The geometry of the uncovered set in the three-voter spatial model

1=,~

Mathematical Social Sciences 14 (198"7) 175-183 "~nrth- Holland

THE GEOMETRY THREE-VOTER

OF T H E U N C O V E R E D

SE T IN T H E

SPATIAL MODEL

Richard HARTLEY (;eneral t:'lectric C.R.D., PO Box 8, Schenectady, N } 12301, U.S.A.

D. Marc KILGOUR* lJet~arlmetzl o / ,¢lat/zemalics lt'~l,/rid Laurier University, l|aterloo, ()ntari(~. (~,,,lag/a N21 3('5 ( o m m u n i c a t e d by F.W. Roush Received 18 June 1986 Revised 1 August 1986 Recent work in social choice theory has suggested that the covering relation and its set of maximal elements, the uncovered set, can help to explain majority rule processes. Roughly, one alternati\e covers another if any alternative majority preferred to the first is also maiority preferred Io the second. This study focuses on the identification of the uncovered set m a simple spatial model of majority preference, characterizes the uncovered set geometricalb, and connects this characterization to previous results.

h e y words." Majority rule; coveiing relation; uncovered set; spatial m o d e l

O. Introduction

The general objectives of the formal theory of voting, originated bv Black (1958) and Farquharson (1969) are to describe, analyze, and explain voting processes. Much of this theory has focused on the pure majority preference relation, the basis for most actual voting processes. It was soon found, however, that the theory of pure majority preference was extremely difficult to square with observations of the real world. Because of the phenomenon of cyclical majorities (intransitivity of the majority preference relation) and other properties, majority preference frequently leads, in :heory, to 'chaos'. However such instability is rarely observed in the real world (Tullock, 1981). This situation has led to the search for deeper structures implied ov majority preference, and, in particular, to the covering relation and the uncovered set. In the original definition of Miller (1980), alternative x covers alternative v if x defeats y and, as well, defeats every alternative that v defeats. The uncovered set * l). Marc Kilgour gratefully acknowledges the financial support of the Nalural Science, and Engineerng Research Council of Canada under Grant No. A8974.

~)165-4896/87/$3.50 ,'7 ~ 1987, Elsevier Science Publishers B.V. (North-Hollandl

176

R. Hart&v, D.M. Kilgour

"

The geometry of the unco~'ere,'/ set

is the set o f alternatives not covered by any other. Miller (1980) p r o p o s e d tile ull~ covered set, which is never e m p t y , as a reasonable general solution set for majority voting processes. Shepsle a n d Weingast (1984) and McKelvey (1983) subsequently s h o w e d that, indeed, m a n y different institutional or a g e n d a procedures always I~lV¢ rise to o u t c o m e s in the u n c o v e r e d set. These findings have m a d e the uncovered ~¢1, and its refinements, the most p r o m i s i n g current a p p r o a c h to the theory of pur~ m a j o r i t y preference. Llnfortunately, the u n c o v e r e d set appears to be very difficult to identify wheel there are infinitely m a n y alternatives. T h e p u r p o s e of this paper is to develop a corn,, plete geometric p r o c e d u r e for locating the u n c o v e r e d set in what is probably the simplest non-trivial m o d e l of this sort - the spatial m o d e l with three voters. The spatial m o d e l of voting can be traced back to Downs (1957); a recent expm~, tion is Enelow and Hinich (1984). It operates by identifying preferences with distaqc¢. in a simple geometric picture. Briefly, alternatives are m o d e l e d as points in a 'policy space', usually r~". Each voter (agent) is identified with an 'ideal' point in tl~e policy space, and of two alternatives prefers the one which is closer to the ideal point. One alternative is m a j o r i t y preferred to a n o t h e r if m o r e ideal points are closet to it than to the other. This is the spatial m o d e l with euclidean (L 2) preferences. O u r objective, then, is to find the u n c o v e r e d set in a spatial model with three voters. We take the policy space to be R e. (Some results on this p r o b l e m were oh. rained by Shepsle and Weingast (1984) and McKelvey (1983~: some o f these arc repeated below in order that this p r e s e n t a t i o n be self-contained. Further properties o f the u n c o v e r e d set have been given by Feld el al. (1986).) Millet- (1980) conjectured that the u n c o v e r e d set is a 'relatively small' subset of the convex hull o f the agent ideal points. If the three ideal points are collinear (or not all distinct), the uncovered set consists exactly o f the m e d i a n ideal point. T h u s we assume below three (distinct) non-collinear ideal points. Later we consider how the uncovered set changes as tile three ideal points a p p r o a c h collinearity.

1. Preliminaries First we present the definitions and b a c k g r o u n d results which are used later. Some o f o u r definitions differ slightly f r o m the s t a n d a r d f o r m u l a t i o n s ; the differences are discussed shortly. Let o~,t~2,...,u ~ be voter ideal points in {>'. For x, ve[~ ''*, define the (strict) majority preference relation, P, by x P y iff d(x, o i ) < d ( y , ~,) for m o r e t h a n half of the ideal points, o i. (Here, d(-, • ) is euclidean (k 2) distance in P,".) If it is n o t true that x P y , write y R x . The relations P and R are not orders as they are not transitive. T h e covering relation is a transitive subrelation of P. For .v, y e D m say that x covers y x C y iff for all z such that zRx, z,Py also.

R. Hurtl
177

;i~ce xRx, x C v implies that xPv. It is easy to verify that C is a transitive partial ,~dcr. I o p a r a p h r a s e the d e f i n i t i o n , x covers v if a n y alternative \,,hich is at least L, p r e f e r r e d as .v is strictly p r e f e r r e d to v. T h e uncovered set, /7(-. ~s the set ol all ~aximal e l e m e n t s with respect to lhe covering relation, C. II should bc n o t e d that o u r definition o f m a j o r i t y p r e f e r e n c e and co~ering arc not ,~,_,icallv identical to those o f McKelvev {1983) or Shepsle and Weingasl (t984) for patial models. H o w e v e r the m a j o r i t y p r e f e r e n c e definitions coincide for almost all Lllcrnatives (in the sense o f Lebesgue m e a s u r e ) and the definitions o f c m e r i n g d i f f e r , , c n l i a l l v o n l y in the substitution o f z.Px for c.R.v. These c h a n g e s :tltect only the ~ , u n d a r i e s o f the u n c o v e r e d set - specifically, the u n c o v e r e d set in our t o r m u l a t i o l l , alx~avs closed, w h e r e a s it is usually neither closed nor open in the c o n v e n t i o n a l ~,mlework. As well, our definitions simplify c o n s i d e r a b l y the exposition beloxv. ~,nother partial o r d e r is the P a r e t o relation: x is u n a n i m o u s h ' prc terred to v, or ,v, iffd(x,~ai) is a ,.Ironger relation t h a n C, i.e. that x > v implies _vQv, so that /_/CC P~'). :knother im!~wtanl p r o p e r t y o f > , not s h a r e d by partial orders in general, is I. I) [ / y ¢ PO, then there exists .re P O such t/tat _\>. v.

P r o o f . Since v is not m a x i m a l , there exists v ' > v. l.cl

S

{ x ¢ >'" I Vi : d(x, ui)_< d(),', t~i)}.

!his set is clearly closed a n d b o u n d e d , hence c o m p a c t . T h u s there exists a point, ,-,~¢.S', which ininimizes the valtle o f f ( x ) - vi': l d(x, I~.), since c o n t i n u o u s f u n c t i o n s :~ttain their c x t r e m a on c o m p a c t sets. This xo satisfies & ~ e P ( ) a n d .v~ :, v, as n-my ~asily be verified. ~' ] T h e following p r o p o s i t i o n s~Lmmmrizes and extends our results. 11.2) UCc_ PO, a n d t f 3'¢ UC, then there exLTts x e P O such that .v( 'v.

P r o o f . F h e first s t a t e m e n t has a l r e a d y been s h o w n . S u p p o s e x C v . If x ' ¢ PO, then, t,,, (1.1), there is an x e P O such that x > . v ' . T h e n x C v ' , so x C v by transitivity. I ) e n o t e by C(x) the set o f points covered by x, and by UC(x) the c o m p l e m e n t o f ~'(_v). A c c o r d i n g to (1.2), the set o f all c o v e r e d points is U , , l,~ ('(-\), or equivalently L/C =

["'l UC(x).

i 1.3)

x ¢ PO

Now for . r e 6~m, d e n o t e by IV(x) tile set {3'e fg"]3'Rx}. Noticing that the q u a n lifier ' f o r all z.' in the d e f i n i t i o n o f the covering relation c o r r e s p o n d s to a set inter-

R. Hart/ey, D.M. Kilgour

178

/

The ge¢*metr!: of [he unc~vered vet

section, we can write C ( x ) = ~=~ we,-) W(=), where the bar denotes c o m p l e m e n l a tion. This is equivalent to

UC(x) :

U

w(z).

(I

=.e I4(_v1

T o g e t h e r with (1.3) this gives a f o r m u l a for the u n c o v e r e d set.

2. The three-voter problem N o w we specialize to the case of n = 3 voters with non-collinear ideal points in the plane, D2. In this case, the P a r e t o set, PO, is simply the (closed) triangle with lhc three ideal points as vertices. To avoid a proliferation of subscripts, these ideal points will be d e n o t e d A, B and C. For x • PO, the general form of W(x) is shown in Fig. 1. We will use lhe n o t a t i o n fA(X), or simply f~,, to represent the closed disk b o u n d e d by the circle with center A passing t h r o u g h x (the 'little circle at A'}. T h e n W(x) can be described as a union of three 'petals', W t x ) = 17.~U T~U7~ , where TA=fB(x)f"lfc(x) and similar definitions hold for T~ and Tc. (In future w¢ will a s s u m e without c o m m e n t that similar definitions hold for quantities involving B and C as hold for A.} F'rom (1.4) we have the f o r m u l a

uc x)= U w{z)u U u' z u U w(z. :.¢ T.,,

: : e T,~

:~ ~

To simplify this expression we notice that if, for all voters u,, d(v,,3,)>_d(ui, y ' l , then W(y)D W(y'). This is clear, since if xRy' then xRv. Denote by x A the point ol W(x) furthest f r o m point A. N o w until future notice, we m a k e the simplifying a s s u m p t i o n that P i s an acute-angled triangle. The point x A is then the reflection ol x in the side BC. As such, il is not only further f r o m A, but also at least as far from B and C, as any other point of TA (see Fig. 1). C o r r e s p o n d i n g statements are also true for petals Tu a n d 7~.. It follows f r o m (2.1), therefore, that (2.2}

g C ( x ) = W { x & ) U IJT(XB) U W ( X c ) .

B

xC

,,

A

C

xB

Fig. 1. W(x), for x e P O .

R. Hartl
/"

179

\

t'ig, ,~ ~ LIC(.v), f o r x c P O .

l ) e n o t e by F A ( x ) , or simply F A, the closed disk with center A a n d b o u n d a r y passing t h r o u g h x. x (the 'big circle at A'). E x p a n d i n g (2.2) fully, we get

UC(x) = ( F A CI (fB U f c )) U ( F B I"1 (f~, U f c )) U (F c, ('1 (./],,, U j i ~1).

( 2.3 )

T h e resulting area is s h o w n in fig. 2. By s t r a i g h t f o r w a r d e x p a n s i o n , or bv examin a t i o n o f fig. 2, it is seen t h a t (2.3) is equal to

UC(x) = ((Ka ('1FB) U ( F B ( / F c ) U ( F a (-I F B )) N (f,\ U /;~ U f~ ) = ( F A U/B)('1 (S~ U F(:) I"1( F A UFB) ('1 (fA U f ~ U.t;_ ).

(2.4)

f a k i n g the intersection over all x in P O , we see that U C = S.~,n S B N S c (-'1 r ] •, c

where

(fA(x) Oft~tx) U.li'(x)),

PO

so= ["1 (Fdx)U&tx)). .v e P O

In o r d e r to e v a l u a t e the last t e r m in this expression, notice that ,(,dx)Uft~(.v)Ufc(x) = _. _ ~ 2 ~ ,[ve ly
N

* ~ PO

[] (e2 {),e

R2

r e PO _

~.2 v c

= PO,

PO

i)'<-v})

180

R. ttartle.v, D.M. Kilgour / The geometry qf the uncovered ~et

by (1.1). Finally, U C = S A f-) S B f"l S c f') P O .

3. Determination o f S c

Note that any point, y, outside of FA(x)UFB(X) satisfies d ( A , . v ) + d ( B , yt d ( A , X A ) + d ( B , xu). Lel: r be the m i n i m u m value of r ( x ) = d ( . 4 , X A ) + d ( B , xl~) t,,, x ¢ P O , and let E c be the closed ellipse (with interior) with foci A and B and mai~, axis equal to r, that is, the b o u n d a r y of E c is the locus of points v such lhal d(A,y)+d(B,y)=r. T h e n , since d ( A , y ) + d ( B , y ) > r , we see that y C E c . In other words, E c c _ S c. The reader m a y guess that E c = S c, but this is not in fact th~ case. H o w e v e r we will show that E c f - I P O = S c F h P O , and hence that U( EA O E u N E c N P O . C o n s i d e r fig. 3. The points A' and B' are the reflections of vertices A and B ~t~ BC and AC respectively, Q and R are the intersections of A ' B ' with CB a n d CA awi Q ' and R' are the reflections of these points. Given a point x in . ABC, we see tha~ d ( A , x A) = d(A',x) a n d dtB, xu) = d(B',x). Hence d(A, XA) + d(B, xu) = d(A ', x) + d(B', x). This q u a n t i t y is clearly m i n i m i z e d for x lying on the line QR. By e l e m e n t a r y trigom~ metry, the m i n i m u m ~alue is d ( A ' , B ' ) = (a2-~ b : - 2 a b cos (3(~)) l -~, where (~7 is lh¢' interior angle at C, and d(B, C ) = a , etc. Let v = y ( x ) be the intersection of the bou1~ A'

B !

]Zig. 3. Construction locating some points of C(x).

R. Hartl
Kii.gour ,, [Tie ¢,eotnt'tCv ~{.I ttl~" un('~v~,r(,~[ ~c ~

I Sl

darics of FA(.V) and IV~t(x). The union, bi~xCv)U/=~(.v), has a cusp al ) (.\). (See figs. 2 and 3.) Now as x travels from Q to R along QR, we see thai d(A, y(x)) + d(B, v(x)) = d(A, a',~ ) + d( B, .vt3) = d(A',.v) + d(B', x) = d(A', B') =/ I~ other words, y(x) travels along the b o u n d a r y of the cllipse~ Zi,.. It is easily ~crilied that y ( Q ) = Q' and ) ( R ) = R', and that lhe line QR maps Io tile segment of Ihc ellipse Q ' R ' , so that all points in P O E(. will be s~vepl out t~v ~he cusp. ] h i s shoxvs that P O D E ( . = P O D S c as required. In the above discussion, it was implicitly a s s u m e d that A ' B ' intersect~ ~LABC. If ( 60 °, then clearly the line A ' B ' will pass t h r o u g h C, and if (~>6I):. then A ' B ' will pass outside the triangle. In either case, .v ( is the unique point in A B C minimizing d(A,x,,,)+ d(B, xt~). T h e n using r = d ( A , C ) + d ( g , C ) , il follows that the ~h,:He of P O lies inside E ( . In this case also, P O ( q & . POFhL, .

4. Obtuse-angled Iriangles It the triangle is not acute, it may h a p p e n that .v~, the point of li~ furthest from B, is not the reflection o f x i n AC. To see how this m a y occur, consider fig. 4. Here, V is lhe point on BC such that C A V and C A B are s u p p l e m e n t a r y (~um to 180°). If x lies inside of ~ C A V , then the furthest point of TB from B will the reflection of .v as in the acute-angled case. If, however, .v lies in zLABV, then the extension of P,A will meet the b o u n d a r y of Tu at a point on the b o u n d a r y o f . l , ( \ ) , the point q~, as s h o w n in fig. 4. This new point is clearly the point of 7i~ lurlhest from B. In this case,

U

:; ~ T .

U "'(.:)

- e Arc

/

where Arc is the arc from ~:t~ to xB. This set is shown in fig. 5 and is equal to /:~(.v) C ' l ( f ~ ( . v ) g f c ( x ) ) - : ~ , where 44 is the union of the two regJon~ c u t o f f by Arc,

-C

,~XB xB Fig..4. H'(.vt, f o r , v e P O , when AB(7 is obtuse.

182

R. Hartley, D.M. Kilgour

l'he ,geometry o f the uncovere~i xet

,i'4

~B

Fig. 5, The set U { W(.:):.:e Tl~ } in an obtuse-angled case.

the arc XBX~, and its reflection in BC. Since b o t h o f lhese regions lie outside P(), we see that

U w(z)APO= W(XB)NPO. ~.¢TB

This o b s e r v a t i o n allows the analysis to proceed as before a n d we obtain the same expression (2.5) for UC. As before, Sc('IPO will be the set of points lying inside PO and the ellipse E c traced out by the point v(x) as x moves along the locus o f points which m i n i m i z e the value o f r(x) = d(A,XA) + d(B, XB). There are two possibilities for this locus d e p e n d i n g on whether the angle A{ZB in fig. 4 is acute or obtuse. Note that A',)B = t~ + 2C.

Case 1. If 13+2C>_90 ° , then the line A ' B ' c o n s t r u c t e d as in fig. 3 passes to the right of V in fig. 4. T h e value of r(x) is m i n i m i z e d along this line, and the analysis o f Sc is the same as for acute-angled triangles.

Case 2. If i~ + 2 C < 9 0 ° , then the line A ' B ' passes t h r o u g h the interior of triangle 5 A B V . For x in this region, however, the point XB is as s h o w n in fig. 4. In this case,

r(x) = d ( A , x A) + d(B, XB) = d(A', x) + d(B,x B) = d(A', x) + d(A, B) + d(A, x). This is clearly m i n i m i z e d for points, x, on the line A A ' p e r p e n d i c u l a r to BC. The m i n i m u m value will be d ( A , B ) + d ( A , A ' ) = c + 2 c s i n l~.

R. HartIcv, D.M. Kilgour / The geometry q f the uncovered ~.;

183

5. Summary and conclusions Ftw the spatial model of three voters, A, B and C, with euclidean preferences and pure m a j o r i t y rule, the u n c o v e r e d set is U C = P O (-1 E A CI 1[ B ("1 E c ,

where E ( is an ellipse with foci A and B and with m a j o r axis given b\:

a+b,

if ( ~ > 6 0 ~'

c+2csin B

if 2,C + i ~ < 9 0 °

c+ 2csin A

if 2,C + ~ < 9 0

(b 2+a2

°

2abcos.(3C)) 1/2, otherwise.

Similar f o r m u l a e define EA and EB. It is n o t e w o r t h y that POC'IE{= PO if C>_60 °. Finally, we consider what h a p p e n s to the u n c o v e r e d set as the three ideal points, :\, B, and C, a p p r o a c h collinearity. S u p p o s e , in 2 . A B C , A increases while b and c are held fixed. As A becomes large, g and C a p p r o a c h 0 °, so that the m a j o r axis of E{. a p p r o a c h e s c, and E(. a p p r o a c h e s the line segment AB. Similarly E'B approaches the segment A C , and their intersection contains, in the limit, only the point A. This c o n f i r m s the k n o w n result (Miller, 1980) that when the three voter's ideal points are coil)near, the u n c o v e r e d set is the m e d i a n ideal point. Because a voter's ideal point remains in the u n c o v e r e d set precisely ~ h e n the angle s u b l e n d e d by the other two ideal points is at least 60 ° , we conchtde that Miller's (1980) c o n j e c t u r e that the u n c o v e r e d set excludes the m o r e ' e x t r e m e ' alternatives is c o n f i r m e d in this model.

References I). Black, The Theory of Committees and Elections (Cambridge University Press, Cambridge, 1958). -\. I)owns, An Economic Theory of Democracy (Harper and Row, New York, 1957}. J.M. Enelow and M.J. Hinich, The Spatial Theory of Voting: An Introduction I{:ambridge University Press, Cambridge, 1984). R. Farquharson, Theory of Voting (Yale University Press, New Haven, 1969). S. Feld, B. G r o f m a n , R. Hartley, D.M. Kilgour and N. Miller, The uncovered set in spatial voting games, Theory and Decision, to appear (1987). R.D. McKelvey, Covering, dominance and institution free properties of social choice, Amer. J. Political Sci. 30 (1986) 283-315. N.R. Miller, A new solution set for tournaments and majority voting, Amer. J. Political Sci. 24 (1980) 68 96. K.,\. Shepsle and B.R. Weingast, Uncovered sets and sophisticated voting outcomes with implications for agenda control. Amer. J. of Political Sci. 28 (1984) 49-74. G. J ullock, Why so much stability'?, Public Choice 37 (1981) 189-202.