Outcomes of voting and planning

Journal

of Public

Economics

16 (1981) 1-15. North-Holland

OUTCOMES

OF

Condorcet,

VOTING

Publishing

AND

Company

PLANNING

Weber and Rawls locations Pierre HANSEN*

Institut d’Economie Scientifique et de Gestion, Lille, France Facultt Universitaire Catholique de Mom, Belgium

Jacques-FranGois

THISSE

SPUR, Universik Catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium Received

March

1980, revised version

received January

1981

The paper is concerned with the comparison of the locations of a facility resulting from voting and planning procedures. It is shown that a voting location exists and coincides with that minimizing the total distance covered by the users to go to the facility when the transportation network contains no cycles. For a general network, least upper bounds on the ratio of the social values of the voting and planning locations are given in the cases where the total or the maximal travelled distance is minimized.

1.

Introduction

It is common in public choice theory that decisions regarding collective the goods are outcomes of voting and planning procedures. Not surprisingly, voting solution is seldom optimal from the standpoint of the social objective function, while the planning solution frequently discontents a large number of voters. But when do both solutions coincide? And if they do not, how large is the divergence between them ? The present paper is devoted to answering these questions in the particular case of locating a public facility. The problem is to place a facility somewhere in a transportation network along the routes of which the users are established. The benefit enjoyed by a user is a decreasing function of the distance travelled. Accordingly, it is of interest to a user to be as close as possible to the facility. Given that the to be impossible to meet population is spread over space, it appears simultaneously the wishes of all. A particular procedure is therefore required for choosing a ‘compromise’ among the candidate locations. In a democratic society, the location is expected to result from a voting process: the selected *The authors thank C. d’aspremont, C. Tapiero and one referee for helpful suggestions. Research supported by the Fonds de la Recherche Fondamentale Collective (Belgium) under Grant 2.4530.80.

0047~2727/81/000@0000/$02.50

0

1981 North-Holland

2

P. Hansen and J.-F.

Thisse, Voting and planning

location has to satisfy the property that no other site is preferred by a majority of users. In a centralized world, the benevolent dictator will resort to a different approach. Being familiar with location theory, he might choose to minimize the total distance covered by the users in going to the facility. This is the utilitarian criterion when the utility of a user is described by minus the distance travelled from his location, and a surrogate objective for this criterion in the (probable) case when the user’s utility is more general but unknown by the planner. Alternatively, it might be supposed that the benevolent dictator is concerned with the welfare of the most remote users: he will then choose a point which minimizes the largest distance covered by a user. This is the maximin criterion or a surrogate for it. In this paper, we provide a comparison of the locations associated with these procedures. The paper is organized as follows. The model is introduced in the next section. We compare in section 3 the locations obtained by voting and planning with the utilitarian objective. Cases where the users go an equal and unequal number of times to the facility are successively considered. Section 4 pursues a similar comparison when planning is made according to the equity criterion. Some remarks complete the paper.

2. The model and some preliminary

results

The model used in this paper is a classical one in location theory. The transportation side is described next. We call network any subset N of R2 which satisfies the following properties: (i) N= Uy= 1 hi([O, I]), w h ere n is a positive integer and injection from [0, l] in R*, i= 1,. . ., n; (ii) hi(8)#hi, (0’) for any if i', with i, I’m (1,. . .,a} and @,Q’E]O, l[; (iii) N is connected.

hi a continuous any

8#@,

with

The set of vertices associated with N is given by V={VEN; !liE{l,...,n}/v =hi(0) or v=h,(l)}. A subset h,([O,l]) of N is called an arc and any connected subset of hi([O, 11) a sub-arc. A route linking H,EN and njEN is defined as a smallest connected subset of N containing ni and nj; it is denoted by R{ni,nj}. In other words, a route is formed by the union of a finite number of linked arcs and sub-arcs. The shortest distance in N between ni and nj, denoted d(ni,nj), is defined by the length of the shortest route linking n, and nj; d is a metric on N. The metric space (N,d) is called transportation space. For a more detailed presentation of the above concepts, the reader is referred to Dearing and Francis (1974). Intuitively, however, let us say that routes are used to represent highways, roads or railways in the real space, while vertices correspond to agglomerations, cross-roads or railways connections. Finally, location of the facility is allowed not only at

P. Hansen

and J.-F.

Thisse,

Voting and planning

3

vertices but also at any point along a route. The demand side of the model is as follows. There is a finite set U of users; the cardinal of U is denoted IUI. A user UE U is decribed by his location V,E V and by his utility function -d(v,, .). We now turn to the concepts of equilibrium and optimal associated with the voting and planning processes respectively.

Definition

1.

A point CEN is said to be a Condorcet

Definition

2.

A point

F(w)=

Definition

3.

point (CP) iff

w E N is said to be a Weber point (WP) iff

C d(v,,w)SF(n)= UEU

A point

locations

C d(v,,n), USU

VnEN.

r E N is called a Rawls point (RP) iff

G(r)=maxd(v,,r)~G(n)=maxd(v,,n), UEU ueu

VnEN.

The CP is a point such that there exists no other location which is strictly closer to an absolute majority of users; it corresponds in terms of location to the concept of equilibrium introduced by Plott (1967) in voting theory. The WP is a point where the total distance covered by all users, or equivalently the average distance covered by one user, to go to the facility is minimized. It has been introduced in location theory by Weber (1909) but has in fact a very old tradition in geometry going back to Fermat [see Kuhn (1967) for an historical sketch]. In terms of welfare, the WP can be viewed as the utilitarian location to the extent that it maximizes the sum of the individual utility functions. Finally, the RP is a point for which the largest distance covered by a user is minimized. It is equivalent to the absolute center concept introduced by Hakimi (1964), and its name refers to the maximin criterion proposed by Rawls (1971) in equity theory. In those definitions, every user has been assumed to go to the facility the same number of times. Of course, this is an extreme case which occurs only for certain facilities (like a school). The general case arises when the number of visits changes with the user. In that formulation, we denote by h, the number of visits effectuated by Mr. u; h, E [ 1, h,,,] with h,,,= max,, u h,. The definitions of the CP, WP and RP can be extended to deal with that situation. Consider first the voting process. If each user is endowed with a number ‘of votes equal to his number of visits, a CP is defined as a point

P. Hansen and J.-F. Thisse, Voting and planning

4 c E

N for which h,,S* c h,, c usu u~(u;d(u.,n)
VnEN.

Formally, this amounts to replacing user u by h, fictitious users having a single vote. If, on the contrary, the voting procedure is based on the ‘oneman-one-vote’ principle, the definition of the CP remains unchanged. Let us now come to the planning procedure. Clearly, the definition of the WP can be extended in the following manner: F(w)=

c h,d(u,,w)sF(n)= usu

c h,d(u,,n), UPU

VnEN.

The case of the RP requires additional comments. Indeed, even with different frequencies of visit, it may be relevant in some location problems to keep the distance travelled by the farthest user unweighted. Such is the case when users may choose to go to the facility (like a public library or a swimming pool) or when the facility supplies an emergency service (like a resuscitation center). Then equality of access per visit, and not the equality of access over the time period, turns out to be the appropriate objective. On the other hand, for services whose frequentation is not (fully) decided by the user (like a hospital or a city hall), we want to have equality in the total time consumed by the users to reach the facility during the whole period. Then the distance weighted by the number of visits is to be considered. In that case, the RP is defined as follows: G(r)=maxh,d(v,,r)SG(n)=maxh,d(v,,n), usu usu

VncN.

Propositions 1 and 2 given below are the best localization theorems for the CP and WP. (By that, we mean that it is not possible to reduce the set to be explored for obtaining a CP or a WP within the present general context). Proposition to K

1.

Ifthere exists a CP and if 1U 1 is odd, then any CP belongs

Proof. Let c be a CP belonging to N - I’0 Denote by zliE I/ and VIE I/ the end points of the arc containing c, and by Ui the set of users for whom d(U,,c)=d(U,,Ui)+d(Ui,C). If IL++J-U,I, th en c would not be a CP since d(v,,q) IU,I, uj beats c. Consequently, we must have luil =lU-Uil. But th en, as U is odd, we come to a contradiction and c must belong to V. QED

P. Hansen and J.-F.

Thisse,

Voting and planning

5

Interestingly, it may obtain that no CP is contained in I/ when the number of users is even [see Wendell and McKelvey (1981) for a counter-example]. Proposition 2. There exists a WP and V contains such a point. Furthermore, any WP belongs to V when IUI is odd. For a proof, the reader is referred to Hakimi (1964). Note, in passing, that the second part of the proposition resembles proposition 1. No result comparable with the above ones holds for the RP. On the contrary, it is easy to see that an RP is usually not at a vertex but is situated between two vertices.

“6 “4

Fig. 1.

Generally, the existence of a CP cannot be established [see Demange (1981)]. The following counter-example illustrates this problem. The network is depicted in fig. 1; there are six vertices and three users located in u2, uq and us respectively. The lengths of the arcs are as indicated. Some simple calculations show that vj, with j= 1, 3 and 5, are preferred to ui, with i=2, 4 and 6, by a majority of users. Hence, we may limit ourselves to consider {v1,u~,u5}. But then, it is easy to verify that we are in the situation described in the Condorcet paradox, so that no CP exists.

P. Hansen and J.-F.

6

3. Comparison

of the Condorcet

Thisse,

Voting and planning

and Weher locations

It is a well-established fact that the non-existence of a voting equilibrium crucially depends on the intransitivity of the majority rule. Expressed within the framework of our model, this amounts to saying that the non-existence of a CP is related to the degree of cyclicity of the transportation network (see also the counter-example of fig. 1). Ultimately, the absence of cycles in N should therefore be sufficient to guarantee that such a point exists. To show it, we need the following concept. Definition

between

4. A network N is called a tree-network any pair of points of N.

iff there is a single route

It is a simple matter to find examples of real networks having the above property. Results obtained for tree-networks thus appear to be relevant for location problems. Furthermore, as shown by the following theorem, the interest of these networks is also to yield a very neat comparison of the CP and WP. Theorem

1.

the same for

Assume all users.

that N is a tree-network Then,

and that the number

Proof.

(i) Let w be a WP. 0 E U exist such that d(v,,n)
Assume

that

it is not

a CP. Then

ti~N

VUEU

is

and

(*)

and lUl>lU-OI. If w and ti belong to the same arc, set G=n; denote the adjacent vertex of w such that d(w,n)=d(w,

of visits

a WP is a CP, and conversely.

if not, let U

v)+d(v,n).

Let us show that d(v,,w)=d(v,,v)+d(v,w),

Suppose,

on the contrary,

V~EU.

(**I

that tin 0 exists such that

d(v,,w)=d(v,,v’)+d(v’,w),

where v’ is an adjacent vertex of w which containing w and 6. Hence, we would have d(v,,w)
does

not

belong

to the

arc

P. Han.srv~ trnd J.-F.

which amounts d(u,,

Thisse,

Ibring

to w) < d(u,,

ii)

(***)

since a single route exists between c’,-and fi and passes through (***) contradicts (*), so (**) holds. Furthermore, by the triangle d(u,,iT)~d(u,,w)+d(w,v),

Using

7

und planning

w and z?. But inequality, (

VUEU-0.

****

1

(**) and (****), we then obtain

since


Ifl( > IU - 01,

a contradiction. (ii) Let c be a CP. Assume that it is not a WP. In this case, Finn exists such that F(fi)
and consider the route R{c,ti} linking c and fi. As I?,ER{c,ii}, 0~10, l] may be found which verifies d(c, ~)=Od(c, ii). Given that N is a tree-network, it follows from lemma 2 of Dearing, Francis and Lowe (1976) that (1 -B)d(v,,ii),

d(c,,v)~~d(u,,c)+

V’UE U.

Hence F(L?)SfTF(c)+


(1 -&F(n) since

8>0

and

F(E)
by i7 the set of users for whom d(u,,c)=d(u,,v)+d(v,c).

Thus, we have F(v)-F(c)=

2 [d( u,,V)-d(u,,c)]

usd

=d(v,c).[/U-UI+/].

+

1 C4qo9-4c,,c)l lieu-u

8

P. Hansen and J.-F.

Thisse,

Voting and planning

As F(G)
by definition

of 0, d(u,,fi)

QED

This theorem has two main implications. Firstly, it has been shown that a CP always exists when the network admits no cycles and when the users’ utility functions are distance-dependent. Moreover, it is not difficult to verify that the majority rule associated with this framework is transitive. Hence, theorem 1 constitutes an extension of the Black theorem to the extent that this one has been proved for a very particular tree-network, namely a linear segment [see Black (1948)]. Secondly, the perfect convergence of the voting and planning procedures has been established provided the number of visits to the facility is the same for everybody. This identity remains true if each user is endowed with a number of votes equal to his number of visits when the frequencies of visit are not uniform. Therefore, in both cases, decentralization does not induce any loss of social welfare and ‘the optimal location turns out to be sustained by the vote of the users. Two remarks are still in order. First, theorem 1 does not rest on the assumption of a planar network; it holds in R”. Then, the hypothesis of a tree-network is by no means necessary for the identity of the CP and WP. Some other structures are considered in a separate paper by Hansen, Thisse and Wendell (1981). Theorem 1 ceases to be true in the case of a general network. Indeed, we already know that the existence of a CP is no longer guaranteed. Furthermore, even when a CP exists, a strong divergence between the two points may be observed. The next theorem gives the maximum value of the bias.

Theorem 2. Assume that a CP exists same for all users. Then F(c)/F(w)g3.

and that the number

of visits is the

This theorem is a particular case of theorem 4 given below, so that its proof is omitted. It clearly shows that, in the case of a general network, an important loss in transportation may result from selecting the facility location by means of a vote of the users. This is reinforced by the fact that the bound 3 is the best possible. To see it, consider the following example. The network is represented in fig. 2; there are m +4 vertices and 2m+ 1 are located at vi; users u=m,...,u=2m+l are users. Users I*= 1,. . ., u=m-1 located one at each of the vertices v2,. . ., I+,,+~. The lengths of the basic routes are indicated in the figure, where E>O is arbitrarily small. It is then easy to verifq:.that the WP is unique and given by v1 while the CP is at V m+4. Consequently, we have F(w)=m+2 and F(c)= (2--c)+ (1 -~)(m1). Hence

P. Hansen and J.-F.

F(c)/F(w)=(2-E)+

Thisse,

(1 -E)x

9

Voting and planning

m-l

for large values of m. As E can be made as small as desired, approached as nearly as wished.

the bound

can be

Vm+4

Fig. 2.

Let us now examine the case where the number of visits varies from one user to another. Then it is no longer true in general that the WP coincides with the CP when the latter is selected according the one-man-one-vote principle. Of course, a CP always exists as implied by theorem 1, but now a large divergence with the WP may arise. Again, this divergence may be bounded as is shown by the following result. Theorem

3.

Assume

that N is a tree-network.

Then, F(c)/F(w)sh,,,,,.

Proof. Let u,={u;d(v,,w)=d(u,,c)+d( c, w)} and U, = (u;d(u,,c)=d(v,, w) +d(w, c)}. As N is a tree-network, there exists a single route R{c, w} linking c and w. Furthermore, for any UE U - U, -U,, a vertex &,ER{c, w} may be found such that

and

d(u,,c)=d(u,,v,)+d(v,,c).

P. Hansen and J.-F.

10

Thisse, Voting and planning

Then. we have

c zz

hJ(~,,c)+

USUl

c

h,d(~,,C)+ ueL’-L’,

UEUZ

c ueU1

V(~,,w)+

c ueu,

2 h,d(u,>c)+ UEU,

h,d(%,w)

<

2

hAvow)

cl, -u2

C uEu-UrU2

1 h,d(c,,,rt~) +

USC12

1

UEU~C’,-“z

h,[d(u,,q,)+d(v,,c)] h,[d(u,,v,)+d(~~,W)]’

of U, and U,,

1

k,d(&i,c)

USU-LJ-02

h,d(c, w)+

=.&

h,d(u,,r) -iI2

1 h,[ld(u,,w)+d(w,c)l+ UEU?

C h,d(w,c)+ u.fJ2

+ utv-

c hL4~,,c)+4c,w)l+ USUl

by definition

c

c

h,d(q,,

w) ’

as F(c)‘F(w)’

”

utu-U1--Li2

1

h,d(w>c)

since d(t&,c)

=

1 h,d(w,c) lleU1

’

and d(C,,,w)>O

for all UEU-U,

-L’,>

given that h, E [ 1, h,,,],
QED

as IU,I~III-U,I.

This bound is the best possible. Consider, indeed, a network containing a single route linking or and ul. Users u= 1,. . . u=m are located at ur and u u=2m+l at u,; h,=h,,,z2, with u=l,...,u=m, and h,=l, with =m+l,..., u =2m+ 1. Clearly, the WP is given by v1 and the CP by ~1~. u=m+l,..., Hence, F(w) = (m + 1 )d(u,, uz) and F(c) = h,,,md(u,,

F(c)/F(w) = h,,,

for large values of m.

.,,si

I:,), so that

P. Hansen and J.-F.

For general Theorem

4.

networks

w)

11

5 1 + 2h,,,.

F(c)/F(w)

d(v,,

Voting und planning

we have the next result.

Let U = {u;d(v,,c)sd(v,,

Proof.

Thisse,

w)} d(v,,

=min

and let ri be defined by

w).

“EL’

By the triangle

inequality,

we have

d(z~,,.r.)~d(~,,,~~)+d(M’,C,)+d(v,-,C),

VUEU-U.

Hence,


c htAv,,c)+

1

h,d(v,,w)+“~~_nh,Cd(w,v,)+d(v,,c)]

usu-I.7

c utu

h,d(v,>w)+

c W(v,>c) + UElPCC

C UEU-ii

h,Cd(w,

v,-) + d(v,,

h,d(v,,w)

c)]


c “EC

by elementary

h,d(v,,

w)

arithmetic

taking

d(v,,

-

w)jU

5 1 + 2h,,, c

d(v,,w)

utii 5 1 + 2h,,,----‘V

u

’

into account

F(w)zF(c),

by definition

01

’

of 0

and as h, ECLLJ,

given the definition

from where the result follows as 1011 IU - 01.

of v,-,

QED

Again, the bound is the best possible as shown by an example similar to that presented after theorem 2, in which users located at uI make h,,, visits and those located at v~,. . .,v,+~ one visit. 1 in theorem 3 and 4 yields the We immediately see that setting h,,,= bounds obtained in theorems 1 and 2 respectively. Note, however, that theorem 1 is not limited to the statement F(c)/F(w)= 1 but also shows that a

12

P. Hansen and J.-F.

Thisse,

Voting and planning

CP exists and is identical with a WP. Clearly, even for moderate values of lo), an extremely large increase in the value of the social h,,,= objective may arise when voting procedure is used. h max (e.g.

4. Comparison

of the Condorcet and Rawls locations

We now suppose that the planner is concerned with the fate of the worstoff users. Even in the situation described in theorem 1, i.e. N is a treenetwork and the number of visits is identical among users, an RP is generally not a CP nor is a CP an RP. To illustrate, consider a network consisting of a single route with users at both endpoints; the RP will then be the middle point of this route regardless of where the CP is situated. The ratio G(c)/G(r) can however be bounded. In the following theorems, we consider directly the general case where the number of visits depends on the user. Theorem

5.

Assume

that N is a tree-network. h,d(u,,r)=h:d(u,*,r)

Proof

Let G(r)=max,,, = h,**d(v:*, c). We have. C(c) = h,**d(u,**, 2 h;*d(v,**,

by the triangle

inequality

G(c)/G(r)Sl+h

Then

G(c)/G(r)s

and

1 +h,,,.

G(c)=max,,Uh,d(o,,c)

c) r) + h:*d(r,

c),

so that d(r, c)

“““h:d($,

r) ’

Assume that h,*d(v:,r)
for any UE U. But then, we have

for any u E U, and c is not a CP. The statement from the inequality d(r, c) 5 hzd(c,*, r).

then follows

from (*) and

QED

P. Hansen and J.-F. Thisse, Voting and planning

13

A network formed by a single route with user 1 at vi and users 2 and v2 provides an example in which the bound is attained. Suppose, indeed, hi = h,,, and h, = h, = 1. The RP is situated at a distance equal to +h,,,))d(vi,r,) from vl, while the CP is at v2. Consequently, = h,,,d(vl, u2) and G(r) = (h,,,/(l + h,,,))d(u,, u,), so that C(c)/G(r) + h,,, . Let us now consider

the case of a general

Theorem

5 1 + 2h,,,.

6.

G(c)/G(r)

3 at that (l/(1 G(c) = 1

network.

Proof. Let G(r)=max,,u h,d(v,,r)=h:d(u,*,r) and G(c)=max,,uh,d(v,,c) = h,**d(u,**, c). By definition of c, UE U exists such that d(v,, c) id(u,,

r).

h,**d(v:*,

h:*d(v;*,

(*)

We have c)s

r) + h,**d(r, u,) + h,**d(v,,

given the triangle 5 h,**d(v;*,

c),

inequality,

r) +2h:*d(u,,

r),

by (*).

Hence. we obtain G(c)/G(r)

5 1+ 2

h:*d(v,,

5 1+ 2h,,,

r)

G(r) h,.d(v,, r)

5 1+ 2h,,,,

G(r)

’

since h,z

by definition

1,

of G(r).

QED

An example depicted in fig. 3 shows that this bound can be approached as closely as desired. There are seven vertices and the distances are as shown on the figure. There are five users, located at vi, us, vq, u5 and vg respectively; hi = h,,,> while h, = 1, with u = 3,. . ., 6. It is then easily seen that u2 is the RP and u, the CP. Hence, G(r)= 1 and G(c)= 1 + (2-.s)h,,, so that the ratio G(c)/G(r) can be made as close to 1+2h,,, as wished for small enough E. Interestingly, while the bounds obtained for a general network are the same for both planning procedures (see theorems 4 and 6) the bound associated with the RP is greater than that for the WP in the case of a treenetwork. To conclude, note that we have the inequalities G(c)/G(r)s2 and G(c)/G(r)s 3 when the distance travelled is not weighted.

14

P. Hansen and J.-F.

This-e,

Voting and planning

Vl

Fig. 3.

5.

Final remarks

one-man-one-vote (i) The above analysis suggests that the prevalent system could be far from ideal for selecting one location by voting. Indeed, endowing users with a number of votes proportional to their frequentation of the facility yields much lower discrepancies with respect to the optimal location, at least from the worst case point of view. However, implementing such a system could be impaired by the difficulties associated with revelation of the true number of visits made by the users. (ii) We observe that in all the situations described, the equal users are unequally treated: the access to the facility is not uniform. Such a property reminds us of the result obtained by Mirrlees (1972) in urban economics, where it is shown to be not optimal for identical people to have the same level of utility. Basically, the reason is that ‘equal’ users are not completely equal since they are ‘differentiated’ by their location [see also Rushton, McLafferty and Ghosh (1981)]. (iii) Those who are familiar with location theory know that another approach to the transportation side of the model can be proposed: any point of the plane is a possible location and the distance is derived from a norm. The most popular is the l,-norm defined by

with pz 1, x and y being points of R*. In this context, the existence of a CP seems to be improbable, which suggests to use other concepts of equilibrium [see Kramer (1974) and Demange (1981)]. Nevertheless, theorem 3.2 of Wendell and Thorson (1974) guarantees the existence of such a point when

P. Hansen and J.-F.

Thisse,

Voting and planning

15

the norm is rectilinear (p= 1). Interestingly, in this case, the CP proves to be a WP, and conversely, since it is a bi-dimensional median. For p> 1, Wendell and McKelvey (1981) have shown that a point c E R2 - {u,; u E U} is a CP if and only if there exists a mapping T from U onto U such that T[(T(u)] =IA satisfy a certain and c E CD,,,+,.,I for any UE U, i.e. the users’ locations property of symmetry about c [see also Demange (1981) for a condition which guarantees the acyclicity of the relation Q defined by xQyol{ue u;d(x,tl,)n/2]. Even in that case, the CP is not a WP, except when c is the middle of the interval [cU, ur&j for any UE CJ. Otherwise, as theorem 2 does not make use of the network structure, the bound given there still holds. (iv) Our comparison of the CP and WP can receive a nice interpretation in terms of game theory. Indeed, a CP corresponds to a non-cooperative location in the voting game while the WP can be viewed as a cooperative one in which players agree to maximize their joint benefit. Theorem 1 then states a sufficient condition for the two equilibria to be the same. Moreover, theorem 2 provides us with a measure of the maximal loss incurred in the non-cooperative situation. To the best of our knowledge, similar results are infrequent in game-theoretic models.

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