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Games and Economic Behavior
www.elsevier.com/locate/geb
Priorities in the location of multiple public facilities
✩
Olivier Bochet
a,b,c,∗
, Sidartha Gordon
d,caUniversity of Bern, Switzerland bMaastricht University, Netherlands cCIREQ, Canada
dUniversité de Montréal, Canada
a r t i c l e
i n f o
a b s t r a c t
Article history:
Received 8 October 2009 Available online 21 June 2011 JEL classification:
D60 D63 D70 D71 H41
Keywords:
Multiple public facilities Priority rules
Hierarchical rules
Object-population monotonicity Sovereignty
Strategy-proofness
Generalized median voter rules No-show paradox
A collective decision problem is described by a set of agents, a profile of single-peaked preferences over the real line and a number of public facilities to be located. We consider public facilities that do not suffer from congestion and are non-excludable. We characterize the class of rules satisfyingPareto-efficiency,object-population monotonicityandsovereignty. Each rule in the class is apriority rulethat selects locations according to a predetermined priority ordering among “interest groups”. We characterize the subclasses of priority rules that respectively satisfyanonymity,avoid the no-show paradox,strategy-proofnessand population-monotonicity. In particular, we prove that a priority rule isstrategy-proof if and only if it partitions the set of agents into a fixed hierarchy. Any such rule can also be viewed as a collection of generalized peak-selection median rules, that are linked across populations, in a way that we describe.
2011 Elsevier Inc. All rights reserved.
1. Introduction
We consider a generalization of the unidimensional voting model studied by Black (1948) and Moulin (1980). A collective decision problem is described by a set of agents, a profile of single-peaked preferences over the real line, and a numberk of public facilities to be located – e.g. public libraries. Each public facility is non-excludable and does not exhibit congestion. Each agent has single-peaked preferences over locations, and compares finite sets of locations by comparing only their best element. We search for a rule that decides on how to locate the facilities, in any possible collective decision problem. We are interested in a set of desirable properties that a rule should satisfy. In addition toPareto-efficiency(henceforthefficiency),
✩ We thank Anna Bogomolnaia, Lars Ehlers, Bettina Klaus, Matthew Jackson, François Maniquet, Hervé Moulin, Jorge L. Garcia Ramirez, Jim Schummer, Yves Sprumont, William Thomson and two anonymous referees for helpful comments. Part of this paper was written when Olivier Bochet visited Université de Montréal, and when Sidartha Gordon visited Maastricht University and the University of Bern. Financial support from CIREQ and METEOR is gratefully acknowledged. Olivier Bochet thanks, respectively, the Netherlands Organisation for Scientific Research (NWO) and the Swiss Science Foundation (SNF) for their support under grant VENI-451-07-021 and grant 100014-126954. Sidartha Gordon thanks the FQRSC (Québec) for financial support.
*
Corresponding author at: University of Bern, Switzerland. E-mail address:[email protected](O. Bochet).we search for rules that satisfy a new property, object-population monotonicity, that we introduce in this paper, as well as sovereignty(see for instance Barberà et al., 1991).1
“Public facilities” should be interpreted in a broad sense. They could be actual facilities, such as wireless towers, to be located in space. They could be varieties of software to be provided to a community of users. In this example, the “locations” represent different program varieties. They could also be the official languages for a linguistically diverse society. Finally, they could be telecom repeaters.2To fix ideas, let us use the latter example to explain the properties we are interested in.
Consider a given community which has to decide where to locate telecom repeaters to ensure mobile phone receptions – where people actually use mobile phones the most. Assume that the community has, for exogenous reasons,krepeaters to locate. How should the location of thekrepeaters be decided? In addition, how should (if at all) the repeaters be relocated if the community expands and/or if new repeaters are available? We would like to have a method for selecting the location of telecom repeaters that respects a few desirable properties.
First, we would like the chosen set of locations to be efficient. That is, there should exist no other set of locations of cardinalitykwhich all members of the community weakly prefer and at least one member strictly prefers, to the location set selected by the rule.
Second, we would like our rules to treat agents in a fair way when both the social endowment, i.e. the number of facilities, and the population vary. Simultaneous variations of this type have ambiguous effects on the opportunity set of the first-comers. On the one hand, the addition of new facilities increases this set. On the other hand, the arrival of new agents who may have rights over the new and old facilities tends to reduce it. We are interested in situations where the first effect clearly dominates the second. Namely, if the new resources are sufficient to saturate the preferences of the newcomers, the opportunity set of the first-comers is enlarged. Our property requires that this enlargement be reflected in the welfare variation of all first-comers. Underobject-population monotonicity, when such a change in social endowment and population occurs, all first-comers should weakly benefit from the change.
Third, sovereignty requires in the problem of locating a single facility, that any location choice x initially made for a society, could be overruled in favor of any other location y provided that some appropriately (possibly large) group of newcomer whose common preferred location is yis added.3
On the one hand, in the situation of a population and resource increase, the second property protects the rights of the first-comers. On the other hand, in the situation of a population increase, the third property protects the rights of the newcomers. A potential conflict between the two properties is avoided, because the second property has bite only when the new resources alone suffice to fully satisfy the newcomers. The possibility that newcomers have rights over the old and new facilities is thus left open. Alternatively, the second property can be seen as a veto power of the first-comers on location changes, but this power is limited to situations where the newcomers can be fully satisfied without hurting the first-comers. As a result, the two principles can be conciliated.
We provide a complete characterization of the class of rules that jointly satisfy these three properties (Theorem 1). These form a restricted class of priority rules. Apriority ruleoperates as follows. First, it partitions the community into unanimous groups, i.e. groups of citizens who share the same preferred location. Second, it ranks these unanimous groups according to some predetermined priority order that can depend on the identities of their members and their full preferences over all locations. The rule then selects the preferred location of the top k groups in the priority order. Priority rules form a rich set. In particular, it includes an interesting subclass ofmajoritarian rulesthat rank unanimous groups according to their cardinality (Bochet et al., 2010).
In addition we provide additional characterizations (Theorem 2, Propositions 1, 2, and 3) with requiring in addition either that a rule (i) avoid theno-show paradox(Moulin, 1988) – an agent should not gain by withdrawing from the vote – (ii) be anonymous – i.e. the names of agents do not matter, (iii) bestrategy-proof – immune to the manipulation of preferences (iv) satisfy population-monotonicity – if new agents arrive initial resources population are kept fixed, all agents initially present should weakly lose. Unfortunately, (v) no priority rule selects the median voter (Black, 1948) when a single location needs to be selected and the number of agents is odd.
The priority rules that are strategy-proof form an interesting subclass ofhierarchical rules (Theorem 2). Each such rule partitions the population into a fixed hierarchy of priority levels. Alternatively, a hierarchical rule can also be described as a restricted and linked collection ofgeneralized peak-selecting median rules(Moulin, 1980). We provide a complete description of the linkage across populations imposed by our properties on such a collection.
The problem of locating a single facility is well studied in the voting literature. Moulin (1980), Ching (1997), Barberà and Jackson (1994) among others, have studied the strategic properties of rules for locating a single facility. Other scholars study rules that satisfy normative properties. The principle of solidarity says that when circumstances change, all agents not
1 The term “axiom” is also often used in place of “property”. We use both in the paper. This should cause no confusion.
2 A repeater is an electronic device that receives a signal and retransmits it at a higher level and/or higher power, or onto the other side of an obstruction, so that the signal can cover longer distances. (Source, wikipedia.) We thank an anonymous referee for suggesting this example.
responsible for the change should be affected in the same direction.4Thomson (1993), Ching and Thomson (forthcoming), Vohra (1999) and Klaus (2001) investigate the two main formulations of this principle in this context.5
The problem of locating two facilities is studied first by Miyagawa (1998, 2001). Miyagawa extends preferences over single locations to finite sets of locations according to the “max-extension”. This extension compares finite sets of loca-tions by comparing only their best element. Ehlers (2002, 2003) also studies this problem, but extends preferences in a “lexicographic” manner. This extension compares finite sets of locations by comparing their best element first. When a tie between the best elements occurs, it compares their second best element, then the third and so on. Both Miyagawa (1998, 2001) and Ehlers (2002, 2003) follow the normative route. They look for rules that satisfy eitherreplacement-domination orpopulation-monotonicity. Bogomolnaia and Nicolò (2005) introduce congestion effects in the two facilities problem.6One important difference between our work and the papers mentioned in the last two paragraphs, is that they study a fixed number of facilities, while we allow this parameter to vary. We formulate a property,object-population monotonicity, that links together the problems of locatingkfacilities whenkvaries and characterize rules that locate any number of facilities. In this respect, our work is similar to Barberà and Beviá (2002, 2005) and Ju (2008). However, they focus onconsistency, a property which says that when onek−1 facility and its “users” are simultaneously removed, the remaining location should remain unchanged. These authors prove that there is a large class of rules for locating one facility that can be extended to rules for locatingkfacilities in a consistent manner, and provide an algorithm to compute the latter from the former. None of the properties we consider in this paper is related toconsistency, but all the rules we characterize are indeed consistent, as a consequence of the properties we use. Thus the rules we characterize here form a subclass of theirs.
Our main contribution is the introduction of theobject-population monotonicityproperty,which has not been studied in any context, and the description of the structure of the rules that satisfy it, together with other requirements, in the context of the location of multiple public goods. These rules have a simple parametric structure. The structure, in turn, facilitates further analysis: It is relatively easy to characterize the set of priority rules that satisfy an additional property. The new property is meaningful in the context of the location of public facilities, but it is also interesting on its own, and could be further investigated in other contexts.
The plan of the paper is as follows. In Section 2, we present the model. In Section 3, we present our main properties. In Section 4, we introduce priorities and priority rules and provide examples. In Section 5, we present our main characteriza-tion and verify that our properties are independent. In Seccharacteriza-tion 6, we study subclasses of priority rules that satisfy addicharacteriza-tional properties. Finally, we offer some concluding remarks in Section 7.
2. The model
There is a countably infinite set N of potential agents. A population N is a finite and non-empty subset of N. The population is collectively endowed with a number k of identical public facilities, each to be located on the real line R. A typical location onR is denoted by x. An assignmentis a menu of locations, i.e. a finite subset X⊂R. A k-assignment is an assignment of exactlyk facilities, i.e. a subset X⊂R such that|X| =k. Let
X
k be the class of all k-assignments. In particular, a 1-assignment is a single locationx∈R, so thatX
1=R. LetX
≡!k!1X
k be the class of all assignments.Apreferenceover
X
is a weak ordering overX
. Each agenti∈Nhas a preference Ri overX
. For each preference Ri, letPi andIistand for the strict ordering and the indifference relation associated with Ri, respectively. We restrict attention to the class
R
ofsingle-peakedpreferences overX
, defined by the following two conditions. The first condition is the common single-peakedness notion, for preferences over single locations on the real line. The second condition extends preferences from single locations to menus.7A preferenceRi is single-peaked if the following holds:
(i) There is a location p(Ri), such that for all x,y∈R satisfying either x<y!p(Ri)or p(Ri)"y>x, we have y Pi x. The location p(Ri)is called thepeakof preference Ri.
(ii) For all X,Y∈
X
, we let X Ri Y if there isx∈X such that for all y∈Y, we havex Ri y.For each population N, a preference profile for N is a list RN=(Ri)i∈N∈
R
N. More generally, a preference profile is a preference profile for some populationN.8For each profileRN and each subpopulation K⊆N, let RK denote the subprofile (Ri)i∈K.For each profileRN∈R
N, letp(RN)be the set ofpeak locationsfor RN, i.e. p(RN)≡{p(Ri): i∈N}.For eachk>0,4 Replacement-domination(Moulin, 1987) operates on preference profiles for a fixed population. It says that when the preferences of one agent change, all the other agents, whose preferences are kept fixed, should be affected in the same direction.Population-monotonicity(Thomson, 1983a, 1983b) operates on preference profiles for a variable population. It says that when new agents join the economy, all the agents who were initially present should all be affected in the same direction. For a survey on these properties, see Thomson (1995, 1999a, 1999b).
5 Gordon (2007b) studies the solidarity principle in the location of a public facility on a cycle. Gordon (2007a) generalizes this literature on the solidarity principle to a more abstract non-geometric setting that contains location problems of single and multiple facilities as special cases.
6 Jackson and Nicolò (2004) also study location problems with congestion, but restrict attention to the single facility case.
7 There are different ways to extend preferences over points to preferences over sets. Consistent with the definition of a public facility used in this paper, we consider the max-extension of preferences used by Miyagawa (2001).
let
P
k be the set of preference profilesRN with a number of distinct peak locations greater than or equal tok, i.e. such thatk!|p(RN)|.A problem is a pair(k,RN)such thatkis a positive integer, andRN∈
P
k.9For eachk"1, a k-ruleis a mapping fk:
P
k→X
k. A ruleis a sequence f = {f1,f2, . . .} of k-rules. For each problem(k,RN), the rule f prescribes an assignment in
X
k.10 For eachk"1, the set of allk-rules isX
kPk.Therefore, the set of all rules is!k∞=1X
kPk.3. Main properties
Our first property is the usual efficiencyrequirement. For each RN∈
R
N, and each k-assignments X,Y, we say that X weakly Pareto-dominatesY for profile RN if X Ri Y for eachi∈N. This is denoted by X RN Y.A rule f satisfiesefficiencyif, for each problem(k,RN), there is nok-assignment X such that X RN fk(RN), and X Pj
fk(RN)for some j∈N.
Next, we would like our rules to treat agents in a fair way when both the social endowment and the population vary. The principle of solidarity says that when such changes lead to either an enlargement or a reduction in the opportunity set of the first-comers (in utility space), since none of these agents is responsible for the change, the effect of this variation on their welfare should have the same direction. If their opportunity set grows, they all should weakly gain. If it is reduced, they all should weakly lose. An increase in the social endowment alone enlarges the first-comers’ opportunity set, and thus all first-comers should weakly gain from it. When new agents, who may have rights on the facilities, enter the economy, while the social endowment is kept fixed, this reduces the first-comers’ opportunity set, and thus all first-comers should weakly lose from this change.11
When both new resources and new agents are brought into the economy, both effects are at work and the overall effect of the change on the first-comers’ opportunity set is in general ambiguous. However, there are cases where the positive effect clearly dominates the negative one. Namely, if the new resources are sufficient to saturate the preferences of the newcomers, the first-comers’ opportunity set grows. As a result, the solidarity principle requires that when such a change occurs, all first-comers should weakly benefit from it.
In the context of our model, this idea translates as follows. If new agents and new facilities to be located are brought into the economy, and the number of new facilities is at least as large as the number of distinct peaks in the preference profile of the newcomers, all first-comers should weakly benefit from this change. We formulate a streamlined and weaker version of this requirement, that only applies to changes where all newcomers have the same peak and exactly one facility is brought.
A rule f satisfies object-population monotonicity if, for each problem (k,RN) with k<|p(RN)|, for each peak-unanimous profile RM∈
T
such thatN∩M=∅, we have fk+1(RN,RM) RN fk(RN).A stronger version of this property that would require that the first-comers weakly gain whenever the number of facil-ities is greater than or equal to the number of distinct peaks in the preference profile of the newcomers may seem more natural, but within the set of rules that satisfy our other two main properties, the weaker and the stronger version of the property turn out to be equivalent.
Our third property, sovereignty, only restricts the one-facility component f1 of a rule, as the population varies. It says that the choice ymade by the rule f1 for a given profileRN can be reversed in favor of any given locationxprovided that some appropriately selected peak-unanimous profile RM is brought into the economy in support of x. In fact, our version of the property is stronger, as it requires that there be infinitely many such peak-unanimous profiles RM. We comment on this below.
A rule f satisfiessovereigntyif, for each profile RN, each locationx∈R\f1(RN), and each population L, there exists a peak-unanimous profile RM∈
T
such thatM is disjoint from bothLandN, that satisfies f1(RN,RM)= {x} =p(RM).Notice the requirement in the definition of sovereignty that M be disjoint from both L and N. In other words, there exists an infinity of groups M disjoint fromN such that f1(RN,RM)=p(RM). This introduces a (weak) idea of anonymity
in the limits imposed to the power of agents in N: the ability to counter decisions taken for N is not the privilege of a specific unanimous group M.
9 The restrictionk!|p(R
N)|allows us to focus on non-trivial cases. Whenk>|p(RN)|, it is possible to locate one facility at each peak location, so that the welfare of each agent is maximized. Locating the remaining facilities does not affect any agent’s welfare.
10 Our definitions rule out locating more than one facility at the same point. Under single-peaked preferences, and for the class of problems we consider,
Pareto-efficiency would exclude duplication anyway.
11 The implications for our model of the conditions that consider changes in resources alone and changes in population alone are discussed in Sections 5.2
4. Priority rules
4.1. Definition
Let us introduce a class
F
of rules which will play an important role in our results. A profile RM ispeak-unanimousif allthe preferences of this profile have the same peak, i.e. p(RM) is a singleton. Let
T
be the set of peak-unanimous profiles.For any two peak-unanimous profiles RL and R!M, we say that RL and R!M arenon-overlapping if they have distinct peaks
and disjoint populations, i.e. p(RL)"=p(R!M)andL∩M=∅.
We now introduce the class of priorities over some non-empty subset
S
ofT
. Let % be a binary relation overS
. The binary relation % is almost complete if for all RL,RM ∈S
, we have (RL% RM or RM % RL) if and onlyif (RLandRM are non-overlapping).12 It is almost transitive if for all RK,RL,RM ∈
S
, such that RK and RM arenon-overlapping, (RK%RL andRL%RM) implies that RK%RM. The binary relation % is apriorityover
S
if it is asymmetric,almost transitive and almost complete.13 For each non-empty
S
⊆T
, letPS be the set of priorities overS
.For each profileRN, the peak-unanimous subprofileRM of RNismaximalifp(RM)∩p(RN\M)=∅.Any two distinct
max-imal peak-unanimous subprofiles are non-overlapping. It follows that the collection of maxmax-imal peak-unanimous subprofiles of some profile is strictly ordered by any priority (see footnote 13). We are now ready to define the family of priority rules, parametrized by the setPT.For each % ∈PT, the priority rule f associated with % is defined as follows. Let (k,RN) be
an arbitrary problem. Then the priority% strictly ranks the maximal peak-unanimous subprofiles in the decomposition of RN and fk(RN) selects the peak locations of the topkmaximal peak-unanimous subprofiles for%.In more precise terms, fk(RN)is thek-assignment such that fk(RN)⊆p(RN), and for all two maximal peak-unanimous subprofiles RM and RL in RN, ifp(RM)⊆ fk(RN)and p(RL)!fk(RN), then RM%RL.Let
F
be the set of priority rules.4.2. Examples
We now provide a few examples that illustrate the size and diversity of the set of priorities. In each of the examples, the priority agrees with some primary criterion (a weak ordering) according to which all unanimous profiles are ranked. In the first two examples, this criterion fully defines the priority. In the other examples, the primary criterion is not always decisive and a tie-breaking priority defines the priority within each “indifference class” of the primary criterion. On each such class the tie-breaking priority can be, for example, one of the priorities in Example 1 or 2 and can vary from one class to the other.
Example 1(Location priority).The left-peaks priority % is such that, for all non-overlappingRL, RM∈
T
, we have RL%RMif and only if p(RL) <p(RM). Theleft-peaks rule f is the priority rule defined by %. By analogy, theright-peaks priority %
and the the right-peaks rule f are similarly defined, by replacing “<” with “>”, in the above definition.14 More generally,
an arbitrary priority based on location, and the corresponding priority rule are similarly obtained, by replacing “<” with an
arbitrary strict ordering over locations inR.
Example 2(Group priority).A priority%is agroup priorityif its choice among two unanimous groups only depends on their respective populations. More precisely, for all disjoint population L and M, all RL, R!L, RM, R!M, such that RL and RM are
non-overlapping and R!L and R!M are non-overlapping, we have RL%RM⇔R!L%R!M. In particular, a priority% is a serial dictatorshipif there exists a strict ordering !of all agents in N such that, for all non-overlapping RL, RM ∈
T
, we have RL%RM if and only if there exists i∈L,such that for all j∈M, we havei!j.For example, the strict ordering could besuch that for alli,j∈N, we have i!jif and only ifi>j.
Example 3(Hierarchical priority). A priority % is hierarchicalif the following holds: (i) There is a weak ordering ! of all agents inN, such that, for all RL, RM∈
T
, if there existsi∈L, such that for all j∈M, we have(i! jand not j!i), then RL%RM.(ii) For each!-indifference class K, consider the classK
(K)⊆T
of unanimous profiles such that the agents in M who are ranked highest for!belong to K.On each such classK
(K), the priority%coincides with either the left-peaksor the right-peaks priority.15If each!-indifference class is a singleton, the priority is aserial dictatorship. If there is a single
!-indifference class, the priority is either the left-peaks or the right-peaks priority.
12 In particular, an almost complete binary relation%overT is never reflexive.
13 A priority%is not a partial order, as it is not fully transitive. However, priorities have the following important property. The restriction of a priority% on any setSof peak-unanimous and non-overlapping profiles is a strict ordering. If this set is finite, the priority%has a greatest (or top) element inS. A priority%may have a finite cycle of peak-unanimous profiles. If this is the case, some of them must overlap.
14 Miyagawa (2001) showed that when|N|!4 andk=2, the only mappingsRN→X2satisfyingefficiencyandreplacement-dominationare the left-peaks rule and the right-peaks rule.
In each of the remaining examples, the primary criterion does not fully specify the priority, like the weak ordering!
in Example 3. It partitions the set of unanimous profiles in “indifference classes” within which the priority coincides with some arbitrary tie-breaking priority (such as the left-peaks or right-peaks priorities in Example 3, or some group priority).
Example 4 (Numeral priority).A priority is numeralif there is a strict ordering !over N such that for all RL, RM ∈
T
, we have|L|!|M|⇒RL#RM.In particular, the priority ismajoritarianif the strict ordering is “>” andminoritarianif it is “<”.Example 5(Numeral and location priority).A priority#isx0-centralist-majoritarianif there is a locationx0∈R(the “center”),
a distance D(x,y)onR and an indexu:{1,2, . . .}×R+→R, whereu(n,d)is increasing16 inn and decreasing ind, such
that, for all RL,RM∈
T
, we haveu
!
|
L|
,
D!
p(
RL),
x0""
>
u!
|
M|
,
D!
p(
RM),
x0""
&⇒
RL#
RM.
Example 6(Priority based on indifference curves).Let # be defined as follows. Letx0∈R be a location and for each interval
A⊆R, let
ρ
(A)= |sup(A)−inf(A)|.Then, for all RL,RM∈T
,ρ
# $
i∈M
{
x∈
R
: x Rix0}
%
<
ρ
# $
i∈L
{
x∈
R
: x Rix0}
%
&⇒
RL#
RM.
4.3. Object-population monotonic and sovereign priority rules
We now identify which priority rules satisfy the properties listed in Section 3. First, it is clear that all priority rules are efficient, since they always locate the facilities on distinct peaks. Second, it is also clear that they all satisfy object-monotonicity, since they satisfy the stronger condition that for each problem (k,RN) such that k<|p(RN)|, we have
fk(RN)⊂fk+1(RN).
All rules in Examples 1, 3 and 5 satisfy object-population monotonicity, as well as the majoritarian rules and many group priorities. While many priority rules satisfy the property, not all of them do. For example, any numeral priority rule that is not majoritarian violates the property. Similarly, with some abuse of notations, any group priority rule whose priority is such that there are populationsM,K, H andLsuch that K andH are not overlapping, M#K#L∪M andM#H#L∪M
violates it as well. As we show in Lemma 1, the following is a necessary and sufficient condition on a priority for the rule it defines to be object-population monotonic.
A priority#isalmost monotonicif, there are no four peak-unanimous profiles RM, RK,RH andRL such thatp(RM)=
p(RL), M∩L=∅, RK andRH are non-overlapping, RM#RH#RM∪L, andRM#RK#RM∪L.
Many but not all of the rules in Examples 2, 3, 4, 5 and 6 satisfy sovereignty. In particular, with some abuse of notation, a group priority satisfies it if (i) for all H,K such that H#K, and for any population L, there exists M disjoint from K
andL, such thatK∪M#H.(ii) For eachH, and each population L, there exists a population M disjoint fromL, such that M#H.In particular, any unbounded serial dictatorship, hierarchical or numeral rule satisfies it.17 As we show in Lemma 1,
the following is a necessary and sufficient condition on a priority for the rule it defines to be sovereign.
A priority# issovereign if the following two conditions hold: (i) For all peak-unanimous RH,RK∈
T
such that RH# RK, and for any population L, there exists a peak-unanimous profile RM ∈T
such that M is disjoint from K and L, and satisfies p(RM)=p(RK), and RK∪M #RH. (ii) For each RH ∈T
, each x-=p(RH) and each population L, there exists apeak-unanimous profile RM∈
T
such thatM∩L=∅, and satisfies p(RM)=x, and RM#RH.Lemma 1.(i)A priority rule satisfies sovereignty if and only if its priority#is sovereign.(ii)A priority rule satisfies object-population monotonicity if and only if its priority#is almost monotonic.
Proof. Both of theif implications are clear.Let us prove theonly if implications in these three claims.
Only if implication for (i): Let f satisfies sovereignty and let f be associated with the priority #, let RH and RK be two arbitrary peak-unanimous profiles satisfying RH#RK, and let L be an arbitrary population. Bysovereigntyof f, since f1(RH,RK)-=p(RK), then there exists a peak-unanimous profile RM satisfying M∩(H∪K∪L)=∅, and f1(RH,RK,RM)= p(RM)=p(RK).Since(RK,RM)is then a peak-unanimous profile, we have (RK,RM)#RH. (ii) Next, letRH be an arbitrary peak-unanimous profile, letx-=p(RH), and letLbe an arbitrary population. Bysovereigntyof f, since f1(RH)-=x, then there exists a peak-unanimous profile RM satisfying M∩(H∪L)=∅, and f1(RH,RM)=p(RM). Therefore, we have RM#RH. Therefore# is sovereign.
16 A more general family is obtained ifuis defined as an arbitrary functionu(|N|,p(RN)).
17 A serial dictatorship or a numeral priority#is unbounded if the associated ordering!ofNis such that for eachi∈N, there exists j∈Nwith j!i.
Only ifimplication for (ii): By contradiction, suppose that RM,RK,RH andRLare four peak-unanimous profiles such that
p(RM)=p(RL), M∩L=∅, RK and RH are compatible, RM#RH#RM∪L, and RM#RK #RM∪L. Then either RH#RK or
RK#RH.Suppose, for example, that RH#RK.Then f1(RH,RK,RM)=p(RM) and f2(RH,RK,RM∪L)=p(RH)∪p(RK), in
violation ofobject-population monotonicity. The same conclusion is obtained if we assume instead that RK#RH.Therefore #is almost monotonic. !
5. Main characterization
In this section, we show that efficiency, object-population monotonicity and sovereignty characterize a class of priority rules.
Theorem 1.A rule f satisfies efficiency, object-population monotonicity and sovereignty if and only if it is a priority rule whose priority is almost monotonic and sovereign.
The “if implication” follows from Lemma 1 and the observations in Section 4.3. Before proving the converse, we first present two useful lemmas. The first one states that object-population monotonicity and sovereignty imply the following property.
A rule f satisfies strong sovereignty if for each problem (k,RN), each location x∈R\ fk(RN), and each popula-tion L, there exists a peak-unanimous profile RM such that M is disjoint from both L andN, that satisfies p(RM)= {x}⊆
fk(RN,RM).
Lemma 2.If f satisfies object-population monotonicity and sovereignty, then it satisfies strong sovereignty.
Proof. Let(k,RN)be an arbitrary problem, letx∈Rbe an arbitrary location, and let Lbe an arbitrary population. We will
prove that there exists a peak-unanimous profile RM such that M∩(L∪N)=∅and p(RM)= {x}⊆ fk(RN,RM). If k=1,
thensovereigntyguarantees that this is true. So suppose instead, thatk>1.Let!≡|p(RN)|.We have!!k.
LetRN1, . . . ,RN!−(k−1) be!−(k−1)distinct maximal peak-unanimous subprofiles of RN.Let!N≡N1∪· · ·∪N!−(k−1).We
have|p(R!N)| =!−(k−1). Let!L≡(N\!N)∪L.By sovereignty, there exists a profile RM∈
R
M such that M∩(!L∪!N)=∅,andp(RM)= {x} =f1(R!N,RM).We have|p(RN\!N)| =k−1.Byobject-population monotonicity, appliedk−1 times, we have
in particular fk(R!N,RM,RN\!N) RM f1(R!N,RM). Since f1(RN,RM)= {x} =p(RM), this implies that x∈ fk(R!N,RM,RN\!N),
i.e. p(RM)= {x}⊆ fk(RN,RM). Since M∩(L∪N)=M∩(!L∪!N)=∅, therefore the population M satisfies all the desired
properties. !
The second lemma shows that a combination of some of the properties listed in Theorem 1 implies that each public facility must be located at some agent’s peak location. Before proceeding to the lemma, we first define formally the property
peak-selection.
A rule f satisfiespeak-selectionif for each problem(k,RN), we have fk(RN)⊆p(RN).
Lemma 3.If f satisfies efficiency, object-population monotonicity and sovereignty, then it satisfies peak-selection.
Proof. Let f satisfy the three properties. We prove by induction onk, that each fk satisfiespeak-selection.
Step 1. The rule f1satisfies peak-selection.
Suppose by contradiction that f1 does not satisfypeak-selection. Let RN∈
R
N and let x∈R be such that f1(RN)= {x}and x∈/p(RN). Throughout the proof, for all j∈N, let pj:=p(Rj). By efficiency,there are two agents i,!∈N such that
pi<x<p!, and p(RN) has no element strictly comprised between pi and p!. Let u,v∈R be such that pi<u<x< v<p!. Bystrong sovereignty, there are peak-unanimous profiles RJ and RK satisfying the following conditions. Let RJ be such that J∩N=∅, p(RJ)= {u}, and u∈f2(RN,RJ). Similarly, let RK be such that K∩(N∪ J)=∅, p(RK)= {v}, and v∈ f2(RN,RK). Let A≡ f3(RN,RJ,RK). We will now show that A has at least four elements. First, byobject-population
monotonicity, p(RJ)= {u} and u∈ f2(RN,RJ) imply that u∈A. Similarly, by object-population monotonicity, p(RK)= {v}
andv∈f2(RN,RK)imply that v∈A.
Second, byobject-population monotonicity,since f1(RN)= {x}, then the set f2(RN,RK) has at least one location that is
at least as good asx for Ri.Let y∈f2(RN,RK)such that y Ri x.Efficiencyrequires that y"pi.In particular y<v.Since |f2(RN,RK)| =2, it then follows that f2(RN,RK)= {y,v}. By efficiency,there exists an agent h∈N such that ph"y. By object-population monotonicity, since f2(RN,RK)= {y,v}, then the set Ahas at least one location that is at least as good as yfor Rh.Let y)∈Asuch that y) Rh y. Since ph"y, then in particular y)"y<v.
Third, byobject-population monotonicity,since f1(RN)= {x}, then the set f2(RN,RJ) has at least one location that is at
|f2(RN,RJ)| =2, it then follows that f2(RN,RJ)= {u,z}. By efficiency,there exists an agentm∈N such that z!pm.By
object-population monotonicity,since f2(RN,RJ)= {u,z}, then the set A has at least one location that is at least as good as
xfor Rm.Let z"∈Asuch thatz" Rm z. Sincez!pm, then in particularu<z!z".
In conclusion, y"<u<v<z" are four distinct elements ofA, in contradiction with|A| =3.
Step 2. Let k"1.Suppose that fksatisfies peak-selection. Then fk+1also satisfies peak-selection.
Let RN ∈
R
N. If |p(RN)| =k+1, then by efficiency, fk+1(RN)=p(RN), so the claim is true. Suppose then that|p(RN)|>k+1. We will first show that fk+1(RN)∩p(RN)$=∅. Since|p(RN)|>k+1, then p(RN)! fk+1(RN). Let RM
be a maximal peak-unanimous subprofile of RN, withM⊂N, such that p(RM) /∈fk+1(RN). Consider the profile RN\M.We
have |p(RN\M)|>k. From the induction hypothesis, we have fk(RN\M)⊆p(RN\M). By object-population monotonicity, we
have fk(RN\M)⊆fk+1(RN).Thus fk(RN\M)⊆ fk+1(RN)∩p(RN).Therefore, fk+1(RN)∩p(RN)$=∅.
Let x∈ fk+1(RN)∩p(RN). Let RL be the maximal peak-unanimous subprofile of RN such that p(RL)= {x}. Then
|p(RN\L)|>kandx∈/p(RN\L).From the induction hypothesis, we have fk(RN\L)⊆p(RN\L). Byobject-population
monotonic-ity, we have fk(RN\L)⊆fk+1(RN).Sincex∈fk+1(RN), then fk(RN\L)∪{x}⊆fk+1(RN).Sincex∈/p(RN\L), thenx∈/ fk(RN\L).
Thus,|fk(RN\L)∪{x}| =k+1= |fk+1(RN)|.Therefore, fk+1(RN)=fk(RN\L)∪{x}.Therefore, fk+1(RN)⊆p(RN), the desired
conclusion. !
We are now ready to prove Theorem 1.
Lemma 4.If f satisfies peak-selection, object-population monotonicity and sovereignty, then it is a priority rule.
Step 1. Construction of a candidate priority)from f.
By peak-selection, for each two non-overlapping peak-unanimous profiles RL and RM, we have f1(RL,RM)⊂p(RL)∪
p(RM).Let)be the binary relation over peak-unanimous profiles such that, for each two non-overlapping peak-unanimous
profiles RL and RM, we have RL)RM if f1(RL,RM)=p(RL). By construction, the relation ) is asymmetric and almost
complete. It remains to show that) is almost transitive. Consider three arbitrary peak-unanimous profiles RK, RL and RM
such that RK and RM are non-overlapping. Suppose that RK )RL and RL)RM. Then in particular, RK and RL are
non-overlapping, and RL andRM are non-overlapping. By definition of), we know that f1(RK,RL)=p(RK) and f1(RL,RM)=
p(RL).This andobject-population monotonicityimply that p(RK)⊂f2(RK,RL,RM)and p(RL)⊂f2(RK,RL,RM). By
compat-ibility, we havep(RK)$=p(RL).Therefore f2(RK,RL,RM)=p(RK)∪p(RL). But this andobject-population monotonicityimply
that f1(RK,RM)=p(RK), i.e.RK)RM, the desired conclusion. Therefore)is a priority inPT.
Step 2. Let)be a priority defined as in step1from f.Let RN∈
P
1.Let RM and RLbe distinct maximal peak-unanimous subprofilesof RNsuch that RM)RL. Then f1(RN)$=p(RL).
The proof is by induction on |p(RN)|. For |p(RN)| =2, the claim follows from the definition of ). Let RN∈
P
1 besuch that |p(RN)| =n"3, and suppose that the claim is true for all R"N" ∈
P
1 such that |p(R"N")| =n−1. Since n"3, there is a maximal peak-unanimous subprofile RH of RN distinct from both RL andRM, such that f1(RN)$=p(RH).By theinduction hypothesis, f1(RN\H)$=p(RL).Let RK be the maximal peak-unanimous subprofile of RN, distinct from both RL
and RH, such that f1(RN\H)=p(RK).Strong sovereignty ensures that there exists a peak-unanimous profile RJ such that
N∩ J=∅, p(RJ)=p(RH), and p(RJ)⊂ f2(RN,RJ).By object-population monotonicity,we have p(RK)⊂ f2(RN,RJ).Since
p(RK)$=p(RJ), thus f2(RN,RJ)=p(RK)∪p(RJ). By object-population monotonicity,we have f1(RN)⊂p(RK)∪p(RH).
Therefore, f1(RN)$=p(RL), i.e. the claim is true for |p(RN)| =n. Thus, it is true for any integer value of|p(RN)|, which
proves the claim.
Step 3. Let)be defined as in step1from f.Let(k,RN)be a problem. Let RMand RLbe distinct maximal peak-unanimous subprofiles
of RNsuch that RM)RLand p(RL)⊆fk(RN).Then p(RM)⊆fk(RN).
Let RM andRL satisfying the assumptions of the step. Consider all the maximal peak-unanimous subprofiles of RN that
are distinct from RM and whose peak location is not contained in fk(RN). There are at leastm:= |p(RN)|−k−1 such
subprofiles. LetRM1, . . . ,RMm bemsuch subprofiles. By step 2, we know that f1(RM1, . . . ,RMm,RM,RL)$=p(RL).Therefore
f1(RM1, . . . ,RMm,RM,RL)⊂p(RM1∪···∪Mm∪M).Next, usingobject-population monotonicity k−1 times,we obtain that fk(RN)∩ p(RM1∪···∪Mm∪M)=$ ∅.Since fk(RN)∩p(RM1∪···∪Mm)=∅by definition of the subprofilesRMk, thereforep(RM)⊆ fk(RN), the
desired conclusion. !
Theorem 1 follows immediately from Lemma 1, Lemma 3 and Lemma 4.
5.1. Independence
k=1, and such that fk coincides with the left-peaks rule for allk!2 satisfies all the properties of Theorem 1, but object-population monotonicity. Third, the rule that locates the first facility on the left-majoritarian peak and the remaining facilities at the smallestk−1 positive integers distinct from p(RN) satisfies all the properties of Theorem 1, butefficiency.
Furthermore anefficient andobject-population monotonicrule need not be a priority rule, as shown in the following two examples.
Example 7.Let ! be the strict ordering on R such that, for all x,x",y,y",z,z"∈R satisfying x<x"<−1<y<y"<1<
z<z", we have−1!1!y!y"!x!x"!z!z". Let $be the priority such that, for any two peak-unanimous profiles RL and R"M, we have RL$R"M if and only if p(RL)!p(R"M). Let g be the priority rule defined by $. Let f be the rule such that, for each problem(k,RN), ifk=1 andp(RN)= {−1,1}, then fk(RN)≡{0}, and otherwise fk(RN)≡gk(RN).
The rule in Example 7 isefficientandobject-population monotonic, but it is not a priority rule, since it does not even satisfy peak-selection. Thus, withoutsovereignty, the result in Lemma 3 does not hold. We now show a rule that isobject-population monotonic, and satisfiespeak-selection but is still not a priority rule. Thus, withoutsovereignty, the result in Lemma 4 does not hold either.
Example 8. Let " be the serial dictatorship such that 1"2"3"· · · and let "" be the serial dictatorship such that
2""1""3""· · ·.Let f be the rule such that, for allkand all RN∈
X
k, the rule f coincides with the serial dictatorship"on the set of problems(k,RN)such thatk+1<|p(RN)|and with the serial dictatorship"" on the set of problems(k,RN)
such that|p(RN)|∈{k+1,k}.
It is easy to see that all priority rules satisfy the perhaps more natural, but stronger version ofobject-population mono-tonicity, which requires that if new facilities and new agents are brought into the economy, and if the number of new facilities exceeds the number of new distinct peaks, all agents who were initially present weakly benefit from the change. The rules in Examples 7 and 8 also satisfy this stronger version. Thus strengtheningobject-population monotonicityin Theo-rem 1 in this way does not make the properties redundant either.
It is easy to see that the stronger version of the property is equivalent to the combination of our version together with the following simple condition. When new facilities are brought into the economy, while keeping the agents and their preferences fixed, all agents should weakly benefit from this change.
A rule satisfiesobject monotonicityif, for each problem(k,RN)such thatk<|p(RN)|, we have fk+1(RN) RN fk(RN).
As Theorem 1 shows,object-population monotonicity,efficiencyandsovereigntyimplyobject monotonicityand the stronger version ofobject-population monotonicity. However, anobject monotonic,efficient and sovereignrule need not be a priority rule as shown in the following two examples.18
Example 9.Consider a rule such that, for each population N the rule coincides with the same priority rule f$N across all profiles RN, but the priority f$N varies across populations. For example, if the population is even the priority is the serial
dictatorship that favors lower index agents, whereas if the population is odd, it is the serial dictatorship that favors higher index agents.
Example 10. For each k∈N, let gk:
P
1−→X
be a 1-rule that satisfies for all RN∈P
1, we have gk(RN)⊆p(RN). For example, let each of the gk be such that, if N is odd, gk(RN)=med(p(RN)) and if N is even, gk(RN)=min(p(RN)). LetM1≡N, and for allk∈{1, . . . ,|p(RN)|}, let
Mh+1
=
Mh\
!
i∈
Mh: p(
Ri)
=
gh(
RMh)
"
.
Next, for eachk∈Nand all RN∈
P
k, letfk
(
RN)
=
!
g1(
RM1),
g2(
RM2), . . . ,
gk(
RMk)
"
.
Moreover, object-population monotonicity and peak-selection do not imply object monotonicitynor the stronger version of object-population monotonicity,as shown in the following example. As a consequence, object-population monotonicity and efficiencydo not implyobject monotonicitynor the stronger version ofobject-population monotonicity.
Example 11.Let f be a rule such that:
(i) For any preferences R1and Ri, with i∈{2,3}, with distinct peaks, we have f1(R1,Ri)=p(Ri);
(ii) For any unanimous non-overlapping profiles RM andR!3such that|M|!2 and 1∈/M, we have f1(R!3,RM)=p(R!3); (iii) For any preferences R1, R2andR3with three distinct peaks, we have f2(R1,R2,R3)= {p(R2),p(R3)};
(iv) For any unanimous non-overlapping profiles R!
1, R!!3 and RM, such that M #= {2}, we have f2(R!1,R!!3,RM) =
{p(R!1),p(R!!3)}.
Last, for any other profile, f coincides with the serial dictatorship"such that(i"j)⇔(i<j).
6. Refining the class of priority rules
An attractive feature of the priority rules is their simple structure. As a consequence, it is easy to characterize the priority rules that satisfy additional properties. This in turn leads to refinements of our characterization based on these additional properties.
6.1. No-show paradox
First, we consider a requirement that ensures that no agent gains from hiding from the social planner (Moulin, 1988). For instance, consider a university which has to choosek software licenses from a larger set of available software for its employees. The decision is taken through an online poll, to which employees can freely respond. One would want that no employee has an incentive not to participate in the poll.
A rule f avoids the no-show paradoxif for each problemk!1, each RN∈
P
k, and eachi∈N such thatRN\{i}∈P
k, we have fk(RN) Ri fk(RN\{i}).Not all the rules characterized in Theorem 1 satisfy this property. For example, with some abuse of notation, the group priority such that{2}%{1,3}, and for all disjoint populations M,N satisfying (M,N)#=({1,3},{2}), we have M%N if and only if max(M) >max(N). The associated priority rule satisfies all the properties of Theorem 1, but does not avoid the no-show paradox. The following is a necessary and sufficient condition for a priority rule to avoid the no-show paradox. A priority% is monotonicif, there are no two non-overlapping peak-unanimous profiles RM and RL andi∈M such that RM\{i}%RLand RL%RM. The proof of the following result is straightforward and we leave to the reader.
Proposition 1.A priority rule f avoids the no-show paradox if its priority is monotonic.
Monotonicity implies almost-monotonicity. As an immediate consequence of Proposition 1 and Theorem 1, a rule satisfies efficiency, object-population monotonicity, sovereigntyandavoids the no-show paradoxif and only if it is a priority rule associated with amonotonicandsovereignpriority.
6.2. Anonymity
A rule f satisfies anonymityif, for all k!1, and all RN,R!M ∈
P
k where for all R∈R
, we have |{i∈M: R!i=R}| =|{i∈N: Ri=R}|, we have fk(RN)= fk(R!M). A priority % is anonymous if it satisfies the following condition. For all RM,RN,RM! !,R!N!∈
T
, such that: (i) RM and RN are non-overlapping, (ii) R!M! and R!N! are non-overlapping, (iii) for all R∈R
, we have|{i∈M: Ri=R}| = |{i∈M!: R!i=R}|, (iv) for all R∈R
, we have|{i∈N: Ri=R}| = |{i∈N!: R!i=R}|, the following equivalence holdsRM
%
RN⇐⇒
R!M!%
R!N!.
The proof of the following results is straightforward and we leave it to the reader.
Proposition 2.A priority rule f is anonymous if its priority is anonymous.
As an immediate consequence of Proposition 2 and Theorem 1, a rule satisfiesefficiency, object-population monotonicity,
sovereigntyandanonymityif and only if it is a priority rule associated with an almost-monotonic, sovereign and anonymous
priority.
6.3. Strategy-proofness: hierarchical rules
We now investigate on the priority rules that are robust to the manipulation of preferences. The non-manipulability property we are interested in requires that truthfully reporting preferences be a (weakly) dominant strategy in the associated direct revelation game.
The left-peaks rule, the right-peaks rule and all the serial dictatorships are strategy-proof priority rules. More generally, any hierarchical rule – whose definition can be found in Example 3 – is strategy-proof. We have the following result, the proof of which is relegated to Appendix A.
Theorem 2.A priority1-rule is strategy-proof if and only if it is a hierarchical1-rule. A priority rule f is strategy-proof if and only if it is a hierarchical rule.
All of the hierarchical rules triviallyavoid the no-show paradox.The left-peaks rule and the right-peaks rule are the only
anonymoushierarchical rules, and these are not sovereign.Each majoritarian rule isanonymous, but it is not a hierarchical rule, i.e. it is notstrategy-proof.A hierarchical priority!is sovereign if and only if it is unbounded (see footnote 17). Finally, all hierarchical rules are also robust to joint manipulation of preferences. They satisfy group strategy-proofness, a prop-erty which states that, for each k∈N, each RN∈
P
k, each M⊆N such that M$=∅ and each R&M ∈R
M, we do not have [fk(R&M,RN\M) RM fk(RN)and for somei∈M, fk(R&M,RN\M) Pi fk(RN)].6.4. Population monotonicity
When new agents are brought into the economy, while resources are held constant and the preferences of the agents initially present (the “first-comers”) are kept fixed, the new opportunity set of the first-comers depends on whether the newcomers have rights over the resources. If the newcomers have no rights, the opportunity set of the first-comers remains the same. If the newcomers have rights over the resources, the opportunity set of the first-comers either remains the same or is reduced. It is therefore natural to require that all these agents weakly lose from the change. Formally, f satisfies population monotonicity if for all non-overlapping profiles RM and R&L, we have f(RM,R&L) RM f(RM,R&L).
Ching and Thomson (forthcoming) characterize the family of population monotonic and efficient 1-rules. Each rule in this family is a “target rule” parametrized by some fixed locationλ∈R. Wheneverλis contained in the shortest segment
that contains all peak locations in the profile, the rule selects the locationλ.Otherwise, the rule selects the location in this
segment that is closest toλ.
Miyagawa (1998) and Ehlers (2003) provide interesting characterizations of the efficient and population monotonic 2-rules. Miyagawa considers the same extension as ours of single-peaked preferences to menus, while Ehlers studies the lexicographic variant of it. Although they are quite different, the rules characterized by both of these authors retain a target rule flavor. Each rule in Miyagawa can be described as a hierarchy of two target rules. First a target 1-rule is applied, associated with some fixed parameterλ. If this target rule selects a peak location, one facility is located according to this
target 1-rule. In this case, the second facility is located according to a secondary target rule applied on the peaks located on one of the sides of the first facility. The parameter of this secondary target rule may depend on the location of the first good. In contrast, each of the rules characterized by Ehlers chooses the efficient pair of locations that is closest to some fixed “target” pair of locations along a fixed path in the space of location pairs.
Using Ching and Thomson’s (forthcoming) characterization – briefly stated earlier in this section – it is easy to see that the only priority 1-rule that satisfypopulation monotonicityare the left-peaks and the right-peaks 1-rules. It is also easy to see that the corresponding rules also satisfy this property.
Proposition 3.A priority1-rule satisfies population monotonicity if and only if it is either the left peaks or the right-peaks rule. A priority rule satisfies population monotonicity if and only if it is either the left peaks or the right-peaks rule.
Object-population monotonicityandpopulation monotonicityare orthogonal requirements in the sense that they operate on different population changes that affect the set of opportunities for the agents initially present in opposite ways. The former applies to changes that enlarge this set while the latter applies to changes that reduce it. The three properties of Theorem 1 (which implyobject monotonicity) are even incompatible with population monotonicity, since none of the sovereign priority rules is either the left-peaks or the right-peaks rule. Moreover,population-monotonicityandobject-monotonicitydo not imply
object-population monotonicity,as shown in the following example.
Example 12.Let f andg be respectively the left-peaks and right peaks priority rules. Next, define the rulehsuch that for all RN∈
X
1, we haveh1(RN)=f(RN), and for all p!1 and all RN∈X
2p, we haveh2p(RN)=fp(RN)∪gp(RN), and for all RN∈X
2p+1, we haveh2p+1(RN)= fp+1(RN)∪gp(RN).Results by Gordon (2007a) imply for the model studied in this paper thatpopulation monotonicity andefficiencyimply
6.5. The median
An important rule for selecting a single location in this model is the one which selects the median peak, for each preference profile RN such that |N|is odd (Black, 1948). A natural question is thus whether there exists a priority rule f such that, for any RN such that |N| is odd, the location f1(RN) is the median of the peaks. Since the median is strategy-proof but is not a hierarchical rule, one can easily see from Theorem 2 that the answer is negative. Unfortunately, we are able to show a much stronger negative result.
Proposition 4.Let n!2. There is no rule f satisfying object-population monotonicity and such that, for each RM∈
R
M, with|M| = 2n+1, the location f1(RM)is the median of the peaks of the profile RM.Proof. By contradiction, suppose that f is such a rule.Let N= {1, . . . ,3n+2}and let RN be a profile such that
p
(
R1)
= · · · =
p(
Rn) <
p(
Rn+1) <
· · ·
<
p(
R2n+2) <
p(
R2n+3)
= · · · =
p(
R3n+2).
We consider in turn subsets of N, each of cardinality 2n+1.
Consider first M= {1, . . . ,2n+1}. Since f1(RM) is the median of the peaks in the profile, thus f1(RM)=p(Rn+1).By object-population monotonicity, we have p(Rn+1)∈f3(RN).Consider next, L= {n+2, . . . ,3n+2}. Notice that|L| =2n+1. Since f1(RL)is the median of the peaks in the profile, thus f1(RL)=p(R2n+2).By object-population monotonicity, we have p(R2n+2)∈ f3(RN). Consider now H= {1, . . . ,n,n+2, . . . ,2n+2}. Notice that |H| =2n+1. Since f1(RH) is the median of the peaks in the profile, thus f1(RH)=p(Rn+2). By object-population monotonicity, we have p(Rn+2)∈ f3(RN).Finally, consider J= {n+1, . . . ,2n+1,2n+3,3n+2}. Notice that |J| =2n+1. Since f1(RJ) is the median of the peaks in the profile, thus f1(RJ)=p(R2n+1). By object-population monotonicity, we have p(R2n+1)∈f3(RN).
In summary{p(Rn+1),p(Rn+2),p(R2n+1),p(R2n+2)}⊆ f3(RN).Sincen!2, thereforen+2<2n+1. Thus the inclusion contradicts|f3(RN)| =3. !
7. Conclusion
In this paper we undertake the task of designing rules for the provision of multiple public facilities to populations of agents with single peaked preferences over the real line. To this end, we introduce a new property –object-population monotonicity. We uncover and characterize a rich family of priority rules. Some rules in the family are appealing, for example the majoritarian rules. Unfortunately, no priority rule coincides with the median for each problem of providing exactly one public facility to an odd population of agents. On the positive side, the structure of the rules in the family is very simple, which makes it possible to characterize several subfamilies on the basis of additional properties. In particular, strategy-proof priority rules form the interesting subfamily of hierarchical rules.
Along with recent papers in the literature (e.g., Barberà and Beviá, 2002, 2005), we view our work as a step towards a comprehensive understanding of the implications of normative and strategic properties in the problem of locating multiple facilities. An obvious question is whether it is possible to generalize Moulin’s (1980) generalized median voter schemes to this more complex setting. We only provide a partial answer to this question. The priority structure enables us to generalize only a particular type of generalized median voter scheme. A generalization of the entire family of generalized median voter schemes to the problem of locating multiple facilities requires a flexibility which conflicts with the priority structure, and thus with our main properties.
Our analysis is based on a particular way to extend preferences over single-locations to preferences over multiple lo-cations, the max-extension, introduced by Miyagawa (1998, 2001). The work of Miyagawa seems to indicate that the max-extension is not very tractable, in the sense that it leads to characterizations of rules that are not particularly ap-pealing, such as the left-peaks rule and the right-peaks rule, and other more complicated rules. As our results suggest, this may have more to do with the solidarity properties used there than the max-extension itself. As Gordon (2007a) shows, solidarity properties have strong general implication in all public decisions models, of which the location of multiple public facilities is a special case. Nevertheless, both Miyagawa and us obtain highly discontinuous rules, for any natural topology. This seems to be a regular feature of the max-extension model. Thus, insisting on continuity in this model may be too strong a requirement. Majoritarian rules, for instance, are not continuous.
A natural extension of our work would be precisely to study what happens under other types of preference extensions, e.g. the lexicographic extension. Ehlers (2002, 2003) studies the same properties as Miyagawa (1998, 2001) under the lexicographic extension and obtains strikingly different characterizations. So, it is quite possible that the properties we used here would lead to other families of rules as well, if one were to replace the max-extension of our model with the lexicographic extension, or some other meaningful extension.
