Majoritarian

Contents lists available atScienceDirect

Journal of Mathematical Economics

journal homepage:www.elsevier.com/locate/jmateco

Weighted majoritarian rules for the location of multiple

public facilities

Olivier Bochet

a,b,_⇤

_{, Sidartha Gordon}

c

_{, Rene Saran}

d a_{NYU-Abu Dhabi, United Arab Emirates}

b_{University of Bern, Switzerland} c_{Sciences Po, France} d_{Yale-NUS College, Singapore}

a r t i c l e i n f o

Article history: Received 23 August 2012 Received in revised form 1 August 2013 Accepted 7 August 2013 Available online 24 August 2013

Keywords: Priority rules

Weighted majoritarian rules Object-population monotonicity Sovereignty

Reinforcement Anonymity

a b s t r a c t

We consider collective decision problems given by a profile of single-peaked preferences defined over the real line and a set of pure public facilities to be located on the line. In this context,Bochet and Gor-don(2012) provide a large class of priority rules based onefficiency,object-population monotonicityand sovereignty. Each such rule is described by a fixed priority ordering among interest groups. We show that any priority rule which treats agents symmetrically —anonymity— respects some form of coherence across collective decision problems —reinforcement— and only depends on peak information — peak-only— is a weighted majoritarian rule. Each such rule defines priorities based on the relative size of the interest groups and specific weights attached to locations. We give an explicit account of the richness of this class of rules.

©2013 Elsevier B.V. All rights reserved.

1. Introduction

We consider a generalization of the unidimensional voting model studied byBlack(1948),Moulin (1980) andBarberà and Jackson(1994). A collective decision problem is given by a set of agents, a profile of single-peaked preferences defined over the real line, and a set of pure public facilities to be located on the line.1_As is standard in the mechanism design literature, we look for rules which can solve any collective decision problem.

In this setup,Bochet and Gordon(2012) characterize a rich class of rules based on the combination ofefficiency,object-population monotonicity, andsovereignty. While efficiency is a standard notion, the last two properties are new. Object-population monotonicity states that if newcomers join a collective decision problem and, at the same time, the number of public facilities increases to com-pensate for this arrival, then agents already in the initial problem cannot be hurt. Suppose next that a single facility must be located. Sovereignty states that any location could be chosen provided that

⇤Corresponding author at: University of Bern, Switzerland. Tel.: +33 647076359. E-mail addresses:[email protected](O. Bochet),

[email protected](S. Gordon),[email protected](R. Saran). 1 By pure public facilities, we mean facilities which are non-excludable and do not suffer from congestion.

an appropriately selected, and possibly large, interest group de-fending this particular location is brought into the problem. Each rule which jointly satisfies these three properties is a priority rule that selects locations based on a fixed priority ordering among in-terest groups.

An appealing feature of the class of priority rules is the simplic-ity with which these rules can be described. However, as will be made clear in Section3, the class contains some rules which ei-ther give too much power to some agents, or exhibit inconsisten-cies across specific collective decision problems. We suggest that some order should be put into this class by imposing that a rule treat agents symmetrically —anonymity— and respect some form of coherence across collective decision problems —reinforcement.

Anonymity is a well-known property imposing that agents’ la-bels do not matter. Reinforcement is a property of stability with respect to merging of collective decision problems. It states that if for two problems — differing possibly in the cardinality of the set of agents and their preferences — the rule selects the same locations, then it should be invariant for the new collective deci-sion problem obtained by merging the two initial problems. This property is, however, not new and already appears in the litera-ture on characterizations of scoring rules — see e.g.Young(1975) orMyerson(1995). Along with a natural informational simplicity property —peak-only— any rule in Bochet and Gordon’s class that satisfies anonymity and reinforcement is a weighted majoritarian 0304-4068/$ – see front matter©2013 Elsevier B.V. All rights reserved.

rule. Each such rule defines priorities based on specific weights attached to interest groups. The weight of any interest group rela-tive to another depends on their densities and most preferred lo-cations. The simplest example is the rule which takes into account only the density of each interest group and gives priority to groups with the highest density. We call these simple majoritarian rules. However, the class is much larger. For instance, a rule can assign different weights to different interest groups based on the regions in which their most preferred locations are located. Also, rules can incorporate additional features such as the distance between the most preferred locations of the interest groups and a reference point. So while each rule in our class is ‘‘density-based’’, additional information can be used.

The problem of locating multiple public facilities was first introduced byMiyagawa (1998,2001)in the case of two facilities.

Ehlers (2002,2003),Bochet and Gordon(2012),Heo(2013),Ehlers and Gordon(2011) provide axiomatic characterizations for this model.Umezawa(2012) considers the location of two facilities on a tree network.Barberà and Beviá (2002, 2005)andJu(2008) show the existence of a rule satisfying interesting normative properties. Our main contribution to this literature is the analysis of the implications of the reinforcement axiom in this context and the characterization of the weighted majoritarian rules.

The plan of the paper is the following. In Section2, we introduce the model. In Section3, we introduce the properties we study, the class of priority rules and provide several illustrating examples. In Section4, we prove our central result. Finally, we conclude in Section5by illustrating the richness of the characterized class of rules.

2. The model and notations

There is a countably infinite setNof potential agents. A pop-ulation Nis a finite and nonempty subset ofN. The population is collectively endowed withk 1 identical public facilities, each to be located on the real lineR. A typical location onRis denoted by x. Anassignmentis a menu of locations, i.e., a finite subsetX

_⇢

R. A k-assignmentis an assignment for exactlykfacilities, i.e., a subset X

_⇢

Rsuch that

|

X

| =

k. LetXkbe the class of allk-assignments.

In particular, a 1-assignment is a single locationx

₂

R, so that X1

₌

R. LetX

_{⌘ [}

k 1Xkbe the class of all assignments.

Each agenti

2

Nhas apreference RioverX, which is a weak

ordering (reflexive, transitive and complete) overX. LetPiandIibe,

respectively, the strict ordering and indifference relation derived fromRi. A preferenceRiissingle-peakedif the following hold.

(i) There is a locationp

(

Ri

), such that for all

x

,

y

2

Rsatisfying eitherx

<

y



p

(

Ri

)

orp

(

Ri

)

y

>

x, we havey Pix. The

locationp

(

Ri

)

is called thepeakof preferenceRi.

(ii) For allX

,

Y

2

X, we letX RiYif there isx

2

Xsuch that for all

y

2

Y, we havex Riy.

The first condition is the standard single-peakedness notion for preferences over single locations on the real line. The second con-dition extends the preferences from single locations to menus.2_We

restrict attention to the classRof single-peaked preferences over X.

A preference profile, RN, specifies a population N and the

preferences of all agents inN, i.e.,RN

=

(

Ri

)

i2N

2

RN. For each

profileRNand each nonempty subpopulationM

✓

N, letRMdenote

the subprofile

(

Ri

)

i2M. For each profileRN, letp

(

RN

)

be the set of

peak locationsforRN, i.e.,p

(

RN

)

⌘

{

p

(

Ri

)

:

i

2

N

}

. For eachk 1,

2 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 byMiyagawa(2001).

letPkbe the set of preference profilesRNsuch thatk



|

p

(

RN

)

|

, i.e.,

the number of distinct peak locations inRNis at leastk. Aproblem

is a pair

(

k

,

RN

)

such thatk 1 andRN

2

Pk.3

Aruleis a sequencef

= {

f1

,

f2

, . . .

}

of mappingsfk

:

Pk

!

Xk.

For each problem

(

k

,

RN

), the rule

f prescribes an assignment in

Xk.4For eachk 1, the set of mappingsfkisXPkk. Therefore, the

set of all rules is

Q

1_k=1XPk

k .

3. Main axioms and priority rules

Consider a profileRN

2

RNandx

,

y

2

R. For allX

,

Y

2

X, we say thatX weakly Pareto dominates Y for profile RN, denoted by

X RNY, ifX RiYfor eachi

2

N. Our first axiom is the usualefficiency

axiom.

A rulef satisfiesefficiencyif, for each problem

(

k

,

RN

), there is

nok-assignmentXsuch thatX RNfk

(

RN

), and

X Pjfk

(

RN

)

for some

j

2

N.

A profileRNispeak-unanimousif all preferences of this profile

have the same peak, i.e.,p

(

RN

)

is singleton. LetT be the set of peak-unanimous profiles.

A rulefsatisfiesobject-population monotonicityif, for each prob-lem

(

k

,

RN

)

withk

<

|

p

(

RN

)

|

, for each peak-unanimous profile

RM

2

T such thatN

\

M

=

;

, we havefk+1

(

RN

,

RM

)

RNfk

(

RN

).

A rulef satisfiessovereigntyif, for each profileRN, each

loca-tionx

2

R

\

f1

(

RN

), and each population

L, there exists a

peak-unanimous profileRM

2

T such thatMis disjoint from bothLand

N, andf1

(

RN

,

RM

)

= {

x

} =

p

(

RM

).

On the one hand, in the situation of a population and resource increase, object-population monotonicity protects the rights of the first-comers. On the other hand, in the situation of a population increase, sovereignty protects the rights of the newcomers.

Bochet and Gordon(2012) show that the combination of ef-ficiency, object-population monotonicity and sovereignty charac-terizes a subclass of priority rules. To define these rules, we need to introduce a class of binary relations called priorities over any nonempty subsetSofT. We say that any two peak-unanimous profiles RN and RM are non-overlapping if they have distinct

peaks and disjoint populations, i.e., p

(

RN

)

6

=

p

(

RM

)

andN

\

M

=

;

. The binary relation over S isalmost completeif for allRN

,

RM

2

S, we have

(

RN RMorRM RN

)

()

RNandRM

are non-overlapping .5 _{It is almost transitive if for all profiles} RN

,

RM

,

RL

2

S, such thatRNandRLare non-overlapping, we have

(

RN RMandRM RL

)

H)

(

RN RL

). The binary relation

is

apriorityoverSif it is asymmetric, almost transitive and almost complete.6_{For each nonemptyS}

_✓

_{T, letP}

Sbe the set of priori-ties overS.

For any profileRN, the peak-unanimous subprofileRMofRNis

maximalifp

(

RM

)

\

p RN\M

=

;

. Since any two distinct maximal

peak-unanimous subprofiles are non-overlapping, the set of maxi-mal peak-unanimous subprofiles of some profile can be strictly or-dered by any priority.

3 The restrictionk  |p(RN)|allows us to focus on non-trivial cases. When k>|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.

4 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.

For each

2

PT, thepriority rule f associated with is defined as follows. Let

(

k

,

RN

)

be a problem. The priority strictly ranks

the maximal peak-unanimous subprofiles in the decomposition of RNandfk

(

RN

)

selects the peak locations of the topkmaximal

peak-unanimous subprofiles for . That is,fk

(

RN

)

is thek-assignment

such thatfk

(

RN

)

✓

p

(

RN

)

, and for all two maximal peak-unanimous

subprofilesRMandRLinRN, ifp

(

RM

)

✓

fk

(

RN

)

andp

(

RL

)

(fk

(

RN

)

,

thenRM RL. LetF be the class of priority rules.

We now introduce two properties that a priority ordering may satisfy. A priority isalmost monotonicif there are no four peak-unanimous profilesRM

,

RK

,

RHandRLsuch thatp

(

RM

)

=

p

(

RL

) ,

M

_\

L

₌

_;

,

RKandRHare non-overlapping,RM RH RM_[L, and

RM RK RM_[L. A priority issovereignif the following two con-ditions hold. (i) For all peak-unanimousRH

,

RKsuch thatRH RK,

and for any populationL, there exists a peak-unanimous profileRM such thatMis disjoint fromKandL, and satisfiesp

(

RM

)

=

p

(

RK

)

andRK_[M RH. (ii) For each peak-unanimous profileRH, eachx

6

=

p

(

RH

)

and each populationL, there exists a peak-unanimous

pro-fileRMsuch thatM

\

L

=

;

, and satisfiesp

(

RM

)

=

xandRM RH.

Next, we provide an example of a priority that is not sovereign.

Example 1 (Left-Peaks Priority/Right-Peaks Priority).A priority is theleft-peaks priorityif for all non-overlapping profilesRM

,

RN

2

T, we haveRM RN

()

p

(

RM

) <

p

(

RN

)

. Similarly, is the

right-peaks priorityif for all non-overlapping profilesRM

,

RN

2

T, we haveRM RN

()

p

(

RM

) >

p

(

RN

)

.

⇧

We now stateBochet and Gordon(2012)’s central result.

Theorem 1. A rule f satisfies efficiency, object-population mono-tonicity and sovereignty if and only if it is a priority rule whose priority is almost monotonic and sovereign.

The proof of this result can be found in Bochet and Gordon

(2012). We now give examples of priorities attached to rules de-scribed inTheorem 1.

Example 2 (Hierarchical Priorities).A priority is hierarchical if the following holds. (i) There is a weak orderingDof all agents inN, such that, for all non-overlapping profilesRL

,

RM

2

T, if there ex-istsi

2

L, such that for allj

2

M, we have (iDjand notjDi), then RL RM. (ii) For eachD-indifference classK, consider the classTK of peak-unanimous profilesRMsuch that the agents inMwho are ranked highest according toDbelong toK. On each such classTK, the priority coincides with either the left-peaks or the right-peaks priority. If eachD-indifference class is a singleton, the priority is a serial dictatorship. Also, if there is a singleD-indifference class, the priority is either the left-peaks or the right-peaks priority.

⇧

Note that a hierarchical priority, as described inExample 2, is sovereign (and therefore satisfies all the properties inTheorem 1) if and only if the weak orderingDhas no maximal element.

Example 3 (Simple Majoritarian Priorities).A priority is simple majoritarian if for all non-overlappingRL

,

RM

2

T, we have

|

L

|

>

|

M

|

)

RL RM. For eachn 1, the tie-breaking rule nwithin each class of the formTn

= {

RN

2

T

_{: |}

N

| =

n

}

can be given by any strict ordering over locations inR. For example, we could require nto be the left-peaks priority for alln(left majoritarian priority) or the right-peaks priority for alln(right majoritarian priority).

⇧

Unlike the rules described by simple majoritarian priorities, the rules described by hierarchical priorities allow for an asymmetric treatment of agents, i.e., agents’ labels matter. We would like rules to respect an anonymous treatment of agents’ preferences.

A rule f satisfies anonymityif for all k 1 and problems

(

k

,

RN

)

and

(

k

,

R0_M

)

such that for allR

2

R

,

i

2

M

:

R0_i

=

R

=

|{

i

2

N

:

Ri

=

R

}|

, we havefk

(

RN

)

=

fk

(

R0_M

)

.

Anonymity imposes an additional requirement on priorities. A priority isanonymousif it satisfies the following condition. For all RM

,

RN

,

R0_M0

,

R0_N0

2

T, such that (i)RMandRNare non-overlapping, (ii)R0

M0 andR0N0are non-overlapping, (iii) for allR

2

R, we have

|{

i

2

M

:

Ri

=

R

}| =

i

2

M0

_:

_R0

i

=

R , (iv) for allR

2

R, we have

|{

i

2

N

:

Ri

=

R

}| =

i

2

N0

_:

_R0

i

=

R , the following equivalence holds:RM RN

₍₎

R0

M0 R0N0.

Bochet and Gordon(2012) characterized the subclass of anony-mous priority rules. We state it below and omit its straightforward proof.

Theorem 2. Let f be a priority rule. Then f satisfies anonymity if and only if its priority is anonymous.

Notice that if anonymity is dropped, the class of rules which satisfy efficiency, object-population monotonicity and sovereignty will include rules whose priorities combineExamples 2and3in interesting ways. We give below two such examples.

Example 4 (Majoritarian–Hierarchical Priorities).LetDbe a weak ordering over agents inN, such that for alli

,

j

,

i

<

j

H)

jDiand, in addition,Dhas no maximal element. Construct the partition of

Ninto indifference classesZ1

,

Z2

, . . .

. according toD. That is, for

eachi

,

j

2

Zk

,

i D jandj D i. In addition, for eachi

2

Zk

,

j

2

Z`

withk

<

`, we have

jDi.

A priority is a majoritarian–hierarchical priority if there is an indexudefined on the class of all populations, such that for each populationM

⇢

N

,

uis defined as

u

(

M

)

= |

M

| +

(

1

)

max

{

k

2

N

:

Zk

\

M

6

=

;

}

for

₂

(

0

,

1

)

, and for all non-overlappingRM

,

RN

2

T, we have

thatu

(

M

) >

u

(

N

)

H)

RM RN. For each

v >

0, the tie-breaking

rule vwithin each class of the formTv

= {

RN

2

T

:

u

(

N

)

=

v

}

is

either the left-peaks or the right-peaks priority.

⇧

Example 5 (Hierarchical Weighted Majoritarian Priorities).A prior-ity is a hierarchical weighted majoritarian priorprior-ity if there exists a list of weights

(

!i

)

i2N

2

RNwith

1

X

i=1

!

i

= +

1

,

such that for all non-overlappingRM

,

RN

2

T, we have

X

i2M

!

i

>

X

i2N

!

i

H)

RM RN

.

For cases where equality holds, the tie-breaking rule within each

v

level curve of the form

{

RN

2

T

:

P

_i2N

!

i

=

v

}

is de-termined by some strict orderingBover locations, independent of

v

.7

⇧

InBochet and Gordon(2012), it is shown that the set of pri-ority rules described by hierarchical priorities is equivalent to the set of strategy-proof priority rules. In contrast, there are only two hierarchical priorities that are anonymous: left-peaks and right-peaks priorities. But left-right-peaks and right-right-peaks priorities are not sovereign. Thus, anonymity, sovereignty and strategy-proofness are mutually inconsistent within the class of priority rules. If we do not impose anonymity, then the class of rules characterized by all other properties (i.e. efficiency, object-population mono-tonicity, sovereignty, reinforcement and peak-only) will include

every hierarchical rule whose associated priority is sovereign.8 Adding anonymity excludes rules like serial dictatorship that vi-olate the pure notion of majoritarianism. But it also excludes some rules which balance agents’ priorities with the notion of majoritarianism—e.g. the priority introduced inExample 5.9

We now introduce additional examples of priority rules which also satisfy anonymity.

Example 6 (Two-Regions Majoritarian Priorities). A priority is

two-regionsmajoritarianif there is a locationx0

2

R(that sepa-rates the two regions), and a coefficient x₀

2

(

0

,

1

]

such that, for

all non-overlappingRM

,

RN

2

T, if either (i)p

(

RM

) <

x0



p

(

RN

)

and x0

|

M

|

>

|

N

|

or (ii)p

(

RN

) <

x0



p

(

RM

)

and

|

M

|

>

x0

|

N

|

or (iii)p

(

RM

),

p

(

RN

)

2

(

1

,

x0

)

, orp

(

RM

),

p

(

RN

)

2

[

x0

,

+1

)

and

|

M

|

>

|

N

|

hold, thenRM RN. For each

v

x0, the tie-breaking

rule vwithin each class of the form

Tv

= {

RN

2

T

:

p

(

RN) x0

,

|

N

| =

v

}

[

RN

2

T

:

p

(

RN

) <

x0

,

x0

|

N

| =

v

can be given by any strict ordering over locations inR. For exam-ple, we could require vto be the left-peaks priority for allTv(

left-two-regions majoritarian priority) or the right-peaks priority for all

Tv(right-two-regions majoritarian priority).

⇧

Example 7 (Centralist Majoritarian Priorities).A priority is

cen-tralist majoritarianif there is a locationx0

2

R(the ‘‘center’’) and an indexu

: {

1

,

2

, . . .

}⇥

R+

!

R, whereu

(

n

,

d

)

is weakly increas-ing innand weakly decreasing ind, with limn!+1u

(

n

,

d

)

= +1

, such that for all non-overlappingRM

,

RN

2

T, we have

u

(

|

M

|

,

|

p

(

RM

)

x0|

) >

u

(

|

N

|

,

|

p

(

RN) x0|

)

)

RM RN

.

For each

v

2

R, the tie-breaking rule vwithin each class of the form

Tv

= {

RN

2

T

:

u

(

|

N

|

,

|

p

(

RN

)

x0|

)

=

v

}

can be any strict ordering over locations inR. For example, we could require v to be the left-peaks priority for all Tv (

left-centralist majoritarian priority) or the right-peaks priority for allTv (right-centralist majoritarian priority).

⇧

There are many possible functionsufor a centralist majoritarian priority rule. For example, with

u

(

n

,

d

)

=

8

>

<

>

:

n

+

d ifn



2

max

⇢

n

,

2

+

d ifn

>

2,

where

>

0, the priority rulefbehaves across problems in a way

that is not coherent. That is, if for two problems — differing possibly in the cardinality of the set of agents, and in preferences — the rule selects the same locations, then the selection operated byf may change for the new collective decision problem obtained by merg-ing the two initial problems. For instance, letx0

=

1

,

=

0

.

1 and

consider the problems

(

1

,

RM

)

and

(

1

,

RL

)

withM

\

L

=

;

,

M

=

H

[

K

,

|

H

| =

1

,

|

K

| =

2

,

p

(

RH

)

=

1₂

,

p

(

RK

)

=

3

;

L

=

H0

8 Suppose we do not impose anonymity but add strategy-proofness instead to efficiency, object-population monotonicity and sovereignty. The class of rules characterized by these four axioms coincides with hierarchical rules whose priorities are sovereign. Notice that hierarchical priority rules whose priorities are sovereign satisfy reinforcement and only. Adding reinforcement and peak-only has thus no refining effect on this characterization.

9 The priority inExample 4does not satisfy both anonymity and reinforcement.

[

K0

_,

_|

_H0

_{| =}

₂

_,

_|

_K0

_{| =}

₃

_,

_p

₍

_RH0

₎

₌

1

2

,

p

(

RK0

)

=

3. It is easy to see

that, givenu

,

f1

(

RM

)

=

f1

(

RL

)

= {

1₂

}

. However,f1

(

RM

[

RL

)

= {

3

}

.

We are interested in rules with the following coherence prop-erty: if two problems deliver the same location(s), the problem ob-tained from merging them still delivers the same location(s).

A rulefsatisfiesreinforcementif for allk 1 and each pair of profilesRN,RMsuch thatN

\

M

=

;

, iffk(RN

)

=

fk(RM

)

=

Xthen fk(RN

[

RM)

=

X.

That is, whenever any two different problems

(

k

,

RN) and

(

k

,

RM

)

select the sameklocations, then reinforcement requires

that the location of thekfacilities should not change in the problem

(

k

,

RN

[

RM

)

.

We add one last property that will be used for our main result. A rulefsatisfiespeak-onlyif for allk 1 and problems

(

k

,

RN

)

and

(

k

,

R0_N

)

, ifp

(

Ri

)

=

p

(

R0i

)

for alli

2

N, thenfk

(

RN

)

=

fk

(

R0N

)

.

Peak-only is an informational simplicity requirement which states that only the information regarding the peaks of agents should be used. It is, however, a strong assumption as it ignores every other aspect of agents’ preference orderings. Nevertheless, if we do not impose peak-only, then the class of priority rules char-acterized by all other properties (i.e. efficiency, object-population monotonicity, sovereignty, anonymity and reinforcement) will in-clude rules that put arbitrary weights on preference orderings. This forms a rich class of rules where weights can depend in compli-cated ways on the full preference relations. Examples of such rules are those described by the following priorities.

Example 8 (Symmetry Biased Majoritarian Priorities).We say that a

single-peaked preferenceRiis symmetric if for allx

,

y

2

R, we have xRiy

()

|

x p

(

Ri

)

|



|

y p

(

Ri

)

|

. For any peak-unanimous

profileRN

2

T, let

(

RN

)

be the number of agentsi

2

Nsuch that Riis symmetric.

A priority issymmetry biased majoritarianif there is

>

0

such that for all non-overlappingRL

,

RM

2

T, we have

(

RL

)

+

|

L

|

>

(

RM

)

+ |

M

|

)

RL RM. For eachn 1, the

tie-breaking rule nwithin each class of the formTn

=

RN

2

T

:

(

RN

)

+ |

N

| =

n can be given by any strict ordering over

loca-tions inR. For example, we could require nto be the left-peaks

priority for allnor the right-peaks priority for alln.

⇧

Thus, imposing peak-only excludes ‘‘undesirable’’ rules like those described by symmetry biased majoritarian priorities. Our position here is that peak-only is a relevant requirement when agents’ peaks (but not preferences) are commonly known. Indeed, in many instances, peak information is difficult to manipulate because it reflects some observable attributes — e.g. because it reflects an agent’s address. Under this interpretation, peak-only is an invariance condition with respect to some preference change.10

4. A characterization of weighted majoritarian rules

We now introduce a family of priority rules that we call weighted majoritarian rules. LetR++andQ++respectively be the set of positive reals and the set of positive rationals.

A priority is aweighted majoritarian priorityif there exists an asymmetric and transitive binary relation, i.e. a strict partial order BonR, and a functionq

:

R2

_!

_R

++withq

(

x

,

y

)

q

(

y

,

x

)

=

1

,

q

(

x

,

z

)

=

q

(

x

,

y

)

q

(

y

,

z

),

andq

(

x

,

y

)

2

Q++

()

(eitherx By

10 But then, notice that the requirement of strategy-proofness in this model may not be appropriate. Indeed, if peaks are verifiable, a weakening of strategy-proofness is called for.Sakai and Wakayama(2012) introduce such a weakening,

ory B x), for all distinctx

,

yandz, such that for any two

non-overlapping peak-unanimous profilesRMandRL, we haveRM RL

if either 1. |M|

|L|

>

q

(

p

(

RM

),

p

(

RL

))

, or

2. |M|

|L|

=

q

(

p

(

RM

),

p

(

RL

))

andp

(

RM

)

Bp

(

RL

)

.

Note that the tie-breaking ruleBis only needed if the image ofqcontains at least one rational number; otherwise, the equality

|M|

|L|

=

q

(

p

(

RM

),

p

(

RL

))

does not hold for any two peak-unanimous

profilesRMandRL.

Theorem 3. A rule f satisfies efficiency, object-population mono-tonicity, sovereignty, anonymity, reinforcement and peak-only if and only if there exists a weighted majoritarian priority such that f is a priority rule associated with .

Proof. It is straightforward to prove the if part, i.e., if there exists a weighted majoritarian priority such thatf is a priority rule associated with , then f satisfies all the axioms listed in the theorem. We prove the only if part.

It follows fromTheorem 1that iff satisfies efficiency, object-population monotonicity and sovereignty, then there exists a priority such thatfis a priority rule associated with . We show that is a weighted majoritarian priority.

Pick any two locationsx

,

y

2

Rsuch thatx

>

y.

Step1. LetRM

,

R0N

,

RK

,

R0Lbe four peak-unanimous preference

profiles such thatp

(

RM

)

=

p

(

R0_N

)

=

x

,

p

(

RK

)

=

p

(

R0_L

)

=

y

,

|

M

| = |

N

|

,

|

L

| = |

K

|

, and both pairs

(

RM

,

RK

)

and

(

R0N

,

R0L

)

are

non-overlapping. By efficiencyf1(RM

)

=

f1(R_N0

)

= {

x

}

andf1(RK

)

=

f1

(

R0L

)

= {

y

}

. By peak-only and anonymity, we get thatf1

(

RM

,

RK

)

=

f1(R0_N

,

R0_L

)

⇢

{

x

,

y

}

, where the set inclusion follows becausefis

a priority rule.

Pick an

(

n1,n2)

2

Z2₊

\

(

0

,

0

)

, whereZ+is the set of

nonneg-ative integers. Ifn1

>

0 andn2

=

0, then letRN1 be any

peak-unanimous profile such thatp

(

RN₁

)

=

xand

|

N1

| =

n1. Ifn1

=

0

andn2

>

0, then let RN2 be any peak-unanimous profile such

thatp

(

RN₂

)

=

yand

|

N2

| =

n2. Ifn1,n2

>

0, then let

(

R0_N1

,

R0_N2

)

be any pair of peak-unanimous and non-overlapping profiles such thatp

(

R0_N1

)

=

x

,

p

(

R0_N2

)

=

y

,

|

N1

| =

n1and

|

N2

| =

n2. Define

gxy

(

n1,n2)

=

8

<

:

f1(RN1

),

ifn2

=

0

f1(RN2

),

ifn1

=

0

f1

(

R0N1

,

RN02

),

ifn1

,

n2

>

0.

The argument in the previous paragraph implies thatgxyis a

well-defined function over the domainZ2₊

\

(

0

,

0

)

.

Now, we extend the domain ofgx,yfromZ2₊

\

(

0

,

0

)

toQ2₊

\

(

0

,

0

)

,

whereQ+is the set of nonnegative rational numbers. For any

posi-tive integerndefinegxy n_n1

,

n_n2

=

gxy

(

n1

,

n2

)

. This is well-defined

because for any two n1

n

,

nn2

=

⇣

_n0

1

n0

,

n0 2

n0

⌘

, we have

gxy

⇣

_n1

n

,

n2

n

⌘

=

gxy

(

n1

,

n2

)

=

gxy

✓

_n

⇥

n0

1

n0

,

n

_⇥

n0

2

n0

◆

=

gxy

(

n

⇥

n01

,

n

⇥

n20

)

=

gxy

(

n01

,

n02

),

where the last equality follows from reinforcement. Note that by this extension, gxy is defined for any pair of rational numbers

(

q1

,

q2

)

2

Q2₊

\

(

0

,

0

)

since any such

(

q1

,

q2

)

equals z_z1

,

z_z2 , where

z1

,

z2are nonnegative integers whilezis a positive integer.

Pick any n1

n

,

nn2

,

⇣

_n0

1

n0

,

n0 2

n0

⌘

2

Q2₊

\

(

0

,

0

)

such thatn_n1

<

n_n01₀

andn2

n

=

n0

2

n0. We argue that ifgxy n_n1

,

n_n2

= {

x

}

, thengxy

⇣

_n0

1

n0

,

n 0 2

n0

⌘

= {

x

}

. To prove this, letn˜1

˜ n

=

n0 1

n0 nn1. Now,

gxy

✓

_n0

1

n0

,

n0

2

n0

◆

=

gxy

(

n01

,

n02

)

=

gxy n

⇥

n

˜

⇥

n01

,

n

⇥

n

˜

⇥

n02

=

gxy n0

(

n

˜

⇥

n1

+ ˜

n1

⇥

n

),

n0

⇥

n

˜

⇥

n2

=

gxy n

˜

⇥

n1

+ ˜

n1

⇥

n

,

n

˜

⇥

n2

=

gxy

(

n

˜

⇥

n1

,

n

˜

⇥

n2

)

+

(

n

˜

1

⇥

n

,

0

) ,

where the second and the fourth equalities follow from reinforce-ment. However,gxy

(

n

˜

⇥

n1

,

n

˜

⇥

n2

)

=

gxy

(

n1

,

n2

)

=

gxy n_n1

,

n_n2

=

{

x

_}

(the first equality follows from reinforcement) andgxy

(

n

˜

1

⇥

n

,

0

)

= {

x

}

. Once again, reinforcement implies thatgxy

(

n

˜

⇥

n1,

˜

n

⇥

n2)

+

(

n

˜

1

⇥

n

,

0

)

= {

x

}

and so we are done.

Step2. Define

q+

₍

_x

_,

_y

₎

₌

_sup

_{

_q1

₂

_Q+

_:

_gxy

₍

_q1,1

₎

_{= {}

_y

_}}

q

(

x

,

y

)

=

inf

{

q1

2

Q+

:

gxy

(

q1,1

)

= {

x

}}

.

We argue that

₁

>

q+

(

x

,

y

)

=

q

(

x

,

y

) >

0. It is easy to see

thatq+

₍

_x

_,

_y

_{) <}

₁

_{since sovereignty implies that there exists an}

integern1

>

0 such thatgxy

(

n1,1

)

= {

x

}

and the last result in

Step 1 implies thatgxy

(

q1

,

1

)

= {

x

}

for all rationalq1 n1.

Like-wise,q

(

x

,

y

) >

0 since sovereignty implies that there exists an

integern2

>

0 such thatgxy

(

1

,

n2

)

= {

y

}

. However,gxy

(

1

,

n2

)

=

gxy

⇣

₁

n2

,

1

⌘

and so the last result in Step 1 implies thatgxy

(

q1

,

1

)

=

{

y

_}

for all rationalq1



_n1₂.

It must be thatq+

₍

_x

_,

_y

₎

_

_q

₍

_x

_,

_y

₎

_{because otherwise there}

exists aq1

2

Q+such thatq

(

x

,

y

) <

q1

<

q+

(

x

,

y

)

. Ifgxy

(

q1,1

)

=

{

x

}

, then the last result in Step 1 implies thatgxy

(

q0₁

,

1

)

= {

x

}

for all

q0₁

_>

_{q1and therefore we must haveq}+

₍

_x

_,

_y

₎

_

_{q1, a contradiction.}

Similarly, ifgxy

(

q1

,

1

)

= {

y

}

, thengxy

(

q01

,

1

)

= {

y

}

for allq01

<

q1

and therefore we must haveq1



q

(

x

,

y

)

, a contradiction. Now,

supposeq+

₍

_x

_,

_y

_{) <}

_q

₍

_x

_,

_y

₎

_{and letq1}

₂

_{Q+such thatq}+

₍

_x

_,

_y

_{) <}

q1

<

q

(

x

,

y

)

. By definition ofq+

(

x

,

y

)

it must be thatgxy

(

q1

,

1

)

=

{

x

}

, whereas by definition ofq

(

x

,

y

)

it must be thatgxy

(

q1

,

1

)

=

{

y

_}

, a contradiction. Hence, we conclude thatq+

₍

_x

_,

_y

₎

₌

_q

₍

_x

_,

_y

₎

_.

Defineq

(

x

,

y

)

=

q+

(

x

,

y

)

=

q

(

x

,

y

)

andq

(

y

,

x

)

=

_q(x1_,y).

Next, define the binary relationBas follows. Ifq

(

x

,

y

)

is irrational,

thenxandyare not comparable forB. Ifq

(

x

,

y

)

is rational and

gxy

(

q

(

x

,

y

),

1

)

= {

x

}

, thenxBy, whereas ifq

(

x

,

y

)

is rational and

gxy

(

q

(

x

,

y

),

1

)

= {

y

}

, thenyBx.

Step3. Pick any two peak-unanimous and non-overlapping pro-filesRMandRLsuch thatp

(

RM

)

=

xandp

(

RL

)

=

y. Sincefis a

pri-ority rule associated with , we know thatf1

(

RM

,

RL

)

= {

x

}

()

RM RL.

In Step 1, we have argued thatf1(RM

,

RL

)

=

gxy

(

|

M

|

,

|

L

|

)

=

gxy

⇣

|M| |L|

,

1

⌘

.

By definition of the function q

(., .)

, it follows that if |M_|L||

>

q

(

x

,

y

)

(or equivalently _|M|L|_|

<

q

(

y

,

x

)

), thengxy

⇣

|M| |L|

,

1

⌘

= {

x

}

and thereforeRM RL. Similarly, if|_|M_L||

<

q

(

x

,

y

)

(or equivalently |L|

|M|

>

q

(

y

,

x

)

), thengxy

⇣

|M| |L|

,

1

⌘

= {

y

}

and thereforeRL RM.

Finally, if|M_|L||

=

q

(

x

,

y

)

(or equivalently_||_ML|_|

=

q

(

y

,

x

)

), thenxB

y

()

gxy

(

q

(

x

,

y

),

1

)

= {

x

}

()

RM RL.

Step 4. Next, we argue thatq

(

x

,

z

)

=

q

(

x

,

y

)

q

(

y

,

z

),

8

x

6

=

y

6

=

z. Let

⇣

n1(n)

˜ n1(n)

⌘

1

n₌1be a sequence of rational numbers such

that n1(n)

˜

n1(n) q

(

x

,

y

)

and limn!1

n1(n)

˜

n1(n)

=

q

(

x

,

y

)

. Similarly, let

⇣

_n

2(n)

˜ n2(n)

⌘

1

n=1be a sequence of rational numbers such that n₂(n)

peak-unanimous and non-overlapping profiles such thatp

(

RMn

)

=

x

,

p

(

RLn

)

=

yandp

(

RKn

)

=

z, and

|

Mn

| =

n

⇥

n1

(

n

)

⇥

n2

(

n

)

+

2

⇥

n1

(

n

)

⇥

n2

(

n

),

|

Ln

| =

n

⇥

n2

(

n

)

⇥

n

˜

1

(

n

)

+

n2

(

n

)

and

|

Kn

| =

n

⇥

n

˜

1

(

n

)

⇥

n

˜

2

(

n

)

. Consider the problem

(

1

, (

RMn

,

RLn

,

RKn

))

. We have |Mn|

|Ln|

=

n⇥n₁(n)⇥n2(n)+2⇥n1(n)⇥n2(n) n⇥n2(n)⇥˜n1(n)+n2(n)

>

n₁(n) ˜

n1(n) q

(

x

,

y

)

and |Ln|

|Kn|

=

n2(n) ˜ n2(n)

+

n2(n)

n⇥˜n1(n)⇥˜n2(n)

>

q

(

y

,

z

)

. Therefore, from the argu-ments in Step 3, it follows thatRMn RLnandRLn RKn. Then we must haveRMn RKnsince is a priority, which is almost transi-tive. This implies that|Mn|

|Kn| q

(

x

,

z

),

8

n. However, limn!1 |Mn| |Kn|

=

q

(

x

,

y

)

q

(

y

,

z

)

, and thereforeq

(

x

,

y

)

q

(

y

,

z

)

q

(

x

,

z

)

. We can

sim-ilarly argue thatq

(

z

,

y

)

q

(

y

,

x

)

q

(

z

,

x

)

H)

_q(z1_{,x) q(z}1_,y)q(y1_,x)

H)

q

(

x

,

z

)

q

(

y

,

z

)

q

(

x

,

y

)

and therefore we must haveq

(

x

,

z

)

=

q

(

x

,

y

)

q

(

y

,

z

)

.

Step5. Finally, we argue thatBis asymmetric and transitive. As defined,Bis clearly asymmetric and compares any two distinct lo-cationsxandysuch thatq

(

x

,

y

)

is rational. We show that it is also

transitive. Supposex

6

=

y

6

=

zare such thatxByandyBz. This implies thatq

(

x

,

y

)

andq

(

y

,

z

)

are rational numbers. Letq

(

x

,

y

)

=

n1 ˜

n1 andq

(

y

,

z

)

=

n2 ˜

n2. LetRM

,

RLandRK be peak-unanimous and non-overlapping profiles such thatp

(

RM

)

=

x

,

p

(

RL

)

=

yand

p

(

RK

)

=

z, and

|

M

| =

n1

⇥

n2

,

|

L

| =

n2

⇥

n

˜

1and

|

K

| = ˜

n1

⇥

n

˜

2. Consider the problem

(

1

, (

RM

,

RL

,

RK

))

. We have|M_|L||

=

q

(

x

,

y

)

and

|L|

|K|

=

q

(

y

,

z

)

. Sincex B ywe haveRM RL, and sincey B z we haveRL RK. However, is almost transitive and therefore it must be thatRM RK. This happens only if either |_|M_K||

>

q

(

x

,

z

)

or|M|

|K|

=

q

(

x

,

z

)

andxBz. But||MK||

=

nn˜11

⇥

n2 ˜

n2

=

q

(

x

,

y

)

q

(

y

,

z

)

=

q

(

x

,

z

)

. So it must be thatxBz. ⌅

5. Concluding remarks

Richness: We conclude by illustrating the richness of the class of rules associated with weighted majoritarian priorities. A simple majoritarian priority (Example 3) is a weighted majoritarian pri-ority if and only if it uses the same tie-breaking rule across all in-difference classes and this tie-breaking rule is defined by a strict complete orderBonRsuch that for anynand any peak-unanimous and non-overlapping profilesRL

,

RK

2

Tn, we haveRL nRK

()

p

(

RL

)

B p

(

RK

)

.11The same is true for a two-regions majoritarian priority (Example 6), i.e., it is a weighted majoritarian priority if and only if there exists a strict partial orderBonRsuch that for any

v

and any peak-unanimous and non-overlapping profilesRL

,

RK

2

Tv, we haveRL vRK

()

p

(

RL

)

Bp

(

RK

)

.12Thus, in particular,

11 The correspondingqis such thatq(x,y)=1,8(x,y)2R2_.

12 The correspondingqis such thatq(x,y)=1/ x0ifx<x0y,q(x,y)= x0 ify<x0xandq(x,y)=1 if eitherx,y<x0orx,y x0. Note that if x0is a rational number, the orderBis complete.

the left majoritarian, right majoritarian, left-two-regions majori-tarian, and right-two-regions majoritarian priorities are weighted majoritarian priorities. Similarly, the centralist majoritarian prior-ity (Example 7) withusuch that there exists a decreasing positive function

(

d

)

such thatu

(

n

,

d

)

=

n

(

d

)

is a weighted

majoritar-ian priority if the same tie-breaking rule is used across all indiffer-ence classes and is defined by a strict partial order onR.13 Extensions: It is clear fromExamples 5and8that dropping either anonymity or peak-only from the characterization offered in The-orem 3leads to a non-trivial enlargement of the class of rules. We offer a discussion on this issue in an online supplement where we also provide some partial characterizations of the classes of rules obtained when one drops either of the aforementioned axioms, or both.14

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13 The correspondingqis such thatq(x,y)= (|y x0|)

(|x x0|),8(x,y)2R 2_.