at zero should be assigned nothing. It is still violated by the equal division rule. Weak irrelevance of null agents: For each N ⊆ N and each (R, M) ∈ ΠN , if P j∈N p(Rj ) ≥ M, then for each i ∈ N such that p(Ri ) = 0, ϕi (R, M) = 0.
Note that both efficiency and irrelevance of null agents individually imply weak irrelevance of null agents. Thus, weak irrelevance of null agents can be considered as a mild efficiency property. Moreover, it is actually the first part of irrelevance of null agents. Herrero and Villar (2000) define a notion called dummy. It requires that once the assignment is selected, if a group of agents each of whom receives zero leave, the assignment selected by the rule for the remaining agents should be the same. Dummy can be viewed as a mild “consistency”8 property. Moreover, dummy implies the second part of irrelevance of null agents. It can be easily verified that dummy and weak irrelevance of null agents imply irrelevance of null agents. However, there is no logical relation between these two. Therefore, we consider irrelevance of null agents, which itself has a normative appeal, as a mild mixture of efficiency There is no logical relation between our two fairness notions: the equal-division lower bound and no envy (Thomson, 1994b, 1995). 8 This requirement says the following: suppose that, after an assignment has been made, some people take their assigned workload and leave. For the reduced problem of assigning the remaining workload among the remaining people, we should obtain the same assignment as before. 7
7
and “consistency”. Finally we discuss several robustness properties pertaining to changes in the endowment. These are the ones that are the focus of our study. • The first robustness property is introduced by Herrero and Villar (2000) for this model.9 Suppose that after an assignment is chosen, the endowment increases. As informally discussed in the introduction, two perspectives can be taken in deciding what to do in the new situation. The first one is to ignore the initial assignment, and to reapply the rule to the new problem. The second one is to use the initial assignment as point of departure: we let each agent keep his initial assignment, and only distribute the incremental amount by first revising each agent’s preferences by the amount he was initially assigned and allocate the incremental amount using these revised preferences. The assignment should be independent of the chosen perspective.10 To formally definite the property, we need additional notations: Let (R, M) ∈ ΠN and x ∈ X(R, M). For each i ∈ N, let Ri−xi ∈ R be the relation defined by shifting the preference relation Ri by the amount xi . Formally, for each pair a, b ∈ R, (a − xi ) Ri−xi (b − xi ) if and only if a Ri b. Let R−x ≡ (Ri−xi )i∈N . Ri−xi
Ri −→ 0
0
xi
(a)
(b)
Figure 1: Shifting operation on preferences. In (a), the preference relation of agent i, Ri , is depicted. His assignment is xi . We define the relation Ri−xi for agent i by shifting his preference relation Ri to the left by amount equal to his assignment xi , as depicted in (b). Composition up: For each N ⊆ N , each (R, M) ∈ ΠN , and each M ′ ∈ R+ with M ′ > M, ϕ(R, M ′ ) = ϕ(R, M) + ϕ(R−ϕ(R,M ) , M ′ − M). Composition up is satisfied by the equal division rule and the equal distance rule11 . But it is violated by the other rules introduced above. In fact, it turns out to be incompatible with They call this property “agenda independence”. This property has been studied for various problems under different names (Plott, 1973; Kalai, 1977; Young 1988; Moulin, 2000; Moulin and Stong, 2002; Moreno-Ternero and Reomer, 2012; Abizada and Chen, 2013). 11 Herreo and Villar (2000) characterize the equal distance rule by imposing this property, efficiency, and some other properties. 9
10
8
a very weak efficiency property, and either one of our fairness properties, no-envy and the equal-division lower bound. The next two propositions state these incompatibility results. Proposition 1. No rule satisfies weak irrelevance of null agents, the equal-division lower bound, and composition up. Proof. Suppose by contradiction that there exists a rule ϕ satisfying the three axioms of the proposition. Let N = {1, 2} and R ∈ R N be such that p(R1 ) = 0 and p(R2 ) = 2. Let M = 2 and M ′ = 4. Since p(R1 ) + p(R2 ) = 2 = M, by weak irrelevance of null agents, ϕ1 (R, M) = 0. Thus, ϕ2 (R, M) = 2. Since p(R) = (0, 2) and ϕ(R, M) = (0, 2), we have p(R−ϕ(R,M ) ) = (0, 0). By the equal-division lower bound, ϕ(R−ϕ(R,M ) , M ′ − M) = (1, 1). By composition up, M′ = 2 and p(R2 ) = 2, agent 2 ϕ(R, M ′ ) = ϕ(R, M) + ϕ(R−ϕ(R,M ) , M ′ − M) = (1, 3). Since |N | M′ prefers |N | to ϕ2 (R, M ′ ). This violates the equal-division lower bound. Proposition 2. No rule satisfies weak irrelevance of null agents, no envy, and composition up. Proof. Suppose by contradiction that there exists a rule ϕ satisfying the three axioms of the proposition. Let N = {1, 2} and R ∈ R N be such that p(R1 ) = 0, p(R2 ) = 2, and 1 P2 3. Let M = 2 and M ′ = 4. Since p(R1 ) + p(R2 ) = 2 = M, by weak irrelevance of null agents, ϕ1 (R, M) = 0. Thus, ϕ2 (R, M) = 2. Since p(R) = (0, 2) and ϕ(R, M) = (0, 2), we have p(R−ϕ(R,M ) ) = (0, 0). By no envy, ϕ(R−ϕ(R,M ) , M ′ − M) = (1, 1). By composition up, ϕ(R, M ′ ) = ϕ(R, M) + ϕ(R−ϕ(R,M ) , M ′ − M) = (1, 3). Since 1 P2 3, agent 2 envies agent 1 at ϕ(R, M ′ ). This violates no envy.
Note that in the proofs of Propositions 1 and 2, stating the incompatibility of very mild efficiency requirement, either one of our fairness properties, and composition up, a difficulty occurs when initially there is enough of the good and the endowment increased. Therefore, we define a weaker notion of composition up by focusing on the endowment increases of the following type: when there is not enough, we only consider increases in the endowment such that the direction of the inequality between the sum of the peaks and the endowment doesn’t change.12 The following property is one of our two central notions. It is not surprising that this one-sided conditioning is relevant to the possibility of rules satisfying a property. In fact, it has appeared in several previous studies (Thomson, 1994a, 1995, 1997). 12
9
One-sided13 composition up: For each N ⊆ N , each (R, M) ∈ ΠN , and each M ′ ∈ R+ , X i∈N
p(Ri ) ≥ M ′ > M =⇒ ϕ(R, M ′ ) = ϕ(R, M) + ϕ(R−ϕ(R,M ) , M ′ − M),
• The second robustness property is related to decreases in endowment. Suppose that after an assignment is chosen, the endowment decreases. Once again, two perspectives can be taken. The first one is to ignore the initial assignment, and to reapply the rule to the new problem. The second one is to use the initial assignment as point of departure: we let each agent i ∈ N to tentatively keep his initial assignment xi , and distribute the shortfall. We revise each agent’s preferences by shifting them by the amount he was initially assigned. Since the shortfall is negative and feasibility requires that each agent should receive a nonnegative amount, we need further revision. By changing the sign of the shortfall, we treat it as an endowment. To be consistent, we also need to revise preferences by using the operator “sym”. Then, we apply the rule to this problem and subtract the assignment we obtain from the initial one. Now, we write that the assignment should be independent of the chosen perspective.14 To formally define our second robustness property, we introduce some more notations: Let (R, M) ∈ ΠN . For each i ∈ N, let sym(Ri ) ∈ R be the relation defined as follows: for each pair a, b ∈ R, (−a) sym(Ri )(−b) if and only if a Ri b. Let sym(R) ≡ (sym(Ri ))i∈N .
Ri
sym(Ri )
Ri
−→
0
0
(a)
(b)
Symmetry operation on preferences. In (a), the preference relation of agent i, Ri , is depicted. We define the relation sym(Ri ) for agent i by taking the symmetric image of his preference relation Ri with respect to the origin, as depicted in (b).
Figure 2:
Composition down: For each N ⊆ N , each (R, M) ∈ ΠN , and each M ′ ∈ R+ with M ′ < M, ϕ(R, M ′ ) = ϕ(R, M) − ϕ(sym(R−ϕ(R,M ) ), M − M ′ ). Although the term “one-sidedness” is consistent with the terminology in Thomson (1994a, 1995, 1997), our property is even weaker since we only consider changes that occur on the excess demand side. 14 This property has been studied for various problems under different names (Moulin, 1987; Moulin, 2000; Moulin and Stong, 2002; Abizada and Chen, 2013). 13
10
Among the rules introduced above, composition down is only satisfied by the equal division rule. In fact, this property is incompatible with a very weak efficiency property, and either one of our fairness properties. Proposition 3. No rule satisfies weak irrelevance of null agents, the equal-division lower bound, and composition down. Proof. Suppose by contradiction that there is a rule ϕ satisfying the three axioms of the proposition. Let N = {1, 2} and R ∈ R N be such that p(R1 ) = 0 and p(R2 ) = 2. Let M = 2 and M ′ = 1. Since p(R1 ) + p(R2 ) = 2 = M, by weak irrelevance of null agents, ϕ1 (R, M) = 0. Thus, ϕ2 (R, M) = 2. Since p(R) = (0, 2) and ϕ(R, M) = (0, 2), we have p(sym(R−ϕ(R,M ) )) = (0, 0). By the equal-division lower bound, ϕ(sym(R−ϕ(R,M ) ), M ′ − M) = ( 21 , 12 ). By composition down, ϕ(R, M ′ ) = ϕ(R, M) − ϕ(sym(R−ϕ(R,M ) ), M ′ − M) = (0, 2) − ( 21 , 21 ) = (− 12 , 32 ). This violates non-negativity of assignments. Proposition 4. No rule satisfies weak irrelevance of null agents, no envy, and composition down. Proof. The proof is similar to that of Proposition 3, so we omit it.
Note that as before, in the proofs of Propositions 3 and 4, stating the incompatibility of very mild efficiency requirement, either one of the fairness properties, and composition down, a difficulty occurs when initially there is just enough of the good and the endowment decreased. Therefore, define our weaker notion of composition down by focusing on the endowment decreases of the following type: when there is too much, we only consider decreases in the endowment such that the direction of the inequality between the sum of the peaks and the endowment doesn’t change. The following property is the second one of our central notions. One-sided15 composition down: For each N ⊆ N , each (R, M) ∈ ΠN , and each M ′ ∈ R+ , X i∈N
p(Ri ) < M ′ < M =⇒ ϕ(R, M ′ ) = ϕ(R, M) − ϕ(sym(R−ϕ(R,M ) ), M − M ′ ).
As it is with one-sided composition up, the term “one-sidedness” is consistent with the terminology in Thomson (1994a, 1995, 1997). Our property is even weaker since we only consider changes that occur on the excess supply side. 15
11
Our two one-sided versions of the composition properties, seem to be the most natural way of weakening our composition properties in the sense that we impose them only when “desirable” changes in the endowment occur, i.e. when there is not enough and the endowment increases, and when there is too much and the endowment decreases. The resulting properties turn out to be satisfied by all four rules that we introduced.
4
Main Result
Although our robustness properties are weak, when we impose them together with fairness and a very mild efficiency property, we obtain one single rule. This rule is the uniform rule. Note that efficiency itself is not one of our axioms. Yet, the uniform rule is efficient. Kesten (2006), Ching (2010), and Ehlers (2011) also provide characterizations of the uniform rule, without imposing efficiency.
Theorem 1. The uniform rule is the only rule satisfying irrelevance of null agents, the equal-division lower bound, one-sided composition up, and one-sided composition down. Proof. It is easy to verify that the uniform rule satisfies all the axioms. Conversely, let ϕ be a rule satisfying these axioms. Let N ⊆ N , (R, M) ∈ ΠN , and n ≡ |N|. We distinguish two cases: X Case 1. M ∈ [0, p(Ri )]. i∈N
Let N 0 ≡ {i ∈ N : p(Ri ) ≤ 0}, and n0 ≡ |N 0 | Let p1 ≡ min 0 p(Ri ), N 1 ≡ {i ∈ N : p(Ri ) = p1 }, and n1 ≡ |N 1 |. i∈N \N
Subcase 1.1. M ∈ [0, (n − n0 )p1 ]. X X ϕj (RN \N 0 , M) = M. Thus, for each ϕj (R, M) = By irrelevance of null agents, 0
j∈N \N 0
j∈N \N 0
i ∈ N , ϕi (R, M) = 0. M Since By the equal-division lower bound, for each i ∈ N \ N 0 , ϕi (RN \N 0 , M) Ri n−n 0. M M 1 0 ≤ p , for each i ∈ N \ N , ϕi (RN \N 0 , M) ≥ n−n0 . Thus, for each i ∈ N \ N 0 , n−n0 M ϕi (RN \N 0 , M) = n−n 0 . By irrelevance of null agents, ϕN \N 0 (R, M) = ϕ(RN \N 0 , M). Thus, M for each i ∈ N \ N 0 , ϕi (R, M) = n−n 0.
12
Let p2 ≡
min
i∈N \(N 0 ∪N 1 )
p(Ri ), N 2 ≡ {i ∈ N : p(Ri ) = p2 }, and n2 ≡ |N 2 |.
Subcase 1.2. M ∈ ((n − n0 )p1 , n1 p1 + (n − n0 − n1 )p2 ]. ′ Let M ′ ≡ (n − n0 )p1 . By one-sided composition up, ϕ(R, M) = ϕ(R, M ′ ) + ϕ(R−ϕ(R,M ) , M − M ′ ). By Subcase 1.1, for each i ∈ N 0 , ϕi (R, M ′ ) = 0, and for each i ∈ N 1 , ϕi (R, M ′ ) = −ϕ (R,M ′ ) M′ = p1 . Since for each i ∈ N 0 ∪ N 1 , p(Ri i ) = 0, by irrelevance of null agents, n−n0X X −ϕ(R,M ′ ) ′ −ϕ(R,M ′ ) ϕj (R , M −M ) = ϕj (R |N \(N 0 ∪N 1 ) , M −M ′ ) = M −M ′ .
j∈N \(N 0 ∪N 1 )
j∈N \(N 0 ∪N 1 ) −ϕ(R,M ′ )
Thus, for each i ∈ N ∪ N , ϕi (R , M − M ′ ) = 0. Thus, for each i ∈ N 0 , ϕi (R, M) = 0 + 0 = 0, and for each i ∈ N 1 , ϕi (R, M) = p1 + 0 = p1 . By Subcase 1.1, for each i ∈ N \ (N 0 ∪ N 1 ), ϕi (R, M ′ ) = p1 . By the equal-division lower ′ M −M ′ bound, for each i ∈ N \ (N 0 ∪ N 1 ), ϕi (R−ϕ(R,M ) |N \(N 0 ∪N 1 ) , M − M ′ ) Ri n−n Since 0 −n1 . M −M ′ M −M ′ 2 1 0 1 −ϕ(R,M ′ ) ′ ≤ p − p , for each i ∈ N \ (N ∪ N ), ϕi (R |N \(N 0 ∪N 1 ) , M − M ) ≥ n−n0 −n1 . n−n0 −n1 M −M ′ 0 1 −ϕ(R,M ′ ) Thus, for each i ∈ N \ (N ∪ N ), ϕi (R |N \(N 0 ∪N 1 ) , M − M ′ ) = n−n 0 −n1 . By irrele′ ′ vance of null agents, ϕN \(N 0 ∪ N 1 ) (R−ϕ(R,M ) , M − M ′ ) = ϕ(R−ϕ(R,M ) |N \(N 0 ∪ N 1 ) , M − M ′ ). ′ M −M ′ Thus, for each Thus, for each i ∈ N \ (N 0 ∪ N 1 ), ϕi (R−ϕ(R,M ) , M − M ′ ) = n−n 0 −n1 . (n−n0 −n1 )p1 +M −(n−n0 )p1 M −n1 p1 M −M ′ 0 1 1 = n−n0 −n1 . i ∈ N \ (N ∪ N ), ϕi (R, M) = p + n−n0 −n1 = n−n0 −n1 Let p3 ≡
0
min 0 1
i∈N \(N ∪N
∪N 2 )
1
p(Ri ), N 3 ≡ {i ∈ N : p(Ri ) = p3 }, and n3 ≡ |N 3 |.
Subcase 1.3. M ∈ (n1 p1 + (n − n0 − n1 )p2 , n1 p1 + n2 p2 + (n − n0 − n1 − n2 )p3 ].
··· P Repeating the argument, we obtain that if M ∈ [0, i∈N p(Ri )], then ϕ(R, M) = U(R, M).
Case 2. M ∈ (
X i∈N
p(Ri ), ∞).
Let pk ≡ max p(Ri ), N k ≡ {i ∈ N : p(Ri ) = pk }, and nk ≡ |N k |. i∈N
Subcase 2.1. M ∈ [npk , ∞). By the equal-division lower bound, for each i ∈ N, ϕi (R, M) Ri M . Since M ≥ pk = n n X . Since ϕi (R, M) = M, for each i ∈ N, max p(Ri ), for each i ∈ N, ϕi (R, M) ≤ M n i∈N
ϕi (R, M) =
i∈N
M . n
Subcase 2.2. M ∈ (
X
p(Ri ), npk ).
i∈N
13
′
Let M ′ = npk . By one-sided composition down, ϕ(R, M) = ϕ(R, M ′ ) − ϕ(sym(R−ϕ(R,M ) ), M ′ − M). ¯ ≡ sym(R−ϕ(R,M ′ ) ). Then, for By Subcase 2.1, for each i ∈ N, ϕi (R, M ′ ) = pk . Let R X each i ∈ N, p(R¯i ) = −[p(Ri ) − ϕi (R, M ′ )] = pk − p(Ri ). Since p(Ri ) < M, we have X i∈N
p(R¯i ) =
X i∈N
p(R¯i ) = npk −
X i∈N
k
p(Ri ) ≥ np −
X i∈N
i∈N
′
p(Ri ) > M − M.
¯ M ′ − M) = min{p(R¯i ), λ)}, where λ solves By Case 1, for each i ∈ N, ϕi (R,
X
¯ ϕj (R,
j∈N
M ′ − M) = M ′ − M. Since for each i ∈ N, p(R¯i ) = pk − p(Ri ), we have min{p(R¯i ), λ)} = min{pk − p(Ri ), λ} = pk − max{p(Ri ), pk − λ}. Thus, for each i ∈ N, ϕi (R, M) = pk − [pk − X max{p(Ri ), pk − λ}] = max{p(Ri ), pk − λ}. Let λ′ ≡ pk − λ. Since min{p(R¯j ), λ)} = X
X
j∈N
M − M, we have max{p(Rj ), λ } = [p − min{p(R¯j , λ)}] = npk − (M ′ − M) = M. j∈N j∈N P Thus, for each i ∈ N, ϕi (R, M) = max{p(Ri ), λ′ }, where λ′ solves j∈N ϕj (R, M) = M. X Thus, if M ∈ ( p(Ri ), npk ), then ϕ(R, M) = U(R, M). ′
′
k
i∈N
• On the independence of the axioms in Theorem 1 (1) The equal division rule satisfies all the axioms of Theorem 1 except for irrelevance of null agents. (2) The proportional rule satisfies all the axioms of Theorem 1 except for the equal-division lower bound. (3) Consider the “mixed” rule16 defined as follows: for each N ⊆ N , each (R, M) ∈ ΠN , and each i ∈ N, P U (R, M) if i j∈N p(Rj ) ≥ M, Mixi (R, M) = EDivi (R, M) otherwise. The mixed rule satisfies all the axioms of Theorem 1 except for one-sided composition down. (4) Consider the “reverse mixed” rule defined as follows: for each N ⊆ N , each (R, M) ∈ ΠN , 16
A similar rule where EDiv is replaced with EDist is defined in Herrero and Villar (1998).
14
and each i ∈ N,
RMixi (R, M) =
Ui (R, M)
M |{j∈N :p(Rj )>0}|
0
if
X j∈N
if
X
p(Rj ) ≤ M, p(Rj ) > M and p(Ri ) > 0,
j∈N
if
X
p(Rj ) > M and p(Ri ) = 0,
j∈N
The reverse mixed rule satisfies all the axioms of Theorem 1 except for one-sided composition up. One natural question is whether the uniform rule is still the only rule left when the equaldivision lower bound in Theorem 1 is replaced by another appealing fairness property, no envy. The answer is no. Consider the “modified uniform”17 rule defined as follows: for each N ⊆ N and each (R, M) ∈ ΠN , let pk ≡ maxi∈N p(Ri ), N k ≡ {i ∈ N : p(Ri ) = pk }, and pk−1 ≡ maxi∈N \N k p(Ri ). For each i ∈ N, P P M − j∈N p(Rj ) k k k−1 p(R if , i) + j∈N p(Rj ) < M, i ∈ N , for each j ∈ N , ∞ Pj p |N k | P ModUi (R, M) = p(Ri ) if / N k , for each j ∈ N k , ∞ Pj pk−1 , j∈N p(Rj ) < M, i ∈ Ui (R, M) otherwise. The modified uniform rule satisfies efficiency, irrelevance of null agents, no envy, onesided composition up, and one-sided composition down. Yet, it differs from the uniform rule for a few special cases.
Although when we impose no envy instead of the equal-division lower bound, there is no parallel result of Theorem 1, we do have another characterization of the uniform rule, mainly based on no envy and the two robustness properties. As we know, the uniform rule is the only rule satisfying efficiency, no envy and peak only (Thomson, 1994a). If we weaken efficiency to its very weak form, weak irrelevance of null agents, and impose our two one-sided composition properties in addition to no envy and peak only, we again are left with the uniform rule as the only one satisfying these properties. Comparing to Theorem 1, we find that this result is less appealing, as we replace the equal-division lower bound by no envy, weaken irrelevance of null agents, yet add a strong property, peak only, which 17
We call this rule “modified uniform” because it is the same as the uniform rule except for very few cases.
15
requires the allocation to depend only on the peaks of preferences. Therefore, we list this characterization as a remark, rather than a theorem. Note that the properties involved in the characterization are independent.18
Remark 1. The uniform rule is the only rule satisfying weak irrelevance of null agents, no envy, peak only, one-sided composition up, and one-sided composition down.
References Abizada, A. and Chen, S. (2013), “House allocation when construction schedule is unpredictable”, mimeo. B´enassy, J.P. (1982), The Economics of market disequilibrium, Academic Press, San Diego. Ching, S. (1992), “A simple characterization of the uniform rule”, Economics Letters 40, 57–60. Ching, S. (1994), “An alternative characterization of the uniform rule”, Social Choice and Welfare 11, 131–136. Ching, S. (2010), “An axiomatization of the uniform rule without the Pareto principle”, University of Hong Kong, mimeo. Ehlers, L. (2011), “A characterization of the uniform rule without Pareto-optimality”, Journal of the Spanish Economic Association 2, 447–452. Foley, D., (1967), “Resource allocation and the public sector”, Yale Economics Essays 7, 45–98. Herrero, C. and Villar A. (1998), “Agenda independence in allocation problems with singlepeaked preferences”, mimeo. Herrero, C. and Villar A. (2000), “An alternative characterization of the equal-distance rule for allocation problems with single-peaked preferences”, Economics Letters 66, 311–317. Kalai, E. (1977), “Proportional solutions to bargaining situations: interpersonal utility comparisons”, Econometrica 45, 1623–1630. Kesten, O. (2006), “More on the uniformrule: Characterizations without Pareto optimality”, Mathematical Social Sciences 51, 192–200. 18
The proof and the independence of properties are available upon request.
16
Maniquet, F. (1996), “Allocation rules for a commonly owned technology: the average cost lower bound”, Journal of Economic Theory, 69, 490–507. Moreno-Ternero J. and J. Roemer (2012), “A common ground for resource and welfare egalitarianism”, Games and Economic Behavior, 75, 832–841. Moulin, H. (1987), “Equal or proportional division of a surplus, and other mehods”, International Journal of Game Theory, 16, 161–186. Moulin, H. (1991), “Welfare bounds and other asymmetric rationing methods”, Journal of Economic Theory, 54, 321–337. Moulin, H. (2000), “Priority rules and other asymmetric rationing methods”, Econometrica, 68, 643–684. Moulin, H. and R. Stong (2002), “Fair queuing and other probabilistic allocation methods”, Mathematics of Operations Research, 27, 1–30. Plott, C. R. (1973), “Path independence, rationality and social choice,” Econometrica 41, 1075-1091. S¨onmez, T. (1994), “Consistency, Monotonicity, and the Uniform Rule. Economics Letters 46, 229–235. Sprumont, Y. (1991), “The division problem with single-peaked preferences: a characterization of the uniform allocation rule”, Econometrica 59, 509-519. Thomson, W. (1994a), “Resource-monotonic solutions to the problem of fair division when preferences are single-peaked”, Social Choice and Welfare 11, 205-223. Thomson, W. (1994b), “Consistent solutions to the problem of fair division when preferences are single-peaked”, Journal of Economic Theory 63, 219-245. Thomson, W. (1995), “Population-monotonic solutions to the problem of fair division when preferences are single-peaked”, Economic Theory 5, 229-246. Thomson, W. (1997), “The replacement principle in economies with single-peaked preferences”, Journal of Economic Theory 76, 145-168. Thomson, W. (2003), “Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey”, Mathematical Social Sciences 45, 249-297. Thomson, W. (2011), “Fair allocation rules”, K. J. Arrow, A. K. Sen, and K. Suzumura (Eds.) Handbook of Social Choice and Welfare, Volume 2, Elsevier, 393-506. Young, H.P. (1988), “Distributive justice in taxation”, Journal of Economic Theory 44, 321-335. 17
• • •
We study allocation of a divisible good when preferences are single-peaked. We formulate two new robustness requirements. We characterize uniform rule using our robustness requirements.
S0165-4896(14)00045-6 http://dx.doi.org/10.1016/j.mathsocsci.2014.05.003 MATSOC 1740
To appear in:
Mathematical Social Sciences
Received date: 13 June 2013 Revised date: 17 March 2014 Accepted date: 16 May 2014 Please cite this article as: Abizada, A., Chen, S., A characterization of the uniform rule based on new robustness properties. Mathematical Social Sciences (2014), http://dx.doi.org/10.1016/j.mathsocsci.2014.05.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
Two characterizations of the uniform rule based on new robustness properties∗ Azar Abizada† and Siwei Chen‡ March 17, 2014
Abstract We study the problem of allocating a divisible good among a group of people. Each person’s preferences are single-peaked. We consider situations in which there might be more of the resource to be assigned than was planned, or there might be less of the resource. Two robustness properties are formulated, which we call one-sided composition up and one-sided composition down. We show that only one rule satisfies irrelevance of null agents, the equal-division lower bound, and our robustness properties. This rule is the uniform rule. JEL classification: C71; D63. Keywords: one-sided composition up, one-sided composition down, irrelevance of null agents, uniform rule
1
Introduction
Consider a department chair having to assign teaching duties to faculty. Each faculty member has single-peaked preferences over the workload he receives: he has an ideal workload (i.e. peak of his preferences) and his welfare decreases as his workload moves away from this ideal amount. We look for rules allocating the total workload that satisfy certain desirable properties. We would like to thank William Thomson for his guidance, and invaluable comments. We are grateful to two anonymous referees for their useful comments. † School of Business, ADA University, 11 Ahmadbay Aghaoglu St., Baku AZ1008, Azerbaijan. Email: [email protected]. ‡ Lingnan College, Sun Yat-sen University, Guangzhou, 510275, China. Email: [email protected]. ∗
1
Various properties of efficiency, fairness, and robustness under strategic behavior have been analyzed for the problem where agents have single-peaked preferences (Sprumont, 1991; Ching, 1992, 1994, 2010; Thomson, 1994a, 1994b, 1995, 1997, 2011). Here, we consider the case where the total workload is not fixed. Once teaching duties have been assigned, imagine that too many students have registered for some course. In such a case, the chair has to increase the number of sections for the course. Alternatively, it may be that too few students have registered for some course. Then, some sections have to be canceled. How should we handle these changes? In general, when the circumstances in which a group of agents find themselves change, there might be several ways to handle the change. Each of these ways may well be equally legitimate. Yet, outcomes may differ depending on which perspective is taken. In such cases, no matter what action is chosen, someone maybe unhappy that the other action is not taken. Therefore, the choice of which perspective should be taken may need to have 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, no matter which one is taken, it should produce the same outcome. When opportunities expand, this principle has been investigated under the name of “stepby-step negotiations” (Kalai, 1977) for bargaining, and it is also reminiscent of the “path independence” axiom for choice functions (Plott, 1973). “Lower composition” (Moulin, 2000) and “composition” (Young, 1988) for the adjudication of conflicting claims, “lower composition” (Moulin and Stong, 2002) for queuing problems, “composition” (Moreno-Ternero and Roemer, 2012) for resource allocation with utilities, and “iterative composition up” (Abizada and Chen, 2013) for allocation of the indivisible goods, can all be viewed as expressions of this robustness principle in different contexts. In the context of the adjudication of conflicting claims, queuing problem, and allocation of the indivisible goods, a shrinking of opportunities has been considered too. Axioms pertaining to such situation have been formulated (Moulin, 1987, 2000; Moulin and Stong, 2002; Abizada and Chen, 2013), which can also be understood as expressions of the general principle. For each class of problems, a general principle needs to be adapted in order to best take into account the structure of the class. We propose two new properties that are expressions of the robustness idea to our model. Return to the example. Our first property is related to the possibility that the total workload increases, that is, when new sections have to be offered. Two possible ways of proceeding are (i) to ignore the choice made initially, and to reapply the rule to the new problem, the one with the larger workload; (ii) to use the initial choice as point of departure. For each faculty member we keep his initial workload and
2
“revise his preferences by that amount”. We allocate the additional workload using these revised preferences. The final workload assigned to each faculty member is obtained by augmenting his initial workload by the additional workload computed in this manner. Each of these procedures provides a plausible way of handling the change in the total workload. However, they may affect the welfare of different faculty members differently. If that is so, depending on which procedure is chosen, some faculty members may object that the other one was not chosen. In order to avoid such situations, we require the final assignments to be the same, independently of which procedure is followed. The second property pertains to the opposite change in the total workload, that is, when some courses have to be 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 with the smaller workload; (ii) to use the initial choice as point of departure. Similarly, since each faculty member keeps his initial workload, we should revise his preferences in the new situation. We allocate the cancelled courses by treating them as an “endowment” (changing the sign from negative to positive), using the revised preferences. The final workload assigned to each faculty member is obtained by subtracting the amount computed in this manner from his initial workload. Again, we require the final assignments to be the same irrespective of the perspective that is followed. We are interested in efficiency, which states that there should be no other assignment at which at least one faculty member is better off and no faculty member is worse off. We consider a new property called irrelevance of null agents, which states that whenever there is excess demand, the assignment should not be affected by the departure of faculty members who ideally prefer not to teach. This property can be viewed as a mild mixture of efficiency and “consistency”1 (Thomson, 1994b; Herrero and Villar, 2000). We also consider two standard punctual fairness properties: first, no faculty member should prefer someone else’s assignment to his own (no envy); second, each faculty member should find his or her assignment at least as desirable as an equal share of the total workload (the equal-division lower bound ). Herrero and Villar (2000) is the first paper to study the robustness principle for this model. They propose a robustness property that pertains to situations in which the endowment increases, under the name “agenda-independence”. However, as we will show, this property is incompatible with efficiency and either one of our fairness properties2 . As disWe will illustrate the logical relations among these properties in Section 3 where they are formally defined. 2 They characterize the “equal distance” rule using efficiency, agenda independence, and some other properties. 1
3
cussed above, we also define a similar robustness property which pertains to situations in which the endowment decreases. Again, this property is incompatible with efficiency and either one of our fairness properties. These incompatibilities occur when initially there is too much (the sum of peaks is smaller than than the endowment) and the endowment increases, or when initially there is not enough (the sum of peaks is larger than the endowment) and the endowment decreases. We propose two more reasonable robustness properties, which are conditional upon the endowment remaining on the same side of the sum of the peaks. This kind of conditioning is introduced in this model in Thomson (1994a). The first property applies to the case of “there is still not enough after an increase”, and the second property applies to the case of “there is still too much after a decrease”. These are the central notions of our paper. Our main result is that there is a unique rule satisfying irrelevance of null agents, the equal-division lower bound, and the two robustness properties. It is the rule known as the “uniform rule”, which is the most central rule in the literature. We also find that the uniform rule is the only rule satisfying a weaker version of irrelevance of null agents, no envy, peak only, and the two robustness properties. The uniform rule has already come out of a number of axiomatic studies of this class of problems. Our study confirms its importance. The remainder of the paper is organized as follows. In Section 2 we define the model. In Section 3 we state the axioms and define the rules. In Section 4 we provide our main result and some discussions.
2
Model
There is a finite set of potential agents N = {1, 2, · · · , n}.3 Let N ⊆ N . Each agent i ∈ N has a continuous preference relation Ri defined over R. Let Pi be the strict preference relation associated with Ri , and Ii the indifference relation. The preference relation Ri is single-peaked: there is a number p(Ri ) ∈ R such that for each pair xi , yi ∈ R, if yi < xi ≤ p(Ri ) or p(Ri ) ≤ xi < yi , then xi Pi yi . The number p(Ri ) is the peak of Ri .4 The number p(R ¯ i ) ≡ max{p(Ri ), 0} is the most preferred non-negative amount at Ri . Let R be the set of all such preference relations. Let R ≡ (Ri )i∈N ∈ R N be the preference Most of our analysis is for a fixed population. However, one mild property involves variable populations. Here we allow negative peaks. Consider the example in the introduction. For each faculty member, there is a minimum of 10 hours teaching per week required by the university. This minimum requirement is the origin in our model. Some faculty member may prefer to teach only 8 hours per week. In our model, it means that his peak is at -2. 3
4
4
profile of N . There is an endowment M ∈ R+ of an infinitely divisible commodity that has to be fully distributed among the agents. A problem with agent set N is a pair (R, M) ∈ R N × R+ . Let ΠN be the set of problems with agent set N. An assignment for (R, M ) is a list x = (xi )i∈N ∈ RN + such that P i∈N xi = M. Let X(R, M ) be the set of assignments for (R, M). A rule associates with S S each problem an assignment for it. Formally, it is a mapping ϕ : N ⊆N ΠN → N ⊆N RN + N such that for each N ⊆ N and each (R, M) ∈ Π , ϕ(R, M) ∈ X(R, M).
3
Rules and axioms
We now define various rules that have played an important role in the literature. They will also help us understand the strength of our axioms and establish their independence in our characterizations. First is the rule that assigns each agent an equal share of the endowment. Equal-division rule, EDiv : For each N ⊆ N , each (R, M) ∈ ΠN , and each i ∈ N EDivi (R, M) =
M . |N|
The second rule is the expression for this model of an idea that has played a central role in the theory of fairness, that of proportionality. It assigns each agent an equal proportion of his most preferred non-negative amount. Proportional rule, P rop : For each N ⊆ N , each (R, M) ∈ ΠN , and each i ∈ N P ropi (R, M) =
P
p(Ri ) M p(Rj )
j∈N
M
if
P
j∈N
p(Rj ) > 0,
otherwise.
|N |
The next rule evaluates sacrifices made by agents at an allocation by distances to peaks and it seeks to equate sacrifices, adjustment being made if needed, to respect account our non-negativity constraint.
5
Equal distance rule, EDist : For each N ⊆ N , each (R, M) ∈ ΠN , and each i ∈ N EDisti (R, M) = max{0, p(Ri ) + λ} where λ ∈ R solves
P
j∈N
EDistj (R, M) = M.
The last rule is arguably the most central rule in the literature. It gives agents an equal opportunity by choosing a closed interval, and assigning each one his most preferred point in the interval. When there is not enough, the left endpoint of the interval is 0. When there is too much, the right endpoint of the interval is M. Uniform rule, U : For each N ⊆ N , each (R, M) ∈ ΠN , and each i ∈ N, Ui (R, M) = where λ ∈ R+ solves
P
j∈N
min{p(R ), λ} i
if
P
j∈N
p(Rj ) ≥ M,
max{p(Ri ), λ} otherwise.
Uj (R, M) = M.5
Next, we introduce the axioms, starting with punctual fairness notions. Let ϕ be a rule. • In discussing fairness, equal division is frequently thought of as a natural reference point. equal-division is typically not efficient, but if we think of equal division as specifying ownership rights, an appealing fairness property is that each agent should find his assignment at least as desirable as an equal share of the endowment.6 It is satisfied by the equal division rule and the uniform rule, but is violated by the other two rules that we defined. Equal-division lower bound: For each N ⊆ N , each (R, M) ∈ ΠN , and each i ∈ N, M . ϕi (R, M) Ri |N | • Another way of assessing the fairness of an assignment is to let each agent compare his assignment to everybody else’s assignment. We require that each agent should find his assignment at least as desirable as each other agent’s assignment (Foley, 1967). It too is satisfied by the equal division rule and the uniform rule, but is violated by the other two rules that we defined. It is defined in B´enassy (1982). The first axiomatic characterization of it is provided by Sprumont (1991). This property has been studied for various models (Moulin, 1991; Maniquet, 1996), including our model (Thomson, 1994b, 1997; S¨ onmez, 1994) 5
6
6
No envy: For each N ⊆ N , each (R, M) ∈ ΠN , and each pair i, j ∈ N, ϕi (R, M) Ri ϕj (R, M).7 • The standard property here is that there should be no assignment at which each agent is at least as well off, and some agent is better off. It is violated by the equal division rule, but is satisfied by the other three. Efficiency: For each N ⊆ N and each (R, M) ∈ ΠN , there is no x ∈ X(R, M) at which for each i ∈ N, xi Ri ϕi (R, M), and for some j ∈ N, xj Pj ϕj (R, M). • The next property, which is variable population requirement, is that when there is excess demand, removing the agents with peaks at zero should not affect the other agents’ assignments (Ching, 2010). It is violated by the equal division rule, but is satisfied by the other three. Irrelevance of null agents: For each N ⊆ N and each (R, M) ∈ ΠN with N 0 ≡ {i ∈ P N : p(Ri ) = 0}, if j∈N p(Rj ) ≥ M, then for each j ∈ N \ N 0 , ϕj (R, M) = ϕj (RN \N 0 , M). • A weaker property is that when there is excess demand, each agent with a peak at zero should be assigned nothing. It is still violated by the equal division rule. Weak irrelevance of null agents: For each N ⊆ N and each (R, M) ∈ ΠN , if P j∈N p(Rj ) ≥ M, then for each i ∈ N such that p(Ri ) = 0, ϕi (R, M) = 0.
Note that both efficiency and irrelevance of null agents individually imply weak irrelevance of null agents. Thus, weak irrelevance of null agents can be considered as a mild efficiency property. Moreover, it is actually the first part of irrelevance of null agents. Herrero and Villar (2000) define a notion called dummy. It requires that once the assignment is selected, if a group of agents each of whom receives zero leave, the assignment selected by the rule for the remaining agents should be the same. Dummy can be viewed as a mild “consistency”8 property. Moreover, dummy implies the second part of irrelevance of null agents. It can be easily verified that dummy and weak irrelevance of null agents imply irrelevance of null agents. However, there is no logical relation between these two. Therefore, we consider irrelevance of null agents, which itself has a normative appeal, as a mild mixture of efficiency There is no logical relation between our two fairness notions: the equal-division lower bound and no envy (Thomson, 1994b, 1995). 8 This requirement says the following: suppose that, after an assignment has been made, some people take their assigned workload and leave. For the reduced problem of assigning the remaining workload among the remaining people, we should obtain the same assignment as before. 7
7
and “consistency”. Finally we discuss several robustness properties pertaining to changes in the endowment. These are the ones that are the focus of our study. • The first robustness property is introduced by Herrero and Villar (2000) for this model.9 Suppose that after an assignment is chosen, the endowment increases. As informally discussed in the introduction, two perspectives can be taken in deciding what to do in the new situation. The first one is to ignore the initial assignment, and to reapply the rule to the new problem. The second one is to use the initial assignment as point of departure: we let each agent keep his initial assignment, and only distribute the incremental amount by first revising each agent’s preferences by the amount he was initially assigned and allocate the incremental amount using these revised preferences. The assignment should be independent of the chosen perspective.10 To formally definite the property, we need additional notations: Let (R, M) ∈ ΠN and x ∈ X(R, M). For each i ∈ N, let Ri−xi ∈ R be the relation defined by shifting the preference relation Ri by the amount xi . Formally, for each pair a, b ∈ R, (a − xi ) Ri−xi (b − xi ) if and only if a Ri b. Let R−x ≡ (Ri−xi )i∈N . Ri−xi
Ri −→ 0
0
xi
(a)
(b)
Figure 1: Shifting operation on preferences. In (a), the preference relation of agent i, Ri , is depicted. His assignment is xi . We define the relation Ri−xi for agent i by shifting his preference relation Ri to the left by amount equal to his assignment xi , as depicted in (b). Composition up: For each N ⊆ N , each (R, M) ∈ ΠN , and each M ′ ∈ R+ with M ′ > M, ϕ(R, M ′ ) = ϕ(R, M) + ϕ(R−ϕ(R,M ) , M ′ − M). Composition up is satisfied by the equal division rule and the equal distance rule11 . But it is violated by the other rules introduced above. In fact, it turns out to be incompatible with They call this property “agenda independence”. This property has been studied for various problems under different names (Plott, 1973; Kalai, 1977; Young 1988; Moulin, 2000; Moulin and Stong, 2002; Moreno-Ternero and Reomer, 2012; Abizada and Chen, 2013). 11 Herreo and Villar (2000) characterize the equal distance rule by imposing this property, efficiency, and some other properties. 9
10
8
a very weak efficiency property, and either one of our fairness properties, no-envy and the equal-division lower bound. The next two propositions state these incompatibility results. Proposition 1. No rule satisfies weak irrelevance of null agents, the equal-division lower bound, and composition up. Proof. Suppose by contradiction that there exists a rule ϕ satisfying the three axioms of the proposition. Let N = {1, 2} and R ∈ R N be such that p(R1 ) = 0 and p(R2 ) = 2. Let M = 2 and M ′ = 4. Since p(R1 ) + p(R2 ) = 2 = M, by weak irrelevance of null agents, ϕ1 (R, M) = 0. Thus, ϕ2 (R, M) = 2. Since p(R) = (0, 2) and ϕ(R, M) = (0, 2), we have p(R−ϕ(R,M ) ) = (0, 0). By the equal-division lower bound, ϕ(R−ϕ(R,M ) , M ′ − M) = (1, 1). By composition up, M′ = 2 and p(R2 ) = 2, agent 2 ϕ(R, M ′ ) = ϕ(R, M) + ϕ(R−ϕ(R,M ) , M ′ − M) = (1, 3). Since |N | M′ prefers |N | to ϕ2 (R, M ′ ). This violates the equal-division lower bound. Proposition 2. No rule satisfies weak irrelevance of null agents, no envy, and composition up. Proof. Suppose by contradiction that there exists a rule ϕ satisfying the three axioms of the proposition. Let N = {1, 2} and R ∈ R N be such that p(R1 ) = 0, p(R2 ) = 2, and 1 P2 3. Let M = 2 and M ′ = 4. Since p(R1 ) + p(R2 ) = 2 = M, by weak irrelevance of null agents, ϕ1 (R, M) = 0. Thus, ϕ2 (R, M) = 2. Since p(R) = (0, 2) and ϕ(R, M) = (0, 2), we have p(R−ϕ(R,M ) ) = (0, 0). By no envy, ϕ(R−ϕ(R,M ) , M ′ − M) = (1, 1). By composition up, ϕ(R, M ′ ) = ϕ(R, M) + ϕ(R−ϕ(R,M ) , M ′ − M) = (1, 3). Since 1 P2 3, agent 2 envies agent 1 at ϕ(R, M ′ ). This violates no envy.
Note that in the proofs of Propositions 1 and 2, stating the incompatibility of very mild efficiency requirement, either one of our fairness properties, and composition up, a difficulty occurs when initially there is enough of the good and the endowment increased. Therefore, we define a weaker notion of composition up by focusing on the endowment increases of the following type: when there is not enough, we only consider increases in the endowment such that the direction of the inequality between the sum of the peaks and the endowment doesn’t change.12 The following property is one of our two central notions. It is not surprising that this one-sided conditioning is relevant to the possibility of rules satisfying a property. In fact, it has appeared in several previous studies (Thomson, 1994a, 1995, 1997). 12
9
One-sided13 composition up: For each N ⊆ N , each (R, M) ∈ ΠN , and each M ′ ∈ R+ , X i∈N
p(Ri ) ≥ M ′ > M =⇒ ϕ(R, M ′ ) = ϕ(R, M) + ϕ(R−ϕ(R,M ) , M ′ − M),
• The second robustness property is related to decreases in endowment. Suppose that after an assignment is chosen, the endowment decreases. Once again, two perspectives can be taken. The first one is to ignore the initial assignment, and to reapply the rule to the new problem. The second one is to use the initial assignment as point of departure: we let each agent i ∈ N to tentatively keep his initial assignment xi , and distribute the shortfall. We revise each agent’s preferences by shifting them by the amount he was initially assigned. Since the shortfall is negative and feasibility requires that each agent should receive a nonnegative amount, we need further revision. By changing the sign of the shortfall, we treat it as an endowment. To be consistent, we also need to revise preferences by using the operator “sym”. Then, we apply the rule to this problem and subtract the assignment we obtain from the initial one. Now, we write that the assignment should be independent of the chosen perspective.14 To formally define our second robustness property, we introduce some more notations: Let (R, M) ∈ ΠN . For each i ∈ N, let sym(Ri ) ∈ R be the relation defined as follows: for each pair a, b ∈ R, (−a) sym(Ri )(−b) if and only if a Ri b. Let sym(R) ≡ (sym(Ri ))i∈N .
Ri
sym(Ri )
Ri
−→
0
0
(a)
(b)
Symmetry operation on preferences. In (a), the preference relation of agent i, Ri , is depicted. We define the relation sym(Ri ) for agent i by taking the symmetric image of his preference relation Ri with respect to the origin, as depicted in (b).
Figure 2:
Composition down: For each N ⊆ N , each (R, M) ∈ ΠN , and each M ′ ∈ R+ with M ′ < M, ϕ(R, M ′ ) = ϕ(R, M) − ϕ(sym(R−ϕ(R,M ) ), M − M ′ ). Although the term “one-sidedness” is consistent with the terminology in Thomson (1994a, 1995, 1997), our property is even weaker since we only consider changes that occur on the excess demand side. 14 This property has been studied for various problems under different names (Moulin, 1987; Moulin, 2000; Moulin and Stong, 2002; Abizada and Chen, 2013). 13
10
Among the rules introduced above, composition down is only satisfied by the equal division rule. In fact, this property is incompatible with a very weak efficiency property, and either one of our fairness properties. Proposition 3. No rule satisfies weak irrelevance of null agents, the equal-division lower bound, and composition down. Proof. Suppose by contradiction that there is a rule ϕ satisfying the three axioms of the proposition. Let N = {1, 2} and R ∈ R N be such that p(R1 ) = 0 and p(R2 ) = 2. Let M = 2 and M ′ = 1. Since p(R1 ) + p(R2 ) = 2 = M, by weak irrelevance of null agents, ϕ1 (R, M) = 0. Thus, ϕ2 (R, M) = 2. Since p(R) = (0, 2) and ϕ(R, M) = (0, 2), we have p(sym(R−ϕ(R,M ) )) = (0, 0). By the equal-division lower bound, ϕ(sym(R−ϕ(R,M ) ), M ′ − M) = ( 21 , 12 ). By composition down, ϕ(R, M ′ ) = ϕ(R, M) − ϕ(sym(R−ϕ(R,M ) ), M ′ − M) = (0, 2) − ( 21 , 21 ) = (− 12 , 32 ). This violates non-negativity of assignments. Proposition 4. No rule satisfies weak irrelevance of null agents, no envy, and composition down. Proof. The proof is similar to that of Proposition 3, so we omit it.
Note that as before, in the proofs of Propositions 3 and 4, stating the incompatibility of very mild efficiency requirement, either one of the fairness properties, and composition down, a difficulty occurs when initially there is just enough of the good and the endowment decreased. Therefore, define our weaker notion of composition down by focusing on the endowment decreases of the following type: when there is too much, we only consider decreases in the endowment such that the direction of the inequality between the sum of the peaks and the endowment doesn’t change. The following property is the second one of our central notions. One-sided15 composition down: For each N ⊆ N , each (R, M) ∈ ΠN , and each M ′ ∈ R+ , X i∈N
p(Ri ) < M ′ < M =⇒ ϕ(R, M ′ ) = ϕ(R, M) − ϕ(sym(R−ϕ(R,M ) ), M − M ′ ).
As it is with one-sided composition up, the term “one-sidedness” is consistent with the terminology in Thomson (1994a, 1995, 1997). Our property is even weaker since we only consider changes that occur on the excess supply side. 15
11
Our two one-sided versions of the composition properties, seem to be the most natural way of weakening our composition properties in the sense that we impose them only when “desirable” changes in the endowment occur, i.e. when there is not enough and the endowment increases, and when there is too much and the endowment decreases. The resulting properties turn out to be satisfied by all four rules that we introduced.
4
Main Result
Although our robustness properties are weak, when we impose them together with fairness and a very mild efficiency property, we obtain one single rule. This rule is the uniform rule. Note that efficiency itself is not one of our axioms. Yet, the uniform rule is efficient. Kesten (2006), Ching (2010), and Ehlers (2011) also provide characterizations of the uniform rule, without imposing efficiency.
Theorem 1. The uniform rule is the only rule satisfying irrelevance of null agents, the equal-division lower bound, one-sided composition up, and one-sided composition down. Proof. It is easy to verify that the uniform rule satisfies all the axioms. Conversely, let ϕ be a rule satisfying these axioms. Let N ⊆ N , (R, M) ∈ ΠN , and n ≡ |N|. We distinguish two cases: X Case 1. M ∈ [0, p(Ri )]. i∈N
Let N 0 ≡ {i ∈ N : p(Ri ) ≤ 0}, and n0 ≡ |N 0 | Let p1 ≡ min 0 p(Ri ), N 1 ≡ {i ∈ N : p(Ri ) = p1 }, and n1 ≡ |N 1 |. i∈N \N
Subcase 1.1. M ∈ [0, (n − n0 )p1 ]. X X ϕj (RN \N 0 , M) = M. Thus, for each ϕj (R, M) = By irrelevance of null agents, 0
j∈N \N 0
j∈N \N 0
i ∈ N , ϕi (R, M) = 0. M Since By the equal-division lower bound, for each i ∈ N \ N 0 , ϕi (RN \N 0 , M) Ri n−n 0. M M 1 0 ≤ p , for each i ∈ N \ N , ϕi (RN \N 0 , M) ≥ n−n0 . Thus, for each i ∈ N \ N 0 , n−n0 M ϕi (RN \N 0 , M) = n−n 0 . By irrelevance of null agents, ϕN \N 0 (R, M) = ϕ(RN \N 0 , M). Thus, M for each i ∈ N \ N 0 , ϕi (R, M) = n−n 0.
12
Let p2 ≡
min
i∈N \(N 0 ∪N 1 )
p(Ri ), N 2 ≡ {i ∈ N : p(Ri ) = p2 }, and n2 ≡ |N 2 |.
Subcase 1.2. M ∈ ((n − n0 )p1 , n1 p1 + (n − n0 − n1 )p2 ]. ′ Let M ′ ≡ (n − n0 )p1 . By one-sided composition up, ϕ(R, M) = ϕ(R, M ′ ) + ϕ(R−ϕ(R,M ) , M − M ′ ). By Subcase 1.1, for each i ∈ N 0 , ϕi (R, M ′ ) = 0, and for each i ∈ N 1 , ϕi (R, M ′ ) = −ϕ (R,M ′ ) M′ = p1 . Since for each i ∈ N 0 ∪ N 1 , p(Ri i ) = 0, by irrelevance of null agents, n−n0X X −ϕ(R,M ′ ) ′ −ϕ(R,M ′ ) ϕj (R , M −M ) = ϕj (R |N \(N 0 ∪N 1 ) , M −M ′ ) = M −M ′ .
j∈N \(N 0 ∪N 1 )
j∈N \(N 0 ∪N 1 ) −ϕ(R,M ′ )
Thus, for each i ∈ N ∪ N , ϕi (R , M − M ′ ) = 0. Thus, for each i ∈ N 0 , ϕi (R, M) = 0 + 0 = 0, and for each i ∈ N 1 , ϕi (R, M) = p1 + 0 = p1 . By Subcase 1.1, for each i ∈ N \ (N 0 ∪ N 1 ), ϕi (R, M ′ ) = p1 . By the equal-division lower ′ M −M ′ bound, for each i ∈ N \ (N 0 ∪ N 1 ), ϕi (R−ϕ(R,M ) |N \(N 0 ∪N 1 ) , M − M ′ ) Ri n−n Since 0 −n1 . M −M ′ M −M ′ 2 1 0 1 −ϕ(R,M ′ ) ′ ≤ p − p , for each i ∈ N \ (N ∪ N ), ϕi (R |N \(N 0 ∪N 1 ) , M − M ) ≥ n−n0 −n1 . n−n0 −n1 M −M ′ 0 1 −ϕ(R,M ′ ) Thus, for each i ∈ N \ (N ∪ N ), ϕi (R |N \(N 0 ∪N 1 ) , M − M ′ ) = n−n 0 −n1 . By irrele′ ′ vance of null agents, ϕN \(N 0 ∪ N 1 ) (R−ϕ(R,M ) , M − M ′ ) = ϕ(R−ϕ(R,M ) |N \(N 0 ∪ N 1 ) , M − M ′ ). ′ M −M ′ Thus, for each Thus, for each i ∈ N \ (N 0 ∪ N 1 ), ϕi (R−ϕ(R,M ) , M − M ′ ) = n−n 0 −n1 . (n−n0 −n1 )p1 +M −(n−n0 )p1 M −n1 p1 M −M ′ 0 1 1 = n−n0 −n1 . i ∈ N \ (N ∪ N ), ϕi (R, M) = p + n−n0 −n1 = n−n0 −n1 Let p3 ≡
0
min 0 1
i∈N \(N ∪N
∪N 2 )
1
p(Ri ), N 3 ≡ {i ∈ N : p(Ri ) = p3 }, and n3 ≡ |N 3 |.
Subcase 1.3. M ∈ (n1 p1 + (n − n0 − n1 )p2 , n1 p1 + n2 p2 + (n − n0 − n1 − n2 )p3 ].
··· P Repeating the argument, we obtain that if M ∈ [0, i∈N p(Ri )], then ϕ(R, M) = U(R, M).
Case 2. M ∈ (
X i∈N
p(Ri ), ∞).
Let pk ≡ max p(Ri ), N k ≡ {i ∈ N : p(Ri ) = pk }, and nk ≡ |N k |. i∈N
Subcase 2.1. M ∈ [npk , ∞). By the equal-division lower bound, for each i ∈ N, ϕi (R, M) Ri M . Since M ≥ pk = n n X . Since ϕi (R, M) = M, for each i ∈ N, max p(Ri ), for each i ∈ N, ϕi (R, M) ≤ M n i∈N
ϕi (R, M) =
i∈N
M . n
Subcase 2.2. M ∈ (
X
p(Ri ), npk ).
i∈N
13
′
Let M ′ = npk . By one-sided composition down, ϕ(R, M) = ϕ(R, M ′ ) − ϕ(sym(R−ϕ(R,M ) ), M ′ − M). ¯ ≡ sym(R−ϕ(R,M ′ ) ). Then, for By Subcase 2.1, for each i ∈ N, ϕi (R, M ′ ) = pk . Let R X each i ∈ N, p(R¯i ) = −[p(Ri ) − ϕi (R, M ′ )] = pk − p(Ri ). Since p(Ri ) < M, we have X i∈N
p(R¯i ) =
X i∈N
p(R¯i ) = npk −
X i∈N
k
p(Ri ) ≥ np −
X i∈N
i∈N
′
p(Ri ) > M − M.
¯ M ′ − M) = min{p(R¯i ), λ)}, where λ solves By Case 1, for each i ∈ N, ϕi (R,
X
¯ ϕj (R,
j∈N
M ′ − M) = M ′ − M. Since for each i ∈ N, p(R¯i ) = pk − p(Ri ), we have min{p(R¯i ), λ)} = min{pk − p(Ri ), λ} = pk − max{p(Ri ), pk − λ}. Thus, for each i ∈ N, ϕi (R, M) = pk − [pk − X max{p(Ri ), pk − λ}] = max{p(Ri ), pk − λ}. Let λ′ ≡ pk − λ. Since min{p(R¯j ), λ)} = X
X
j∈N
M − M, we have max{p(Rj ), λ } = [p − min{p(R¯j , λ)}] = npk − (M ′ − M) = M. j∈N j∈N P Thus, for each i ∈ N, ϕi (R, M) = max{p(Ri ), λ′ }, where λ′ solves j∈N ϕj (R, M) = M. X Thus, if M ∈ ( p(Ri ), npk ), then ϕ(R, M) = U(R, M). ′
′
k
i∈N
• On the independence of the axioms in Theorem 1 (1) The equal division rule satisfies all the axioms of Theorem 1 except for irrelevance of null agents. (2) The proportional rule satisfies all the axioms of Theorem 1 except for the equal-division lower bound. (3) Consider the “mixed” rule16 defined as follows: for each N ⊆ N , each (R, M) ∈ ΠN , and each i ∈ N, P U (R, M) if i j∈N p(Rj ) ≥ M, Mixi (R, M) = EDivi (R, M) otherwise. The mixed rule satisfies all the axioms of Theorem 1 except for one-sided composition down. (4) Consider the “reverse mixed” rule defined as follows: for each N ⊆ N , each (R, M) ∈ ΠN , 16
A similar rule where EDiv is replaced with EDist is defined in Herrero and Villar (1998).
14
and each i ∈ N,
RMixi (R, M) =
Ui (R, M)
M |{j∈N :p(Rj )>0}|
0
if
X j∈N
if
X
p(Rj ) ≤ M, p(Rj ) > M and p(Ri ) > 0,
j∈N
if
X
p(Rj ) > M and p(Ri ) = 0,
j∈N
The reverse mixed rule satisfies all the axioms of Theorem 1 except for one-sided composition up. One natural question is whether the uniform rule is still the only rule left when the equaldivision lower bound in Theorem 1 is replaced by another appealing fairness property, no envy. The answer is no. Consider the “modified uniform”17 rule defined as follows: for each N ⊆ N and each (R, M) ∈ ΠN , let pk ≡ maxi∈N p(Ri ), N k ≡ {i ∈ N : p(Ri ) = pk }, and pk−1 ≡ maxi∈N \N k p(Ri ). For each i ∈ N, P P M − j∈N p(Rj ) k k k−1 p(R if , i) + j∈N p(Rj ) < M, i ∈ N , for each j ∈ N , ∞ Pj p |N k | P ModUi (R, M) = p(Ri ) if / N k , for each j ∈ N k , ∞ Pj pk−1 , j∈N p(Rj ) < M, i ∈ Ui (R, M) otherwise. The modified uniform rule satisfies efficiency, irrelevance of null agents, no envy, onesided composition up, and one-sided composition down. Yet, it differs from the uniform rule for a few special cases.
Although when we impose no envy instead of the equal-division lower bound, there is no parallel result of Theorem 1, we do have another characterization of the uniform rule, mainly based on no envy and the two robustness properties. As we know, the uniform rule is the only rule satisfying efficiency, no envy and peak only (Thomson, 1994a). If we weaken efficiency to its very weak form, weak irrelevance of null agents, and impose our two one-sided composition properties in addition to no envy and peak only, we again are left with the uniform rule as the only one satisfying these properties. Comparing to Theorem 1, we find that this result is less appealing, as we replace the equal-division lower bound by no envy, weaken irrelevance of null agents, yet add a strong property, peak only, which 17
We call this rule “modified uniform” because it is the same as the uniform rule except for very few cases.
15
requires the allocation to depend only on the peaks of preferences. Therefore, we list this characterization as a remark, rather than a theorem. Note that the properties involved in the characterization are independent.18
Remark 1. The uniform rule is the only rule satisfying weak irrelevance of null agents, no envy, peak only, one-sided composition up, and one-sided composition down.
References Abizada, A. and Chen, S. (2013), “House allocation when construction schedule is unpredictable”, mimeo. B´enassy, J.P. (1982), The Economics of market disequilibrium, Academic Press, San Diego. Ching, S. (1992), “A simple characterization of the uniform rule”, Economics Letters 40, 57–60. Ching, S. (1994), “An alternative characterization of the uniform rule”, Social Choice and Welfare 11, 131–136. Ching, S. (2010), “An axiomatization of the uniform rule without the Pareto principle”, University of Hong Kong, mimeo. Ehlers, L. (2011), “A characterization of the uniform rule without Pareto-optimality”, Journal of the Spanish Economic Association 2, 447–452. Foley, D., (1967), “Resource allocation and the public sector”, Yale Economics Essays 7, 45–98. Herrero, C. and Villar A. (1998), “Agenda independence in allocation problems with singlepeaked preferences”, mimeo. Herrero, C. and Villar A. (2000), “An alternative characterization of the equal-distance rule for allocation problems with single-peaked preferences”, Economics Letters 66, 311–317. Kalai, E. (1977), “Proportional solutions to bargaining situations: interpersonal utility comparisons”, Econometrica 45, 1623–1630. Kesten, O. (2006), “More on the uniformrule: Characterizations without Pareto optimality”, Mathematical Social Sciences 51, 192–200. 18
The proof and the independence of properties are available upon request.
16
Maniquet, F. (1996), “Allocation rules for a commonly owned technology: the average cost lower bound”, Journal of Economic Theory, 69, 490–507. Moreno-Ternero J. and J. Roemer (2012), “A common ground for resource and welfare egalitarianism”, Games and Economic Behavior, 75, 832–841. Moulin, H. (1987), “Equal or proportional division of a surplus, and other mehods”, International Journal of Game Theory, 16, 161–186. Moulin, H. (1991), “Welfare bounds and other asymmetric rationing methods”, Journal of Economic Theory, 54, 321–337. Moulin, H. (2000), “Priority rules and other asymmetric rationing methods”, Econometrica, 68, 643–684. Moulin, H. and R. Stong (2002), “Fair queuing and other probabilistic allocation methods”, Mathematics of Operations Research, 27, 1–30. Plott, C. R. (1973), “Path independence, rationality and social choice,” Econometrica 41, 1075-1091. S¨onmez, T. (1994), “Consistency, Monotonicity, and the Uniform Rule. Economics Letters 46, 229–235. Sprumont, Y. (1991), “The division problem with single-peaked preferences: a characterization of the uniform allocation rule”, Econometrica 59, 509-519. Thomson, W. (1994a), “Resource-monotonic solutions to the problem of fair division when preferences are single-peaked”, Social Choice and Welfare 11, 205-223. Thomson, W. (1994b), “Consistent solutions to the problem of fair division when preferences are single-peaked”, Journal of Economic Theory 63, 219-245. Thomson, W. (1995), “Population-monotonic solutions to the problem of fair division when preferences are single-peaked”, Economic Theory 5, 229-246. Thomson, W. (1997), “The replacement principle in economies with single-peaked preferences”, Journal of Economic Theory 76, 145-168. Thomson, W. (2003), “Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey”, Mathematical Social Sciences 45, 249-297. Thomson, W. (2011), “Fair allocation rules”, K. J. Arrow, A. K. Sen, and K. Suzumura (Eds.) Handbook of Social Choice and Welfare, Volume 2, Elsevier, 393-506. Young, H.P. (1988), “Distributive justice in taxation”, Journal of Economic Theory 44, 321-335. 17
• • •
We study allocation of a divisible good when preferences are single-peaked. We formulate two new robustness requirements. We characterize uniform rule using our robustness requirements.