Ledyard - The Pure Theory of Two-Candidate Elections

8/3/2019 Ledyard - The Pure Theory of Two-Candidate Elections

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P ubl i c Cho ic e 44: 7 - 41 ( 1984) . 1 98 4 M a r t i n u s N i j h o f f P u b l i sh e r s , D o r d r e c h t . P r i n t e d i n t h e N e t h e r l a n d s .

The pure theory of large two-candidate elections

JOHN O. LEDYARD*

INTRODUCTION

A long-standing goal of political theorists has been the development ofa coherent, consistent, and nonvacuous theory of elections, particular lyof those using majority rule. If possible, such a theory is to be basedon rational individual and group behavior. In spite of extensiveeffort, recent writings (see, for example, Ordeshook and Shepsle, 1982)reveal that many may now be prepared to give up this research program onthe grounds that no such model exists. There appear to be two mainstumbling blocks to a consistent theory based on the rational behaviorof participants: (1) the theoretical proposition that, given anyreal ist ic assumption about the cost of voting, rational voters wi ll notpart icipate in elections, and (2) even i f they vote, majority-ruleequi libr ia rarely exist. The First result is obviously contradicted by

* N o r t h w e s t e r n U n i v e r s i t y . T h e r e s e a r c h f o r t h i s p a p e r h as b e e n s u p p o r t e d b y N S Fg r a n t S E S 8 1 0 6 8 9 6 w h i c h i s g r a t e f u l l y a c k n o w l e d g e d . T h e p a p e r i t s e l f w a s p r e p a r e d f o rp r e s e n t a t i o n a t t h e C o n f e r e n c e o n P o l i t i c al E c o n o m y a t C a r n e g i e - M e l l o n U n i v e r s i t y i n J u n e1 9 8 3 . W i t h o u t t h a t i n c e n t i v e i t m i g h t st i ll b e u n w r i t t e n . I h a v e h a d t h e h e l p a nd a d v i c eo f m a n y p e o p l e - - s o m e o f w h o m d i s a gr e e w i t h t h e c o n c l u s i o n s . A m o n g t h e s e w e r e t hep a r t i c i p a n t s i n w o r k s h o p s a t Ca l T e c h , l o wa , T u l a n e , I n d i a n a , S t a n f o r d , a n d N o r t h w e s t e r nU n i v e r s i t y . T h e e x a c t i n g s t a n d a r ds o f H o w a r d R o s e n t ha l a r e r e s p o n s i b l e f o r m u c h o fw h a t e v e r q u a l i t y i n p r e s e n t a t i o n e x i s t s . I t h a n k h i m f o r t h i s h e l p a n d a d v i c e . P e t e rC o u g h l i n f o u n d e a r l y l a p s e s . A n e a r l y v e r s i o n w a s p r e s e n t e d a t t h e P u b l i c C h o i c e S o c i e t ym e e t i n g s , a s m y p r e s i d e n t i a l a d d r e s s , i n 1 9 8 1 .


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the facts; the second means that the theory as we know i t isfundamentally flaw ed . Faced with these resu lts , those who have notgiven up on po li ti cal theory altogether have gone in two other obviousdirections. They have either given up on "rat ional" behavior (see, forexample, Hinich et a l . , 1972; Coughlin, 1979), or they have given up on"equilibrium" models and have turned to "process" models (for example,Kramer, 1977.)

I t is my belie f that th is ret reat is premature. In particula r, Iintend to show in th is paper that even under assumptions of extremelyrational behavior, i t is possible to combine voters, who may or may notvote depending on the benefits and costs, with candidates who gameagainst one another and end with equil ib ria which not on ly ex is t but

Figure I

NO ABSTENTIONS

ABSTENTIONS

RATIONALCHOICE

TRADITIONAL

THIS PAPER

PROBABILISTICVOTING

COUGHLIN

HINICH, LEDYARD& ORDESHOOK


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also have a remarkable soc ia l -w el f ar e pro pe rty. The approach is as t ra igh t for wa rd extens ion of the now s tandard spat ia l com pet i t ion modelo f e lec t i ons . Voters have pre ferences (u t i l i t y func t ion s ) over i ssues ,candidates choose a pla t form (a po int in the issue space), and thenvoters vo te fo r th e i r most p re fer red ca nd ida te- - i n two-cand ida te e-le c t i o n s - - i f and only i f the expected bene f i ts f rom so doing outweighthe costs. Given th is vo ter be havior , candidates are assumed to maxi-mize expected p lu ra l i ty (a very good approx imat ion of the pro ba b i l i ty ofw inn ing) . A fu l l general e qu i l ibr iu m occurs when no vo ter or candidatew ishes to a l te r h i s s t ra tegy .To show where the theory posed in th i s paper f i t s i n to the l i t e ra -tu re on the theory o f ma jo r i t y - ru le e le c t i o ns , I re fe r the reader toF igure i in which ex is t in g theo r ies are d iv ided in to fo ur "boxes" de-pending upon the assumptions concerning vot in g beh avior . In the tr a -d i t io na l theory i t i s assumed tha t a l l vote (no abste nt ions) and tha tchoice behav io r i s ra t i on a l (some fo rm o f u t i l i t y max im iza t ion) . I t i sth i s theory fo r which e qu i l i b r i a ra r e l y e x i s t . H in ich e t a l . changedboth of these beh avioral hypotheses by al low ing ab sten t ions due toind i f fe ren ce , a l ien at ion , etc . and by modeling the dec is ion to vote aspr ob ab i l i s t i c wh i l e l eav ing the choice o f cand ida te based on u t i l i t y .A lt hough equ i l i b r i a ex i s t i n t h i s m od i f i ca t i on , vo t i ng behavi or issomewhat ad hoc and ce rt a in ly not ra t io na l-ch oic e-b as ed . Coughl inmainta ined the t ra d i t io n a l assumption of no absten t ions but removed thevo te r 's choice of candidate from ra t io na l theo ry . Ins tead, he adoptedthe dec is ion- th eore t ic - f ram ewo rk of Luce (1959, 1977 ) by assuming tha tc ho ic e i s p r o b a b i l i s t i c , w h er e p r o b a b i l i t i e s a re p r o p o rt io n a l t o u t i l i -t y . W ith th i s mode l o f vo te r behav io r , e qu i l i b r i a ex i s t and have noi n te r e s t i ng w e l fa r e p r ope r t y a l b e i t d i f f e r e n t from tha t i n t h i s paper.I t is not known what occurs in Co ugh l in 's models i f abs ten t ions areal lowed. I

The model in th is paper depar ts f rom the t r a d i t io n a l by a l lowing

I S t o p P r e s s : I h a v e r e c e n t l y s e e n C o u g h l i n (~9 8 3 b ) i n w h i c h a b s t e n t i o n s a r e a l l o w e db u t o n l y o n a n a g g r e g a t e b a s i s . I n d i v id u a l b e h a v i o r r e m a i n s u n s p e c i f i e d .


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ra ti on al abstention behavior. 2 We begin by recognizing the obvious fa ctthat when voters make the decis ion to vote, they do not know how manyothers have voted, or plan to vote, or , es pec ia lly, how the others havevoted. They face a decision - - or be tter , a game -- under uncertain tysim ilar in s p i r i t to a sealed-bid auction. In modeling th is simultane-ous decision problem fo r a l l voters we impose as much ra t io na l i t y aspossible -- ra tional cho ice and ra tio na l expectations -- and arr iv e at amodel in which turnout is ne ithe r the 100% nor the 0% that has tra -d it io na ll y been implied by rational-c hoice models. I t is thi s model ofthe vo ters ' behavior which cons ti tu tes the "new" component of the theoryin th is pa pe r. Most of the res t o f our model is standard, although theimpl icat ions derived from th is combination of new and old are not.

In Section I we describe the behavior of a sing le voter in much thesame way as that posed by Downs (1957), Tullock (1967), and others. InSection I I we consider the simultaneous behavior of a l l voters andpresent the equili brium concept F ir s t introduced in Ledyard (1981). InSection I l l we def ine and describe both the behavior of candidates andthe eq uil ibr ium which arises when a l l actors -- candidates and voters - -are combined into a general equi lib rium . In Section IV we explore thewelfare properties of those equ il ib ri a , in Section V we examine theexistence of equil ib rium , and conclud ing remarks are added in SectionVI.

I . THE VOTER

The voter is assumed to choose whether to vote or abstain , as we ll asfo r whom to vote , consistent with the expected u t i l i t y hypothesis. Thismodel has already received much at ten tion in the l i te ra tu re so I w i l lnot dwell on it s ra tionale but w i l l immediately turn to the notation anddefi n it io ns. The interested reader can consult Ferejohn and Fio rina(1974) for a good survey.

21 am finally answering the complaints of $1utsky (1975) about the ad hoc andunrealistic nature of our earlier paper (Hinlch eta l, , 1972). In spite of my efforts heremains unconvinced of the "reality" of the model.


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i i

We assume now th at the re are on ly two ca nd ida tes, A and B. Candi-da te A chooses a pl at fo rm which we den ote by A and ca nd ida te B chooses apla t for m denoted by B. We assume tha t the vo ter knows the can dida tes 'cho ices and has a u t i l i t y f un c t io n over a l l poss ib le p la t fo rm s , R , g ivenby u (R, x ) where x represen ts the app ropr ia te u t i l i t y parameters fo rth is vo ter . We assume througho ut tha t u is cont inuous in R. Wesometimes ca l l x t he " t ype" o f t h i s vo te r . I f t h i s vo te r dec ides to goto the po l l s , he w i l l cas t h is vo te fo r A over B i f and on ly i f u (A , x)> u(B, x ) . We assume the vot er rec eive s no consumpt ion be ne f i t f romv o t i ng . T he r e f o r e , whe the r t h i s v o t e r w i l l v o te i ns tead o f abs t a i n i ngdepends on a s imp le be ne f i t - co s t c a l cu la t io n . The expected be ne f i t sf rom vo t ing a re equa l t o t he p robab i l i t y o f a f f ec t ing the ou tcome t imest he ga in from so do ing . Le t t i n g P be t he p r ob ab i l i t y t ha t t h i s pa r t i c u -la r v ote r w i l l a l t e r the outcome, and assuming th at u(A, x ) > u(B, x ) ,t he expect ed ben e f i t s a re ( P ) ( u ( A , x) - u ( B , x ) ) / 2 . The u t i l i t y d i f f e r -ence is d iv ide d by 2, s ince a vo ter a f fe c t s the outcome only i f hecrea tes a t i e or breaks one. Assuming th at t i e s are broken by a fa i rc o i n t os s , t he ga in fr om e i t he r ev ent i s t he u t i l i t y d i f f e r en c e d i v i dedby 2. We assume th at the v ot er faces a known cos t of vo t in g equal to c> 0 and t h a t t h i s c o s t e n t e rs th e u t i l i t y c a l c u l a t i o n l i n e a r l y .The refo re, i f candida te A wins and the vote r had gone to the p o l l , her ec e i v es u ( A , x ) - c i n u t i l i t y .

I n o r de r t o c omplete t h i s model o f r a t i o na l - v o t i ng behav io r , wemust prov id e a bas is fo r the vo te r 's b e l i e fs about Pa and Pb' where P ji s the p r ob ab i l i t y t ha t c and ida te j e i t h e r t i e s t he o t he r o r l oses byone vo te . We assume, a t t h i s po in t , t ha t t he vo te r knows the p r o ba b i l i -t y t ha t a v o t e r , random ly s e lec t ed from a l l o t he r v o t e r s , w i l l v o te f o rA, vote for B, or ab sta in . (We w i l l see in the next sect io n how thesecan be es t ima t ed . ) Using t hes e p r ob ab i l i t i e s , denot ed, r es pe c t i v e l y ,Qa, Qb, and Qo, where Qa + Qb + Qo = i, i t is a stan dar d e xe rc is e t oc a l c u la t e t he p r ob ab i l i t y o f a t i e when t he r e a re n o t he r v o t e r s . I t i sa lso easy to c a lc u la te the p r ob ab i l i t y t ha t A loses to B by one vo te .Adding these we f i n d th at Pa = f (Q a' Qb) where f ( z , y) =

It,ll n! n-2 k li~-~ II n!zk=O k ! k ! n - 2 k ! zk y k (1 - z -Y ) +s k=O k + l ! k ! n - 2 k - l ! z k y k + l ( l - z - y ) n - 2 k - I


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A symmetric calcula tion yields Pb = f(Qb' Qa)" Gathering th is al ltogether we have described the voter.

A voter with characterist ics (x, c) who is faced with a choicebetween two candidates, A and B, and who thinks the probabili ty that arandomly selected voter wi ll vote for candidate j is Qj wi ll

(a) vo te for A i f c < (Pa/2)(u(A, x) - u(B, x) ) ,(b) vo te for B is c < (Pb/2)(u(B, x) - u(A, x) ),(c) abstain otherwise,

where Pa = f(Qa' Qb)' Pb = f(Qb' Qa) and f is defined above.This model assumes rat ional behavior in the form of expected u t i l i -

ty maximization, no income ef fects, no candidate specific preferencesother than the platform choice, posi tive costs of voting, and knowledgeby the voter of x, c, A, B, Pa' and Pb"

At th is point most wr iters reach an unsettl ing conclusion. "Sincethe expected benefit from voting is obviously small ( i f Qa = Qb and Qo =0 then Pa and Pb are of order of magnitude i/2n -- see Chamberlain andRothschild, 1981), and since the cost of voting is not small, no ra tion-al voter wil l ever vote in large elections. Therefore, something mustbe wrong with the theory." This is not an unreasonable conclusion butthe analysis is incomplete, since i t is based on a part ia l equilibriumview which i s simply not appropriate. I f this voter and others areembedded in a general equilibrium model, the apparent failure of ration-al choice to explain voting disappears. We turn to that task next.

I I . PJ~TIONALVOTERS' EQUILIBRIUM

We now explore what happens when voters take into account the fact thatother voters are also rati onal. The logic is simple and compelling andis contained in Ferejohn and Fiorina (1974). I f everyone is rationaland carries out the par tial equilibrium calculus in the previous sectionthen, presumably, no one wil l vote. But then the probab il ity of a ti eis 1. If thi s is true and i f these same rational , par tial equilibriumnonvoters redo thei r calculus, most wil l find that i t is now de fini te lyin thei r interest to vote; they wil l be able to determine the outcome by


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them selve s. And so on. Somewhere between no one vo t in g and everyonevo t ing l ie s a s i t ua t io n in wh ich some vo te and in wh ich the p r ob ab i l i t yo f a t i e i s cons is ten t w i th those numbers and w i th the be l ie f s o f a l lvo t e r s . I t is t h i s s t a b le , r a t i o n a l , i n t e rme d ia t e s i t u a t i o n t h a t wecap tu re in t he vo te rs ' equ i l i b r ium de f ined be low .

To c lose the pa r t ia l equ i l i b r ium model in the p rev ious sec t ion , i tremains o nly to sp ec if y how a vo te r est im ates Qa and Qb" We assume th ati t is common knowledge among a l l vo ters tha t each vo ter is r a t io na l and,the re fo r e , tha t each fo l lo ws the mode l o f Sect ion I . What is not knownto each vo te r , and never w i l l be , a re the va lues o f the o the rs ' charac-te r i s t i c s ( x , c ) . We do, howeve r, assume tha t t he d i s t r ib u t io n o f thesech a r ac te r i s t i c s i s known to a l l by the dens i t y f unc t ion s h (c ) andg( x) . That is , c and x a re indepe,~dently d i s t r ib u t e d , where g (x) is thep r o b a b i l i t y t h a t a ra nd om ly se le c te d vo t e r w i l l have ch a r a c t e r i s t i c x ,and h(c) is the p ro ba b i l i t y tha t a randomly se lected vo te r w i l l have acost o f vo t i ng equa l to c . 3

Given these dens i t ies , one can compute the p robab i l i t y tha t arandomly se lected vo te r w i l l vo te fo r a cand ida te . We a l ready know tha tt he vo t e r w i l l vo te f o r A i f and o n l y i f h er ch a r a c t e r i s t i c , ( x , c ) ,s a t i s f i e s

c < (Pa/2 ) (u (A , x ) - u (B, x ) ) .

Us ing the d en s i t ie s g and h we can compute tha t the p r ob ab i l i t y o f th isis 4

qa : f X+ ( A , B ) H ( ( Pa / 2 ) ( u ( A ' x ) - u ( B , x ) ) ) g ( x ) d xwhere X+(A,B) : { x ]u (A ,x) > u (B ,x) } and H(r ) = f~ h (c) dc. Wr i t ing th is

3The assumption of independence is made on ly f or ex po si ti on al convenience. The eagerreader can ea si ly show that co rr el at io n between c and x in a dens ity func tion li ke g (x, c)can be accommodated without destroying the results detailed below.

4Note that i f A = B, then Qa = 0 sinc e U(A, x ) = U(B, x ) , and H(O) = O.


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as Qa = t(Pa ' A, B; g, h), i t is easy to show that Qb = t(Pb, B, A; g,h) and Qo = I - Qa - Qb"

We can thus compute Qa and Qb from Pa and Pb" In the previoussection we computed Pa and Pb from Qa and Qb" A fu l ly rational voterwith fu l ly rational expectations wi ll require these calculations to beconsistent with one another and wil l be ab le to compute the values of Qand P for which consistency obtains.

Given the densities on characterist ics, h and g, and given thecandidate platforms A and B, we call a RATIONAL VOTERS'EQUILIBRIUM i f and only i f

Pa : f(Qa'Qb and Pb = f(Qb'Qa and

Qa = t(Pa'A'B;g'h ) and Qb = t(Pb'B'A ;g 'h )'

where f( , ) is define~ in Section I and t( r, s,w;g ,h ) is defined above.As an aside the reader should note that i f we were to model the

voters as playing a game of incomplete information, as is done inmodeling auctions, the three pure strategies would be vote A, vote B,and abstain, and the Bayes equil ib ria of that game would be exactly theRational Voters' Equilibrium defined above. I chose the approach abovefor its expositional simplicity.

To complete th is section, we consider several properties of therational voters' equilibrium.

PROPOSITION 1: (EXISTENCE). I f H(c)~C (that is, i f H is continuous),then a rational voters' equilibrium exists.

PROOF: I f A = B, then Qa = Qb = O, Pa = Pb = 1 is an equilibrium.If A ~ B, t h e n define the functions Pa = NI(Pa,Pb) =f(t (Pa,A,B), t(Pb,B,A)) and Pb = N2(Pa,Pb) = f( t(Pb 'B 'A)' t(Pa 'A,B))" I tis easy to show that NI and N2 are continuous in (Pa,Pb) since f ispolynomial and therefore continuous, while t is continuous in P since His by assumption. Further, NI and N2 map {0,1]x[0, i ] into i t se l f .Therefore, Brouwer's fixed-po int theorem can be applied. There is atleast one pair P* = (Pa*,Pb*) such that P* = N(P*). Let Qa* =t(Pa*,A,B ) and Qb* = t(Pb*'B 'A)" Then (P*,Q*) is a rat ional voters '


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equilibrium.Q.E.D.

PROPOSITION 2: (SYMNETRY). (Pa,Pb,Qa,Qb) is a rational voters' equi-librium given (A,B) if and only i f (Pb,Pa,Qb,Qa) is a rational voters'equilibrium given (B,A).

PROOF: Immediate.Q.E.D.

This is the f i r s t of several propositions concerning the symmetryof the model in this paper. The primary reason fo r symmetry is that wehave assumed that voters care about the platform which candidates adoptand not the name of the candidate.

The next property is of interest for it s implications about thevoting probabilities in equilibrium.

PROPOSITION 3: In any rational voters' equilibr ium, (Pa - Pb) = (Qb -Qa)F where F > = O.

PROOF: Pa Pb = f(Qa,Qb) - f(Qb,Qa) =

ll-n-~ J n! xkyk(l_x_y)n-2k-1(Qb-Qa) Z k=O k!k+1!n-2k- l!Q.E.D.

I t should be noted fo r completeness that F = 0 i f and only i f thenumber of voters is even and Qa + Qb = 1, ( i .e . , turnout is 100%).

Another interesting property of equilibrium is uniqueness, or lackthereof. We have two propositions to present, bo th of which depend onthe turnout probabilities.

Defin ition: (Maximum Turnout Probab ility ). Given thecandidates platforms, A and B, and given the dis tri butionof voters' characterist ics, we can compute an upper l imiton turnout which is independent of the par ticu lar voterequilibrium arrived at. In particula r, le t

M(A,B,g,h) = ~H((I/Z)IIu(A,x) - u(B,x)U)g(x)dx.We call M( ) the maximum turnout probability.


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We have defined M( ) th is way since M is the p robabi li ty that arandomly selected voter wi l l go to the polls i f he thinks the probabi li -ty of a t ie is i . To see this, remember that

Qa+ Qb = fX+(A,B)H((Pa/2)(u(A'x) - u(B,x) ))g(x)dx+ fX+(B,A)H((Pb/2)(u(B,x ) - u(A,x)))g (x)dx .

Let Pa = Pb = I. The observation fol lows immediately, since H is adi st ri bu ti on function and H' ~ O.

M( ) = 0 i f no rat ional voter w i ll go to the po lls even when theprobab il it y of influenc ing the elec tion is 1. For an example, assumeH(O) = 0 and le t A = B.

PROPOSITION 4: (UNIQUENESS 1). I f M(A,B,g,h) = 0 then (1,1,0,0) is theunique rational voters" equilibrium.

PROOF: Under the hypothesis, Qa = Qb = 0 fo r a l l values of Pa and Pbsince H(c) ~ H(c') whenever c > c ' . But i f Qa and Qb are 0 i t fo llowstha t Pa = Pb = i .

Q.E.D.I t would be helpful i f we were also able to exh ib it a proposition

li st in g su ff ic ient conditions for the uniqueness of the voter equ il ib ri -um when the maximum turnout probabil it y is pos it ive. Unfortunately, Ihave not yet discovered such a resu lt . I t is true, however, tha t i f thecandidates' platforms are close enough, then M is near 0 and the equi-lib rium wi l l be both unique and continuous in (A,B).

PROPOSITION 5: (UNIQUENESS 2 ).all x, and H(c)~C1 for a ll c.fo r example, i f A is near B),

Suppose M(A,B,g,h) > O, u( ,x)~C forI f M(A,B,g,h) is near O, (which is t rue,

the equil ibr ium (Pa,Pb,Qa,Qb) is unique


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and is C1 in A and B (f o r A ~ B). 5P R O O F : Let Qa(Pa) = t(P a,A ,B) and Qb(Pb) : t(P b,B,A ).

equ i l ib r ium i f and only i f P solves(P,Q) is an

Pa f(Qa (Pa),Qb (Pb)) : 0 and

Pb - f(Qb(Pb) 'Qa(P a )) = O.equations is

The Jacobian of th is system of

S = I - f l(Qa,Qb)Q~f2(Qb,Qa)Qa f2(Qa,Qb)Qbi - fl(Qb,Qa)Q~Q~ = BQa/BPa : ~fx+hi(Pa/2){u(A,x) - u(B,x)} l g(x) dx/BPa

= fx+h[(Pa/2){U(A,x) - U(B,x)}] g(x) {U(A,x) - U(B,x)} () dx

Since this integral is taken over X+, its value is positive. Similarlyfor Q~. FromLedyard (1981) we know that

fl(X,y)= (y) ZII Ik=l

n! k - l yk ( l_x_y)n-2k - ik + l ! k - l ! n - 2 k - l ! x

_ k=ln! kyk( l_x_y)n-2k-1k!k+ l !n-2k! x

_ n( l_x_y) n-I

5Qa and Qb may have discontinuous derivatives at A : B, I thank Peter Coughlin fornot{ng this in an earlier version.


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f2(x,y) : (x-y) Z k-1n! xk-lyk-l(l_x_y)n-2k.k!k-l!n-2k!

From these i t can be seen that the signs of f i are:

fl(Qa,Qb) f2(Qa,Qb)i f Qa>Qb - +i f Qa=Qb - 0i f Qa


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possible.A f inal comment seems in order about the amount of turnout pre-

dicted by th is model. We have seen that i f the maximum turnout proba-b i l i t y is 0 or i f A = B, then turnout is predicted to be O, Since wehave adopted a ra tional behavior hypothesis, one might suspect that, infa ct , turnout is never posi ti ve. Such a suspicion would be fa lse.

PROPOSITION 6: (POSITIVE EXPECTED TURNOUT). I f the maximum turnoutprobab il it y is posi ti ve , given A and B, then expected turnout is posi-ti ve in any rational voters' equilibrium.PROOF: Remember hat expected turnout is (n + I)(Q a + Qb)" Supposethat Qa + Qb = O. Then Qa = Qb = O. But f(O,O) = i . 'Therefore, Pa =Pb = 1. I t fol lows that expected turnout is then (n + I)M(A,B) > 0which is a contradiction.

Q.E.D.Corollary: I f there is a set of x, with posit ive measure, such thatu(A,x) - u(b,x) # 0 and i f H(c) > 0 when c > 0 ( i . e . , h(c) > 0 fo r al lc _>= 0), then expected turnout w i l l be posi tive in equilibrium. M( )gives an upper-bound on expected turnout.

Thus, contrary to naive expectations based on pa rt ia l equ ilib riumanalys is, a rat ional general equi libr ium consideration of voting be-havior yie lds pos itive turnout in equilibrium unless each voter refusesto vote even when he knows he is the only voter.

I I I . THE CANDIDATE AND THE ELECTION EQUILIBRIUM

In this section, we model how candidates determine thei r platforms and,therefore, the outcome of the elec tion . We begin by considering whatmotivates the candidates. Since this is a stat ic model and since wehave been assuming that platforms w i l l be implemented and that theextent of implementation does not depend on the margin of vi ct or y, i tseems reasonable to assume that these candidates care, ex post, onlyabout winning. The appropriate outcome space then is simply the two-point set [W,L} = [w in ,lose] . The simultaneous choice of platforms bythe candidates determines a probab il it y di st ri bu ti on (Ra,Rb) on th is setand the ra tional , expected uti lity-maxim izing candidate A wi l l choose


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the platform to maximize Ra*V(W + Rb*V(L . Thus, this candidate wi l lalways choose to maximize the probabil it y of winning. We assume thatthese candidates know the model of the previous section, or at least actas i f they know i t . From that model they can determine an elec tion-outcome function, or , more properly, an outcome correspondence, whichmaps pairs of platforms (A,B) into sets of 4-tuples (Pa,Pb,Qa,Qb). I tis possible for candidates to compute various implications of th eirchoices such as the probab il ity of winning.

The probability A winsGiven A,B, h(c), and g(x) , and a rat ional voters' equilibrium of a two-candidate election, the probabil ity that A wins is:

[~] zn-2k+l n+l! (QA)k+r(QB)k(I_QA_QB)n-2k-r+IRa = Zk=O r=l k+r! k! n-2k-r+1!

[ n+.__~lk20 J I k! k~+~-2k+l (QA)k(QB)k(I-QA-QB)n-2k+l

where Qa(A,B) and Qb(A,B) are the appropriate parts of a rational voterequilibrium for A,B.

Although the analysis can be carried out us ing i t , th is is a re-markably unwieldly function. To simpl ify, le t us use an approximationof R which is appropriate for large elections. (See Hinich, 1977.) I fn is large, then Qa-Qb is a good approximation for a candidate to use inplace of R . To see this , le t S = 1 i f voter i votes for A, S = 0 i fi abstains, and S = - I i f i vo tes for B. Then A wins i f and only i f

n+lZ S > O.11 This is true if and only if

n+l( ll (n + I) ) Z S. : S > O.i = ~ 1 Since the S are independently and


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iden ti ca ll y di st ribu ted, i t follows from a Law of Large Numbers that

1 i f Qa > Qblim Pr (S > O) : 1/2 i f Qa = Qbn + ~ 0 i f Qa < Qb

Therefore, maximizing Qa - Qb maximizes (in the l imit ) the probabi li tythat A wins. Us ing th is approximation, we posit the following model ofthe candidates.

An alterna tive ju st if ic at io n for the use of expected pl ur al it y, QA- QB' in place of R is that the equi librium described below is the samein both cases because there are two Candidates and symmetry. Basica lly,we can show that R ~ 1/2 i f and only i f Qa ~ Qb" S ince equi l ibr iawi l l occur only at R = 1/2 or Qa = Qb, the two object ives produce thesame equi libr ium. We adopt Qa - Qb for simplici ty.

The candidates' objectivesIn a large, two-candidate elec tion , each candidate w i l l try to maximizeexpected p lura li ty . In pa rt icular , the object ive function of candidateA is

W(A,B) = Qa(A,B) - Qb(A,B)

and that of candidate B is

V(A,B) = Qb(A,B) - Qa(A,B)

where (Pa,Pb,Qa,Qb) is a Rational Voters Equil ibr ium for the platformchoices A,B. 6 The observant reader wi l l have already noticed a po-tential difficulty with this model--a rational voters equilibrium maynot be unique and, therefore, the mapping W(A,B) may not be a

6The Rational Voters t Equilibrium is defined in Section II.


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func tion . We do have to confront t his problem, but i f W( ) and V( ) areunique, the above objective functions would point ins tan tly to theappropriate behavior for the candidates in their choice of a plat form,since this is a two-person zero-sum game for which game theorists are inagreement about a solu tion concept. We adopt the consensus so lutionconcept with modif ications because of the nonuniqueness.

The candidates' behaviorIn a two-candidate large elec tion , candidates wi l l choose platforms(A*,B*) which sat isf y

min W(A*,B*) ~ max W(A,B*) for all A

min V(A*,B*) ~ max V(A*,B) for al l B

where W(A,B) = [w lIQa - Qb = w fo r some rat ional voters' equi libr ium} ,and V(A,B) = [viIQb - Qa = v for some rational voters ' equi libr ium] . Wecall (A*,B*) a (STRONG) RATIONAL ELECTION EQUILIBRIUM.

I f W( ) and V( ) are single-valued, th is de fi n it io n corresponds tothe noncooperative equilibrium (or maximin solution) of thi s game. Dueto the modificat ion, we have called th is a strong equilibrium since, i fthe candidates choose these s trateg ies, then even i f candidate A couldchoose from the mult ip le set W( ) aft er changing her strategy , she coulddo no better than now. Weaker eq uil ibr ia may also ex is t since a r is k-averse candidate might choose to play a strategy A*, even though maxW(A,B*) > min W(A*,B*) fo r another A, in order to avoid a possible lossi f min W(A,B*) < max W(A*,B*). I have chosen the stronger version,since a more strategic candidate would notice that even i f such a lossoccurred he could regain at least a payoff of 0 by choosing A = B*.Thus, no outcome which yields less than 0 to some candidate shouldsurvive as an equilibrium. A strong equilibrium has the property thateach player receives O.PROPOSITION 7: (VALUE). I f (A*,B*) is a strong rat ional e lect ionequi libr ium then W(A*,B*) = V(A*,B*) = 0

PROOF: Given any B, since candidate A can always choose the same


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platform, B, i t must be true that min W(A*,B*) > O. Also, minV(A*,B*) >_O. But i t is easy to see that i fwEW(A*,B*) then -w~V(A*,B*) . Therefore, min W(A*,B*) > max W(A*,B*)

= O.Q.E.D.

This means that (A*,B*) is a strong equil ibrium i f and only i f maxW(A,B*) < 0 for a l l A and max V(A*,B) < 0 for al l B. Even i f thereare "weaker" eq ui li br ia , one suspects that only strong equ il ib ri a arepermanent. We, therefore, concentrate on them.

IV. EQUILIBRIUMAND OPTIMALITY

In this section we show tha t i f u t i l i t y functions are concave and havecontinuous der ivat ives in A, and costs are dist ribu ted from zero, thenal l equ il ib ri a can be characterized in a remarkably simply manner; thecandidates choose the same plat form, the chosen platform maximizesfu(A,x)g (x)dx, and no one votes. Thus, i f an equi libr ium ex ists there

is a simple maximization problem by which i t can be computed. We giveseveral examples at the end of th is section.

To show these properties of equilibrium, we need to establ ishintermediate resu lts. The f i r s t of these occurs because of the symmetryof the model; candidates are essent ia lly anonymous in al l respectsexcept their platforms.

PROPOSITION 8: (SYMMETRY). I f (A,B) is a strong rational elec tionequilibrium then so are (A,A), (B,A), and (B,B).

PROOF: Since (A,B) is an equilibrium, W(A,B) = 0 = V(A,B) fromproposit ion 4. For all D and w i f w~W(D,B) then w ~ O. For a ll D andv i f vEV(A,D) then v ~ O. Further, we know that w~W(A,B) i f and onlyi f -w~V(A,B) i f and only i f w~V(B,A). Now suppose thatw~V(B,D) for some D. Then, -w~V(O,B) which implies that w~W(D,B) and

therefore w ~ O. Thus, (B,B) is a strong rational election equi li b ri -um. The rest follows in a similar manner.Q.E.D.

Now we take up two lemmas which allow us to use calculus in theanalysis of equilibrium.


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LEMMA 1: (A*,B*) is a strong equilibrium i f and only i ffH((Pa/2)ilDil)l(D)g(x)dx ~ 0 fo r all A where D = u(A,x) - u(B*,x) and

where I(D) = 1 i f D > O, I(D) = 0 i f D = O, and I(D) = -1 i f D < O.PROOF: By definition (A*,B*) is a strong equilibrium i f and only i f

~x+H((Pa/2)(liDll))g(x)dx - ~x_H((Pb/2)(IIDil))g(x)dx ~ 0 (1)

for all A. This is true i f and only i f

~H((Pal2)(ItDIL))I(D)g(x)dx + fx_[H((Pal2)(liDi l))

- H((Pb/2)(LiDII))] g(x)DX ~ 0

for al l A. This in turn is true i f and only i f

(2)

~H((Pa/2)(llD11))I(D)g(x)dx < 0 (3)

for all A.

Statement (I ) follows fr om theStatement (2) follows~Hx_((Pa/2)(llDll))g(x)dx to and

tabl ish (3) takes more work. Ileave the converse to~H((Pa/2)(llDll))g(x)d > 0 and that

then be true that

remark after Proposition 7 above.by adding and subtractingfrom the le f t side of (1). To es-wil l prove that (2) implies (3) andthe reader. Assume that(2) is correct for some A. I t must

[H((Pa/2)IDi) - H((Pb/2)iDi)] g(x)dx < O.

Therefore, Pa < Pb" Referring to lemma 3 in Section II we see that Qb


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25

B* and i f u~C I and H~C and i f t h e ir de ri va -

d ~ H ( ( P a / 2 ) I D l l ( D ) ) g ( x ) d x /d A :~h ( (P a /2 ) ID ) ) { (dPa /dA) (D /2 ) + (Pa /2 ) (dD/dA) ] g (x )dx .

PROOF: For any A and x such th a t I(D ) ~ O, we fi n d th at

d ( H ( ( P a / 2 ) I D I l ( D ) ) / d A =h ( (Pa /2 ) ID l ) [D /2 } ( dPa /dA ) + (Pa /2 ) (dD /dA) } .

I t f o l l o w s t h a t e q u a l i t y i s a l s o t r u e i f I (D ) = 0 . 7 F r o m P r o p o s i t i o n 5o f t he p rev ious sec t i on dH/dA ex i s t s f o r a l l x , s i nce A* i s near B* .The Lemma then fo l l ow s from the Lebesque Dominated Convergence Theorem.

Q.E.D.Lemma 2 is va l id even i f A is n-d ime nsional where A is r epla ced by

A fo r i = i . . . . . n .We now have a l l the to o ls needed to e s ta b l i sh the main p r op os i t io n

o f t h i s s e c t i o n .Theorem I : G iven the d i s t r i bu t i o n o f vo te rs ' t ypes , g (x ) and h (c ) , suchth at u~C I , H~C , th e i r d er iv at iv es are bounded, h(O) > O, and u isconcave in A fo r a l l x and s t r i c t l y concave fo r some x . I f (A* ,B *) i s as t rong ra t io na l e l ec t io n eq u i l i b r iu m , then A* = B* , Pa = Pb = i , Qa = Qb: O, and A* maximizes f u( A, x) g( x) dx .

PROOF: From Pr opo s i t io n 8 we know tha t i f (A* ,B* ) i s an eq u i l i b r iu mthen so i s (A* , A* ) . We con cent ra te on the la t t e r . Suppose tha t (A * ,A*)is an e q u i l ib r i u m . We know th at max W(A,A*) < 0 fo r a l l A. From Lemmai i t must be t ru e th a t J = fH ( (P a/2 ) l lD i l ) l (x )g ( x )d x _< 0 fo r a l l A .From Lemma 2 and the f i r s t -o rd er co nd i t io ns fo r m ax imiza t ion i t must bet r u e , t h e r e f o r e , t h a t

7For ar bitr ary funct ions f(x), if f1(x) + a as x + 0 for all seq uenc es of x, thenf'(O) = a. Let A + B so that D ~ O. Then dH/dA + h(O )(~ )(d U(A ,x) /dx ) for all suchsequences.


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dJ/dA = 0 at A = A*. I f A = A* then D = O, Pa = 1, and

h(O) ~ (du(A*,x)IdA)g(x)dx = 0

Since ~u(A,x)g(x)dx is a s t r i c t l y concave function, A* maximizes thatfunction.

To fi ni sh the proof, we need to show that i f (A*,B*) is an equi-librium then A* = B*. Suppose not. From Proposition 8, bo th (A*,A*)and (B*,B*) are equi l ibr ia . Therefore, bo th A* and B* are maximizersof ~u(A,x)g(x)dx. But u is s t r i c t l y concave fo r some x which impliesthat there is a unique maximizer; that is , A* = B*.

Q.E.D.Theorem I fully characterizes the rational-election equilibrium if

i t ex is ts . In tha t equilibrium, even though no one votes -- thus a-voiding a ll the nonproductive costs of voting -- candidates are led toselect a platform which maximizes a social-welfare function, the sum ofvoters' u t i l i t i e s . The existence of voters who are on the margin ofvoting, those with costs near O, leads candidates to take the prefer-ences of these voters in to account. Because of the li ne ari ty of u t i l i t yin the costs of voting, the change in the probab ili ty that a voter w il lvote, due to a change in a candidate's posit ion , is "loc al ly " pro-portional to the extra utility received by the voter if that candidateis elected. I t is always in the in te rest of the candidates to changethei r pos iti on in the di rect ion which maximizes the "aggregate marginalu t i l i t y of the marginal voters." This l ead s them inexorably to apos ition which maximizes the aggregate u t i l i t y of the voters whose costsare minimal. 8

Because of the s imi lar i ty of th is theorem to the fundamental-welfare theorem that competitive-market equi libr ium allocations arePareto-optimal, I am finding i t d i f f i c u l t to refr ain from phrases lik e

81f types and costs are correlated~ that is if the density is g(x,c) instead ofg(x)h(c), then candidates will choose the platform A which maximizes fu(a,)g(x,O)dx.


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" the inv i s ib le hand o f the e lec to ra te . " However , the fac t tha t equ i -l ib r iu m p la t fo rm s maximize a "soc ia l u t i l i t y func t ion" should no t leadthe reader to conc lude tha t e le c t io n -e qu i l i b r iu m a l l oca t ions a re a l soPa reto -opt ima l . The next few examples help to i l lu s t r a t e th is and othe rimplications of the model.

Example I: Suppose there is a one-dimension issue space and that theclass of u t i l i t y functions which any voter can have is given by u(A,x) =-I A- xl . x is usually interpreted to be voter x's ideal platform. Forthis type of example, tra dit ional theory te ll s us that the electionequilibrium wi ll be the ideal platform o f the median voter,A : x* where ~X*g(x)dx = i /2 . Let us calculate the ra ti onal -e -

lection equilibrium. A* wi ll maximize ~u(A,x)g(x)dx = ~-IA-xlg(x)dx.I t is easy to see tha t A* w i l l also be the median of the density g(x ).The two theor ies y ie ld the same predicted-equilib rium platform, althoughturnout is predicted to be 100% by the tra di t io na l theory but 0% by thi stheory.

Example 2: Let us now look at a we ll-u se d example. Su ppo se th a tpreferenc es over a one-dimension issue spa ce are given by the Ty pe -iu t i l i t y f u n c t i o n s , u ( A ,x) = - ( A - x ) 2 . I n t h i s case t r a d i t i o n a l t h eo r ys t i l l p red ic ts tha t the p la t fo rm w i l l be the med ian vo t e r ' s i dea lp la t fo rm . The ra t io na l - e l ec t i on equ i l ib r ium is , however, the meanvo te r 's idea l p la t fo rm. That is , A* maximizes [ - (A x) 2 g(x )dx .D i f f e r en t ia t i n g , one ge ts [ -2 (A - x )g (x )dx = O. F rom th i s , we know tha t~Ag(x)dx = [x g(x )d x or a : [x g( x) dx , the mean o f g (x ) .

Th is s imp le examp le i l l u s t r a t e s tha t the re i s abso lu te l y no th ingsacred about the m edian vo te r. 9 One might ju s t as ea si ly be concernedabout th e mean or , indeed, any ot he r moment. For example, i f u = -( A -x) b then the ( b - l ) s t moment is the eq u i l ib r ium p la t fo rm . The pred ic tedequ i l ib r iu m p la t fo rm depends on the composit ion o f the c lass o f u t i l i t yfunc t ions . An impor tan t im p l ic a t io n o f th is and the pr io r example istha t f un ct i on al forms are importan t. The fun ct ion s - I x - A I and -( x -

gHin ich (1977) , Cough l in and Ni tzan [ lg 81 ) , and Cough l in (1983a) a ls o f ind the medianto be un impor tan t when unc e r ta in t y i5 inc luded in the vo t ing mode l ,


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A) 2 each represent the same ordinal ri sk-f re e preferences on the set ofA. However, they do represent di ff er en t att itudes towards risk anddif fe re nt ind if ference surfaces between c and A. These differences arereflec ted in di ff er en t equi li br ia . The intensi ty of preference for A asopposed to c, as measured by the wil lingness to vote, is what drives theresult.

One other fact to note in thi s example: a mult ip le issue space wi l lnot eliminate th is equi lib rium. I f A and x are, say, n-dimensional,then the equilibrium is the mean of the mul tivaria te di st ri bu ti ong(x). These remarks reinforce the insights of Hinich (1978) tha t (a) inthe presence of uncertain ty functional fo rms are important and (b)quadratic -loss functions can imply tha t the mean ideal platform is thetwo-candidate equi librium. Hinich bases his model on voters ' errors inperceptions of candidates. My model shows that his conclusions holdeven where errors are not present.

Example 3: Fina ll y, le t us look at a simple app lica tion of th is theoryand consider what happens i f the elec tion is held to decide the al lo-cation of a public good and the assignment of the taxes needed to payfor that good. Let u( y, l, x) be the u t i l i t y of voter x for the publ ic-good level, y, when that voter 's income is I. We assume that x and Iare independently dist ri bu ted according to r( x) and s(1). Platformswi ll be of the form ( y , t ( ) ) where the function, t( 1) , indicates the taxto be paid i f income is I. I am assuming that taxes cannot be placeddir ec tl y on the unobservable x. I f the cost of the public good is C(y),we require that f f t(1) r(x) s(1) dldx = C(y) for all platforms - - nodef ic i t or surplus f inancing is allowed. Giv en th is model, we knowthat , in a rat ional- electi on equilibrium, y and t( ) maximize f fu (y, I -t( 1) ,x )r (x )s (1 )d ld x subject to the above constraint. Letting L be theLagrangian mu lt ip li er associated with the cons traint, i t follows fromfirst-order conditions that

d[~f u(y, l-t (1) ,x) r(x )s(1 )dl dx] /dy - L[dC(y)/dy] = O,(1) f(d u(y , l - t(1 ),x ) /d l)r( x)d xs + L(1)fr(x)dx s = 0

f o r a l l I a nd


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C(y) = [ [ t(1 )r(x )s(1 )dl dx.

Let I* solve f( du (y ,I *, x) /d I) r( x) dx = L fr( x)d x. The second e-quation above implies that in equilibr ium I - t(1 ) = I* fo r all I; thatis, t ( I ) = I I* . Everyone's af te r-t ax income wi ll be iden tical --income is redist ribu ted towards the mean. Using the constrain t, we findthat everyone's aft er- tax income is I* = (RM C(y))/NR where N =f S ( 1 ) d l , R = ~ r ( x ) d x and M = f l s ( 1 ) d l . T u r n i n g t o t h e f i r s t o f th ef i r s t - o r d e r c o n d i t i o n s , i t c an be shown t h a t i f d ( d u / d l ) / d x = O, t h a t i si f d u / d l i s c o n s t a n t o v e r a l l x , th e n

andL = d U ( y , l * , x ) / d l

N f ( d u / d y ) / ( d u / d l ) r ( x ) d x : d C /d y.

This is simply the Samuelson-Lindahl f i rs t -order condition for thePareto-optimal al loca tion of the public good: the sum of marginal ratesof substi tut ion equals marginal cost. Thus , we conclude that i f thepost-tax marginal u t i l i t y of income is independent of the vo te r's type,then large two-candidate elections allocate resources efficiently.There are no "free riders " in th is situa tion. I

Examples of u t i l i t y functions fo r which d(du/dl) /dx = 0 can begiven:

U = v ( y , x ) + IU = v ( y , x ) + w ( u , l ) , a n d, a s a s p e c i a l c a s e,u = x i n y + i n I .

I leave i t to the interested reader to show that i f income and typeare not independent, then the efficiency disappears and redistributionwi l l no longer require equal post-tax income. One can also show that i fcosts of voting and income are pos it iv ely correlated, as is sometimesargued, then low-income types wi l l have a larger impact on red is t r i -

IOA side issue: since this case covers utility functions without income effects, itcovers a ll si tu at io ns covere d by the Demand-Revealing Mechanisms, The refo re, i t dominatestha t method fo r social choice,


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bution.These three examples are only a small indica tion of the powerful

use one can make of the ra ti ona l- e le ct io n equil ibrium. I am sure theeager reader can provide many more.

To prove that a l l the above is not vacuous, we move next to thequestion of existence.

V. EQUILIBRIUMAND EXISTENCE

In the tr ad it io na l theory of ma jor ity -ru le equilibrium with no ab-stentions, existence of equi lib riu m is an unusua l occurrence. Oneimpl icat ion is that we cannot re ly on theorems which assume existence.For example, loca l public-goods theories us ing the median voter shouldbe highly suspect; the resu lts are l i k e l y to be vacuous. The e q u i l i b r i -um described in th is pape r, on the other hand, exi st s in many cases.These equ i l ib r ia can po ten t ia ll y provide the foundation fo r many modelswhich make social choices by major ity -ru le elec tions.

In the la st section we proved that i f A* were a ra ti on al -e le ct io nequili briu m, then A* maximized aggregate u t i l i t y . I f we could prove theconverse, that i f A* maximizes aggregate u t i l i t y then A* is a ra tio na lelec tion equ il ib rium , we would be done since the appropriate compactnessand contin uit y conditions which ensure the existence o f a maximum (theWeierstrauss Theorem) are well-known. Unfortunately, the converse isnot true without add itional cond itions on the densit ies g and h. I~ I tis our task to delineate as much as possible the set of d is tr ib u ti on sfo r which the follow ing is true:(S) i f A* solves max ~u(A,x)g(x)dx, then A* is a ra ti on al -e lect io nequil ibrium.

I f we knew fo r wh ich (g,h) the function W(A,B) were concave inA and convex in B, with V(A,B) behaving symmetrica lly, we would bedone, since under these cond itions the game-theoretic so lution to the

lIEconomists will, notice that this phenomenon also arises when considering thewelfare theorems about competitive equilibria.


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candidates' problem is known to ex is t. Unfortunately, one cannot takethis approach. Remember that W(A,B) = Qa - Qb where Qa =~x+H((Pa/2)(D))g(x)dx and Qb = ~x-H((Pb)( -D))gix)dx and where D isconcave in A and convex in B from the concavity of u (leaving aside thebehavior of Pa and Pb for the moment). I f H is a concave function of cthen Qa is concave, but we cannot te l l about Qb which, in this instance,is a concave function of a convex function. I f H is convex, then wehave a symmetric problem since -Qb is concave, but we can say nothingabout Qa" Only i f H is li nea r, bo th concave and convex, can we discussthe concavity proper ties of W. We capture thi s in tui t ion in the nextproposition.

PROPOSITION 9: i f h( ) is the uri~orm density on [O,k}, k > O, then iS)is true.

PROOF: Let J = fH(Pa/2)IDl) l(D)g(x)dx = f( I/ k) (P a/2) ID II (D )g (x )dx =f(i /k ) (P a/2) DG(x)dx. At A*, the maximizer, le tt in g D = u(A,x)u(A*,x), we see that J = O. At any other A, J ~ O. Referring to lemmaI we now conclude that A* is a ra tional -e lect ion equi libr ium.

Q.E.D.Absolute ly no conditions have been placed on g. That is , we need

not worry abou t single-peakedness, symmetry, unimodality, or un id i-mensionality. Any density over concave-ut il ity functions can be ac-commodated. The second th ing to notice is that we have been preciseabout h. An obvious question is whether (S) is true when h is notuniform. The answer is "no" i f we require al l g to be accommodated.

PROPOSITION 10: (S) is true for al l g i f and only i f h is uniform.PROOF: The " i f " is simply Proposit ion 9. We prove the "only if "

statement.Suppose we have a nonuniform h, an A, a g and an A* such that A*

solves max~u(A,x)gdx and such that ~Hi(Pa/2)IDl) l(D)gdx ~ 0 when D =u(A,x) ~ u(A* ,x ). I f there are no such g, h, and A then we are done,since there wi l l then be no g fo r which (S) is true. I f there are sucha g, h, and A for which (S) is true , we can show that there w i l l beanother g fo r wh ich iS) is not t rue , wh ich would prove the pro-posit ion. Thus, i f we show tha t we can perturb g to g' such that ~g'Ddx

0 and ~Hi iPa /2)ID l)l (D)g'dx > O, then we wi l l have proven tha t (S) is


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false and the proposition is true.The per turbat ion works as follows. Let S be the set of x fo r

which D > 0 and H(c) > c/k, le t S2 be those x f or which D > 0 and H(c) c/k, and let S4 be theset of x such that D < 0 and h(c) < c/k. We make g larger on S1 and S3and smaller on S2 and S4 by le t ti ng g' (x ) = g(x) + e when x~S , suchthat ire I dx + ~e2 dx + ~e3 dx + ~e4 dx = O, such that e1~Ddx = e2~Ddx =e3~Ddx + e4fDdx s O, and such that el~H((Pa/2)D)I(D)dx + e2CHldx +e3HIdx + e jHIdx > O. The careful reader can check to see that as longas H is is not uniform, th is perturbat ion w i l l be possible, since thesets S wi l l be nonempty.

Q.E,D.Proposition 10 informs us that i f we want a simple-existence

theorem and we want i t to be appl icable to al l possible preferencepatterns, we must re s tr i ct our attention to uniform d ist ribut ions ofcosts. Suppose, instead, we want a theorem applicable to a ll d i s t r i -butions of costs. The answer is sim ila r to tha t in Proposition 10 --statement (S) is true fo r al l cost di st ribu tions i f and only i f weseverely re st ri ct the possible preference dist rib ut ion g. In order tosee why, let us f i r s t define the derived distr ibution of u t i l i t y di ff er -en ti al s. Let J(r) = ~X(r) g(x)dx where X(r) = [xlu(A ,x) -u(A *,x ) < r ] ,and let j( r) dr = dJ(r). Finally let 1(r) = j( r) - j (- r) for al l r > O.I t can be shown that A* maximizes ~u(A,x)g(x)dx i f and only i f

co~0 r l ( r) d r < 0 for all A. (5.1)I t can also be shown tha t expected p lura l i ty W(A,A*) =

co~0 {H ((Pa/2) r ) / r } r l (r)d r" (5.2)

Statement (S) is true when (5.1) implies tha t (5.2) is less than orequal to O. Therefore, in order fo r statement (S) to be true , thefunction H((Pa /2) r)/ r cannot weight r re la t iv el y more heavily when 1(r)> 0 than when l ( r ) < O. Notice that an equal re lat ive weighting occursexactly when H is uniform. I f we require that (S) be true for al lpossib le h, then we must not allow l ( r ) > O, fo r otherwise there wi l l be


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at least one H which weights l ( r ) inco rre ct ly . We capture a ll of th isin the following:

PROPOSITION 11: (S) is true for a l l h i f and only i f #z l ( r )d r ~ 0 fora l l z ~ O, and al l A at A*.

PROOF: (IF) Let (5.1) be true and le t z = sup { r l l ( r ) > 0} . I fl ( r ) ~ 0 for a l l r, then we are done since #(H /r) r1( r)d r ~ O.Therefore, we consider the case fo r which l ( r ) > 0 fo r some r < ~. I fz = ~, then there is a z' such that #z , l( r )d r > O. But th is is im-possible by hypothesis. Thus, we need only consider cases for which z 1

Let z2 = sup {rlO ~ r ~ z , l ( r ) > 0}. Let I I = [a l ,~ l and 12 =(z2 ,zl ). Since #z l(r )d r ~ O, i t fol lows that #i21(r)dr ~ O.Further, since H is a d is tr ib ut io n function, H(kz) ~ H(kzl) i f z ~ z Iand H(kz) s H( kZ l) i f z ~ z I .

z I~ H ( p r ) l ( r ) d r = ~ H ( p r ) l ( r ) d r

Therefore, letting p = Pa /2 ,

f~ H(pr)1(r)dr ~ H(P(Zl))~l(r)dr ~ O.+ z IOne can it er a te th is proof fo r al l z unt il z = O.

(only i f ) Suppose that f z , l ( r ) d r > 0 for some z* e O. le t H "( z )= 1 i f z ~ z* and = 0 i f z < z* . Then

f { H " (( P a / 2 ) r) / r} l ( r ) d r = f l ( r ) d r > O. But then A* cannot maximizefu(A,x)g(x)dx.

Q.E.D.In the proof of Proposition 11, we used a d is t ri bu t io n of costs,

H" which is very discontinuous at z* . We could, however, have found acontinuous H ' " which is nea r to H" and which is also appropriate.Thus, the above proof would remain applicable with minor adjustments. 12

In th is section , we have proven resu lts only about the extremeli m it s of the set of (g,h) for which rati on al-e lec tion equil ibrium

12As a side note, the cost di s tr ib u ti o n , H ' ' , used in th is proo f, wh ich assumes equalcosts of voting known to a l l , is the same di st ri bu ti on used in Ledyard (1 98 1) , The factthat t h is dis tr ib uti on causes the most d i f f i c u lt ie s for existence par t ly explains the weaktheorem in that paper,


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34exi sts . That is, we have required existence to occur eit her for al l gor fo r a l l h. I f we are wi l l ing to consider only some g or h, we shoulddo be tter . One can show that there is an open set of (g,h) fo r whichexistence obtains. In pa rt icular , i f h is almost uniform or if g isalmost "symmetric," then equil ibrium wi ll ex is t. I suspect that thereis a large set of such (g,h ), but it s precise characterization remainsan open question.

Since the resu lts of Coughlin; of Hinich, Ledyard, and Ordeshook;of Hinich; and of th is paper al l poin t to the fact that multidimensionalelection eq ui li br ia ex ist more often than suspected and that they ra relyinvolve the median voter, one might speculate whether i t is my as-sumption of ra ti on a li ty or the role of uncertain ty which derives theseresu lts. 13 I suspect uncer tainty is the key to existence and that someform of ra t io na li ty is the key to "op tima li ty ." This remains a futureresearch issue.

VI. VARIATIONSON A THEMEAs I have presented t his paper in many places, a number of issues havebeen raised which seem to be easi ly handled wi th in the framework of theabove model. Let me address these var ia tions.

( I) Income effectsIn the analysis of the rational voter I assumed that the cost of vot ingentered the voter' s u t i l i t y function line ar ly . This assumption is notnecessary and can be eliminated. In pa rt ic ula r, le t u(A,O,x) be theu t i l i t y received by the voter i f A wins and th is voter did not vote.Let u(A,c,x) be the u t i l i t y i f A wins and th is voter voted where x and care as in the origina l model. Assume that du/dc exists and is less thanzero (tha t is , an increase in the cost of voting lowers x's u t i l i t y ,ce te ri s par ibus). Although the analysis is messier than above, one canderive simi lar resu lts. For example, at an equi libr ium

1 3These re ma rks a r e mo t i va te d b y a n i n s i g h t fu l o b s e r va t i o n o f Howard Ro se n th a l .


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h(O)~UA{(A,O,x)/-Uc(A,O,x)}g(x)dx = O.

This is ide nti cal to the ea r li e r resu lt i f we "normalize" marginalu t i l i t y by the marginal u t i l i t y of voting costs at O. Th at is, i f anequilibrium exists A* = B* and A* maximizes

f[U(A,Q,x)/UA(A*,O,x)] g(x) dx.

I do not yet know how other resu lts t ra nslat e.tabli sh ing existence appears to be more d i f f i c u l t .

For example, es-

(2) Negative voting costsI am suspicious of anyone who claims to vote no matter what the issuesor how close the elec tion . In almos t every election there aref r i c t io n s , or other phenomena ignored by th is model, which cause d i f f e r -ences among the cand idates and which might l ead low-cOst voters tovote. As fa r as I can t e l l there is s t i l l no common agreement on thefacts about voter behavior. In spi te of my skepticism i t is importantfor completeness of the theory to explore what would happen to theequ ilibrium i f there were indeed voters who derive u t i l i t y from the actof voting i t s e l f . I t is easiest to model these as voters whose cost, c,is negative. In the model of the calculus of vo ting , a voter with c < 0w i l l always vote for his most preferred candidate. With th is in mindconsider now the equi lib rium in which those voters with c < 0 alwaysvote and those voters with c > 0 behave as described above. I f A = Bthen only the voters with c < 0 w i l l vote and, therefore , i f A = B inequil ibr ium i t must be true that A is the ideal platform of the medianvo te r, the median of those who always vote, i f one exis ts . (We knowfrom standard theory tha t existence can be problema tical .) I f c and xare uncorrelated and i f that median platfo rm also maximizesfu(A,x )g(x )dx, then A w i l l be the equ ilibr ium. However, i f the medianei ther does not ex is t or does not equa l the maximizer of aggregateu t i l i t y , th en we must lo ok elsewhere. I t is an open question as towhether an equi librium even exi sts in t hi s sit ua tio n and, if so, what i tis . I am not even sure whether cand idates would choose the sameplatform in equilibrium. All I can conclude so fa r is that " ir ra t io n a l"voters who derive u t i l i t y from the act of voting create an ext e rn a li ty


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36which interferes with the selection by the election of a soc ial ly de-si rab le outcome. Perhaps we should educate voters not to be "c it iz ens"but to be selfish?

(3) Minimax-regret votersSuppose some voters in the electorate use the minimax regret c r i te r ia ofSavage (made popular by Ferejohn and Fior ina 1974, 1975). The analysisremains much the same, but the conclusions are s l ight ly al te red. As Ishowed in Ledyard (1981), i f we replace Pa by I/2 in the model ofra ti onal -vot ing behavior we wi l l have modeled the behavior of a minimax-regret voter. I f one then Follows the model to i ts conclusion, one wi l lsee that , in equ ilibrium A* = B*, no one wi l l turn out, and A* wi l lmaximize fu(A, x)[g(x ) + (i /2 )g *( x) Idx where g( ) is the density of theexpected ut il ity-maximizing voters and g*( ) is the density of theminimax-regret voters. I t appears that because minimax voters do notcare about closeness, they end up being weighted at hal f that of u t i l i t ymaximizers in t hei r effect on the outcome. At the margin, when A isnear B, they react more slowly to changes in platforms and, thus, losetheir effectiveness.

(4) Vote-maximizing candidatesI t is sometimes argued that candidates care about other things than justwinning. This is another of those areas of disagreement in po l i t i ca ltheory. There is no agreement on the Factors which motivate candi-dates. Although i t is obviously of l i t t l e use to a candidate to have alarge vote i f that candidate does not win, some argue that candidatesshould want to maximize votes, not the pr ob ab il it y of winning. Severalof our conclusions change i f that is the case. First , candidates wi l lnot choose the same platform. I f they did, one of them could increasehis votes (from O) by simply moving away from the other candidate. (Ofcourse, th is could lead to an election loss .) In equil ibrium, i f oneexists, turnout will occur with vote-maximizing candidates.

(5 ) TurnoutA major issue raised by many who see th is model fo r the f i r s t time isthe lack of turnout in equi librium. Whi le I see th is as good (thedeadweight loss of voting costs is avoided), many see this as a pre-


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di ct ion of the model c le arl y contradicted by the facts. I t must beremembered that , because of the many poss ible f r i c t i o ns , actual e-lections w i l l ra re ly match th is theory. Among other things, most e-lec tions are he ld to decide several contests simultaneously and p o l i t i -cal ac t iv is ts , ignored in my model, operate to in te rfer e with the natu-ral forces. I t is true that single-issue elections with few ac ti vi st sand with low stakes have l i t t l e turnout . Examples abound but the normalschool-tax e lect ion is the obvious one. In a New Hampshire town anelection was he ld to f i l l the school board. On ly one slate was on theba l lo t . No one voted. I am not sure why the judges did not wri te inthe i r own names, but the moral is clea r; when there is no cho ice i t paysnot to go to the p oll s . 14

I am not sure what an appropriate example is for the model in th ispaper, but the fo llowing provides ease of computation. Let u = -(A-x) 2,H( ) = l-ex p( -ac) , and le t g( ) = (R)(b) exp( -bx). Going through theappropriate manipulations one ca n, somewhat te dious ly , discover th atgiven the platforms A and B, the maximum turnout M( ) is

1 + (b/(aD-b))((exp-2aDS) - ((2aD/(aD+b))exp - bS)).Here, D = (A-B)/2 and S = (A+B)/2.

I t can be eas il y shown that M is near 1 i f D,S,a,b are large. M isnear 0 i f D,S,a,b are near O. I have no idea what "reasonable" valuesof these parameters are. Does anyone want to guess?

I f one wishes to estimate equi lib rium turnout, given A and B, onemust solve the fol lowing equation; le t M(a,b,D,S) by the equation above,then solve N = M(a/N,b,D,S) for N. N/R w i l l th en be an estimate of thepercentage turnout since 1/N estimates Pa"

These are but a few of the pos s ib i l i t ie s fo r refinement of themodel. Oth er s fo llow which I think are as important, but of which Iknow litt le:

] 4 St o p P r e ss : F o r t h o s e w ho b e l i e v e t h e p r o b a b i l i t y o f a t i e d e l e c t i o n i se m p i r i c a l l y z e r o , l e t me r ep o r t the outcome o f the 1983 e le c t i on fo r the Boar d o f T r us teeso f Pasadena Community Co l le ge : Ger tmen ia , 2592 and Mi e le , 2592. Th is fo l low ed ar e c o u n t . B o th c a n d i d a te s a rg u ed a g a i n s t d r aw i n g l o t s , t h e l e g i s l a t e d a c t i o n , a s b e i n gundemoc r a t i c .


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38(a) Three-candidateelections (and multiple-candidate)(b) Pol it ical act ivists and parties(c) Candidatechoice and the role of primaries(d) Intertemporal considerations(e) Representativedemocracy and the responsiveness of the system(f) Multiple, simultaneous elections(g) Empirical estimation


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39REFERENCES

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Journal of Economic Theory, 25: 152-162.Coughlin, P.J.

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Ferejohn, J.A. and Fiorina, M.P.(1974) The Paradox of Not Voting: A Decision Theoretic Analysis,

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Ledyard - The Pure Theory of Two-Candidate Elections

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