A Downsian Spatial Model with Party Activism
Author(s): John H. Aldrich
Source: The American Political Science Review, Vol. 77, No. 4 (Dec., 1983), pp. 974-990
Published by: American Political Science Association
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A Downsian Spatial Model withPartyActivism
JOHNH. ALDRICH
of Minnesota
University
parallels
A unidimensionalspatial model is proposed in thisarticle.Althoughitsformalstructure
thespatial model of electoralcompetition,thismodel examinesthedecisionsof individualsas they
choose whetheror not to become activistsin one of twopoliticalparties.An individual"calculus of
of the "calculus of voting."
participation"is developed thatis similarto thespatial interpretation
This calculus is then generalizedby examiningconditionsthat may hold for aggregateactivism
probabilities,and therelationshipbetweenthe twoforms is investigated.Some resultsare thenpreof activistsin thetwoparties. Theseresultsin generalconform
sentedwhichconcernthedistributions
to the existenceof "partycleavages," in whichthereare two stable (equilibrium)distributionsof
and relativelydistinctive
activists,such thatthetwoparties' activistsare relativelycohesiveinternally
externally.Finally,some suggestionsare offeredabout how thismodel can be combined withthe
spatial model of candidatecompetitionto provide a more completemodel of elections.
explore the relationshipsin other articles, my
questionshereare different.
The questions I ask concern political parties
and actual or potentialactivistsin one or theother
of the two parties.Specifically,givena two-party
system:a) Who is more or less likelyto become
involvedwitha politicalparty,and in whichparty
are theylikelyto be active?b) Whatare theconsequences of thesechoices; thatis, how are thepartisan activistsdistributedacross the space? Are
these distributionsstable and enduring?and c)
What consequences mightthe answersto questions (a) and (b) have for parties,policies, and
elections in the short and long runs? In this
article,I proposea model to answerquestions(a)
and (b). In the concluding section (and more
brieflybelow), I will suggestsome of the ways in
which question (c) might be explored and
answered.
Keepingin mindthatthe one long-term
goal is
to developa "threeactor" spatialmodel(i.e., one
withcandidates,citizens,and activists),themodel
of partisan activistsI propose will parallel the
usual spatial model of electionsas closelyas possible. The connections between the model of
activistsand the electoralmodel willnot be made
here. Still,it is importantto notethattheremight
Received:May 10, 1982
be some fairlystraightforward
connectionsbeNovember
Revisionreceived:
22, 1982
tweenthe two models.
Acceptedforpublication:
May3, 1983
The principalconclusionsI will reach here are
An earlierversionofthisarticlewas delivered
at the
of activistsin thetwo parties
1981AnnualMeetingof theMidwestPoliticalScience thatthedistribution
Ohio. I will be relativelycohesive within each party,
Association,April 16-18,1981, Cincinnati,
wouldliketo thankPeterAranson,RobinDorff,Mor- relatelydivergentbetweenparties,and generally
risFiorina,GaryMiller,theanonymous
and in stable equilibrium.These conclusionsseem to
reviewers,
MikeMcGinnis
fortheircriticism
andsugges- be close to the descriptionof the long-observed
especially
tions.Supportforthisresearchwas providedby the partycleavagesin theUnitedStates.At leastsince
National Science Foundation,Grant #SS-8105848.
Whereastheresearch
hasbenefited
fromthisindividual the time of McClosky, Hoffman, and O'Hara
and institutional
I
alone
am responsible
for (1960), research has shown consistentlythat
generosity,
Democratic partisans and activistsdifferfrom
itsremaining
errors.
The spatialmodel of electoralcompetitionproposed byDowns in 1957has generateda greatdeal
of subsequentanalysis. Davis and Hinich (1965)
were the firstto formalizeDowns's model and
extenditto includemorethana singlecompetitive
dimension; they, Ordeshook, McKelvey, and
many others have provided numerous further
extensionsand modifications(for reviews, see
Davis, Hinich, & Ordeshook, 1970; Riker &
Ordeshook, 1973). In short, there has been a
cumulation of knowledge about these sorts of
models. Throughoutthisdevelopment,therehas
been virtuallyunwaveringattentionto a closely
intertwined
set of questions: a) How do rational
citizensdecide how and whetherto vote in twocandidate (or two-party)elections?; b) What
spatialpositionswould rationalcandidateschoose
to offercitizens?;and c) What position(s),ifany,
would win and hence become the social choice?
I propose to use the spatial frameworkto
analyze a differentset of questions. Although
thesequestionsdo bear on thetraditionalvoting/
social choice questions of the spatial model of
electoral competition,and although I hope to
974
1983
A Downsian Spatial Model withPartyActivism
their Republican peers in predictable,enduring
ways, especiallyalong the line of the New Deal
cleavage. Page (1978) linksthesefindingswiththe
positions of presidentialnomineesand partisan
cleavages in the electorate. Aranson and
Ordeshook (1972; see also Coleman, 1972) illustratea morespecificallyspatialmodelingrelationship. There theyshowed thatif partisanactivists
are distributedn ways ratherlike those derived
here, candidateswho mightwant to convergeto
each other (and the policy center) to win the
generalelectionmightneed to divergealong the
lineof partycleavageto obtainthesupportof partisanactivistsin orderto receivethenominationin
thefirstplace. In effect,I answerin thisarticlethe
question of why partisanactivistsmightbe distributedas theseauthorsassumed themto be.
I call the model a "Downsian spatial model"
because, like Downs's own work, the model
assumes a single dimension. I also use the bynow-standard Davis, Hinich, and Ordeshook
(1970) formalizationof Downs. In the next section, I propose a spatial "calculus of participation" that parallels the Riker-Ordeshook(1968)
calculus of voting (see especially Riker &
Ordeshook,1973,or Davis, Hinich& Ordeshook,
1970, for its spatial interpretation).In this
calculus, the question is whetheror not one
becomes an activist.It does not ask how many
resources,such as time,effort,and money,any
activistcontributes;it mightbestbe thoughtof as
modelingeitherthe decisionto initiate(or maintain) a partisan affiliationor perhaps to contribute some small amount without worrying
about how much. This activismcalculus is then
generalizedto a probabilisticformulation,parallelingMcKelvey'sgeneration(1975) of the Davis,
Hinich, and Ordeshook spatial model of elections. This formulationalso permitsan interpretationin termsof how much the individualcontributesas an activist.Withthisindividual-choice
machinery,I thenturnto such macrolevelquestions as aggregatedistributions
and stability.
975
model assumes a single such dimension, as
assumed here:
Al: The space is a unidimensionalcontinuum,
XeR'.
For mathematicalconvenience,I assume that
thereare an infinitenumberof citizens,with a
typicalcitizenbeingdenotedas i. All i have preferencesdefinedoverall pointsinX, and each i has
a point of maximumpreferenceor utility,i's
"ideal point" in X, denotedxi:
A2: There are an infinitenumberof citizens;
A3: For all i, thereexistsa well-defined
utility
functionoverX, say U,(x): R1 - R1, and, for
all i, thereis some point of maximumutility,
xi, xi E X.
The set of all xi definesthedistribution
of ideal
points,fixi). I assume thatall xi are real numbers
(i.e., no one mostprefersan infinitely
positiveor
negativeamountor degreeof X), so thatthereis
some smallestand largestideal point value of X
(say, pointsa and b, respectively).2 I assume that
fixi) is nonzero, continuous,and differentiable
over the range(ab):
A4: fixi), thedistribution
of ideal pointsin X,
is a continuous,nonzero,differentiable
function over the range(a,b), a,b e R1.
At this point the descriptionof a "Downsian
spatial model" is complete. The model of electoral competitionwould add such variables as
candidates, theirpreferences,and voting rules.
The model of partyactivism,however,will turn
on different
variables.
The Individual
"Calculus of Participation"
Davis, Hinich,and Ordeshook(1970) put their
assumptionsabout citizens' voting behavior in
The Definitionof the
termsof the Riker-Ordeshookcalculus of voting.
DownsianSpatial Model
Rikerand Ordeshook(1973) argue thattheircalculus ought to extend to political participation
What makes these models "spatial" is that more generally(see especially chap. 3). I will
choices are based on the evaluation of points follow theirsuggestion,but firstadd
a bit more
as context.
along one or morecontinua,usuallyinterpreted
The
Downsian
dimensions.'
policy(or ideological)
The sortof activistI seekto modelis thatof the
mass activist,the party's rank and file. In para spatialdimension
is interpreted
'Ordinarily
as a
2The assumptionthat ideal points are real numbers
dimensionof publicpolicy(althoughDowns's own
thana specifically (i.e., noninfinite)is actually rather common (cf.,
examplewas moreof an ideological
I willfollowthisconvention,
but Kramer, 1977, who uses this assumption,indicatesits
policydimension).
generalinotethattheformal
definition
specifies
noparticular
in- commonusage, and calls themultidimensional
zation "Type I utilityfunctions").
terpretation.
976
The AmericanPoliticalScienceReview
ticular,I am notmodelingthebehaviorof theelite
activist,the careerpartyofficialor the actual or
potentialcandidate.Rather,I am consideringthe
person who might performany of the usual
measuresof campaignparticipationforthe party
(from identification to contributingmoney,
attendingrallies, and stuffingenvelopes). One
is thatthe potencrucialpresumption,therefore,
tial activistis similarto what economistscall a
"price taker." That is, theytakethepositionsand
goals of the partyas given,independentof their
behavior, just as consumerstake the prices of
goods as fixed(and, just as the actions of conso, too, do
sumersdo affectpricesinfinitesimally,
the actions of activistsaffectthe positionsof the
I am not studyingthe
parties infinitesimally).
behavior, therefore,of those activists whose
behaviormaybe predicatedon theirpotentialfor
changingthe partymaterially.I also ignorethe
careerist,whose cost-benefitcalculationsare so
vastly different.Activists here are partisans
because they desire to see the party realize its
goals; theywantto supporttheparty,not change
it.
I also assume that activistscontributeto only
one party,not to both simultaneously.Finally,
the decisionto become an activistis a potentially
long-termcommitment,a "standing decision."
To be sure, the decisionto become involvedin a
specificcampaignwilldependupon theparticular
set of candidatesstandingforelectionat thatparticulartimeand place. But, the"partisan" partof
the decision emphasizesthat the "standingdecision" depends on whereand for what the party
stands.
calculusis moreobviousThe Riker-Ordeshook
lyrelevantin thissetting.Since theyshow thatno
rationalcitizenwould vote for the less preferred
candidate, theircalculus swingson the turnout
decision,leadingto theirfamousequation:
R = PB + D - C, where
R = "Reward" forvotinginsteadof abstaining,
P = (Subjective) probabilitythat one's vote will
''count,'"
B = Differentialbenefitsfrom seeing the preferredcandidateelected,
D = "Citizen duty" (or positiverewardsassociated withvoting,independentof theoutcome),
and
C = Costs of voting.
To translateinto thiscontext,let us beginwith
theindividual'sutilityfunction,U,(x). As already
indicated,I assume that theyexistand thatthey
have a singlemaximum.The Davis, Hinich, and
Ordeshook formulationof the Downsian voting
modelinitiallyspecifiedcitizenutilityfunctionsas
Vol. 77
strictlyquadratic (cf. Davis & Hinich, 1965).
Later work examined "quadratic-based" utility
space, thisrequirefunctions.In a unidimensional
mentis that the utilityfunctionis unimodaland
symmetricabout the mode, xi. At times, the
spatial voting model examined unimodal utility
but
functionsthatwerenotnecessarilysymmetric,
were convex (i.e., monotonicallyand marginally
decreasing).Here, I willassume(unlessotherwise
noted) that all utilityfunctionsare quadraticbased. This assumptionmeans that theyare unibut that theycan be any
modal and symmetric,
particularformconsistentwiththose constraints
(e.g., they could be strictlyquadratic or they
could be shaped like a normalcurve):3
Ul: For all i, U,(x) is unimodaland symmetric
about xi, and it is of thesame functionalform
forall i.
The utilityfunctionis used to evaluatethecomof thetwoparties.In thespatial
parativeofferings
votingmodel, the comparisonis of the two candidates' positionsin X. Here, I assume thatthere
are two parties,say 0 and A, and thatall i see and
agree upon the positionon X thatrepresentsthe
policygoal of each party,saypoints0 and i,&,0, t e
X. The originof thesetwopointswillbe discussed
in a later section(briefly,I will assume that the
partyposition is based on the preferenceof its
"typical" activist):
U2: Each partyentersinto each citizen'sutilityfunctionas a singlepointin X, the "goal"
or "position" of the party,i.e., as U,(0) and
U1(&) forparty0 and I.
The comparabletermto the Riker-OrdeshookB
term,then,is the differencein the evaluationof
the two partypositions,as definedin U2.
The P termin the calculus of votingis often
as theprobabilitythatone's singlevote
intepreted
will make or break a tie betweenthe two candidates. In thecontextof partisanactivity,itis more
reasonable to adopt a less specificinterpretation
of P. Afterall, thereare any numberof offices
and electionsoccurringat numeroustimes.Thus,
theP-analog here
one mightreasonablyinterpret
as the perceivedlikelihoodthatone's activitywill
helpthepartyachieveitsgoals. In fact,evenlooking at a single office, one mighttake a more
in thissectionaredenotedbyU's (in'Assumptions
thatthey
steadoftheA 's of lastsection)to emphasize
them
concernindividualutilityfunctions,
separating
madeinthenextsection,
about"parfromassumptions
thatroughly
ty supportfunctions"(S assumptions)
parallelthese.
1983
A Downsian Spatial Model withPartyActivism
generalinterpretation,
sinceit is likelythata winner would be more effectivein implementing
the
party'sgoals withmorevotes (or a largerplurality)thanwithfewer.We can denotetheseas, for
example,poo, indicatingtheprobabilitythatparty
o willachieveits goals if individuali "abstains"
fromactivity(chooses alternativeaction o).
The C and D termswill be divided a bit differently
here,althoughthedivisionis a matterof
convenienceand not a new approach. There are,
of course, real costs associated with being an
activist, as well as psychologicaland decision
costs (relevanteven for becomingan identifier).
The costs of contributing
scarceresourcesare undoubtedlyhigherthan the costs of voting,alone.
Here, I differfrommanyspatialvotingstudiesin
themto be variablecosts,althoughitis
permitting
reasonable to assume that such costs vary inof i's ideal point,thepositionsof the
dependently
two parties,and of X in general.
The D termrefersto the benefitsof being an
activistwhichare independentof outcome.In the
votingcalculus, these are called "citizen duty,"
sincemanyfeelit theirdutyto vote,regardlessof
who winsor loses. In moregeneralterms,D type
benefitscan be consideredanyrewardone obtains
thatcomes independentof the outcome. We can
thinkof twoclassesof thesebenefits.The firstare
any of the"duty" typeof rewards,i.e., thosethat
are independentof outcome and of the goals of
thetwopartiesand theirevaluationbyi. For party
activists,such goals include the social, solidary
typeof benefits(Clark & Wilson, 1961), such as
the numerous rewards,mementos,and opportunitiesto meetthe party'spublic figures.All of
these and more can be consideredroughlyto be
independentof the positionsof the two parties.
What I assumeabout thesecostsand benefitsis
that "costs" can be considered"net costs"; that
is, the usual costs less these social and other
benefits,called ci. The notationservestwopoints.
First, I assume that, for all i, ci is greaterthan
zero. That is, costs are not outweighedby the
"particularized" or "private" benefitsof participation.Second, ci variesfrompersonto person
(but is the same for all i consideringactivityin
party0 as fori beingactivein party I):
977
in partyactivitiesthatis
benefitsforparticipating
relatedto the positionsof thetwo parties.Some,
possiblymany,may feelthattheyare participating in a good and worthwhile
cause. This motivationis politicaland willbe feltevenif theactivity
is unsuccessful(and even if i believesit would be
so in advance). But, how good and worthwhile
the
cause is dependsupon how i evaluatesthatcause.
Assume thatthereis anotherutilityfunctionfori
with the same ideal point location that is also
quadratic-based (but possibly of a different
form),say U1I(x).5 Then, if i is an activistin, say,
party e, i receives U,'(0), merelyfor being involvedwiththatparty.But, how good and worthwhilethatgoal is also may depend in part on the
other party's position. Afterall, workingfor a
good cause may be less worthwhileif no one is
seekinganythingotherthanthatsame end. Since
the positionof the otherpartymay be relevant,
the appropriateformulationwould be a comparison of the utilityassociated withthe two parties,
in theevaluationof goal of the
i.e., thedifference
partyin whichone is active,less thatof the other
party'sgoal. However,the otherparty'sposition
may not be as important.Let us weighttheother
party'sgoal by a scalar, say a: 0 6 a < 1. If a is
zero, thepositionof thesecondpartyis irrelevant.
If a is one, the positionof the second partyis as
importantas the first.To summarize:
U4: For all i, thereexistsanother(unimodal,
symmetricabout xi) utilityfunction,U1'(x),
and
U5: For all i, thereis an intrinsic
utilityforbeing active in party 0 (1) that is U1'(0) aUi'(i,) [or Uj'(so) - aUi'(0)].
Of course, i receivesthe utilityin U4-U5 if and
onlyif i is active.
Combiningour utilityassumptions,U1-U5, we
can specifytheexpectedutilityforbeingactivein
party 0, in I, and in neither.Let Aj denote
choosingactivity
j:
1. EU,(A0) = [(p00)U,(0)+ (pto)U,(i&]
+ [UJ'(0) - aU1'(i6)] - ci
U3: For all i, thereexistsa "net costs" term, 2. EU,(A;) = [(pO)U1(0) + t;O)U,0A
ci, such that ci > 0, where ci is equal for
+ [U,'(0) - aU,'(0)] - ci
activityin party0 and in party'.4
There remains a possible source of intrinsic
'For example,thetwo functions
coulddifferby a
or thetwocould
constantbutbe otherwise
identical,
thesamelevel be completely
different
4Obviously,thelastpartofU3 assumes
(althoughunimodaland symIn effect,
ofcommitment
tothetwoparties.
(e.g., Ui(x) mightbe shapedlikea
then,italso metric)functions
assumesthatprivate
benefits
provided
bythetwopar- normal distribution,whereas U1'(x) is strictly
tiesareequallyvaluable.
quadratic).
978
The AmericanPoliticalScienceReview
3. EU2(Ao) = [(pOO)U,(O)+ (poo)U<@]
Finally, we assume, of course, that all i are
rationalexpectedutilitymaximizers:6
U6: For all i, i will choose the maximumof
equations (1-3) above.
The Calculus and Activism
The "calculus of participation" follows as
closelyas possible the Riker-Ordeshookcalculus
of voting in a Downsian framework.At this
point,we mustnow connectitwithchoices.I continue to assume that the individualis choosing
whetheror not to become an activistand not consideringhow muchthe individualcontributes.
Here, I willshow thatthisformulationleads to
resultssimilarto thoseobtainedby Davis, Hinich,
and Ordeshook. Before doing so, we mustmake
some kind of assumptionabout the particular
probabilitytermsin equations 1-3. Specifically,I
betweenthe two
assume thatthereis a symmetry
parties;contributionsare equally valuable to the
recipientparties.I also assumethatbeingactivein
partye helpsthatpartyto achieveits goals more
than "abstaining," which in turnhelps partye
more than beingactivein the otherparty:
U7:
a) (pt - poe) = (p7o - Poo);
b) (pee - poo) = (POo - Poo);
(POO - PRO) = (PtO - Poo);
c) Poe > Poo > Poe; Pop > Poo > Po.
With theseassumptions,it followsthatno one
becomes an activistin the partyofferingthe less
the moredistant,position:
preferred,
Prop. 1: For all i, ifassumptionsU1-U7 hold,
and if 1x1- ji < 1x1- kj for {jk} =
then EU,(Aj) > EUL(Ak), and i does not
choose Ak.
Proof: See appendix(forall proofs).
This propositionis, of course, the same sort of
resultthatRikerand Ordeshookobtained.It says,
simply,thatone is activein the closerpartyor is
not activein eitherparty.As withthe calculus of
Vol. 77
voting, the interestingquestion is when one is
activeat all.
One is activein thecloserpartyratherthannot
active at all when the followingconditionholds,
which is shown to parallel the famous RikerOrdeshookformula:
Prop. 2: One is activein thepartywhoseposition one preferswhenEUi(Aj) > EU,(AO) or
(wherep = (pjy - pjo)): p[U1(j) - U1(k)] +
[U, '(j) - aUi '(k)] - ci > O or: P B + D - C
>0.
The two formsof abstentionthathave received
the most studyin the spatial model of electoral
competition are termed "indifference" and
is the basis of absten"alienation." Indifference
tion if turnoutdeclinesas the utilitydifferential
betweenthe two candidatesdecreases.This basis
of abstentionis usuallytied to theB term,which
In this model of parmeasuresthat differential.
tisan activity,there are two sources of indifference,theB termand (possibly)theD term.
Abstentionis said to be due to alienationif
as the
of votingdecreases
a citizen's
probability
candihe associateswithhispreferred
[utility]
of votand thathisprobability
date[decreases]
increases].(Davis,
ing increasesas this[utility
in
(1970),p. 437,emphasis
Hinich,& Ordeshook
original)
Here we can definethetwo puretypesof "abstention" as:
U8: "Abstention" fromactivismis said to be
policyrelatedif it is due to eitherindifference,
alienation,or both:
a) Abstentionis due to indifferencealone,
given that partyj is closer to xi than is k, if
either:
(1) U,'(x) = 0 for all i, so that: EU,(Aj) EUI(Ao) = P[UQ{) - U,(k)] - ci or
(2) a = 1 forall i so that:EU,<Aj) - EU1<AO)
= P[Ui(j) - U!(k)] + [Ui'(j) - Ui'(k)] - c,.
b) Abstentionis due to alienationalone ifP =
0 and a = 0 for all i, so that EUi(Aj) - c.
EU,(A0) = Ui')
c) Abstentionis due to alienationand indifferencecombinedifP * 0 or a ? 0, and if a
* 1, and UI'(x) * 0 forall i, so thatthe full
equation, above, is relevant.
6For those who prefer the Ferejohn and Fiorina
A More GeneralModel
(1974) minimaxregretformulationto the expectedutilitycalculus, note thatthe minimaxregretdecisionrule
The Davis, Hinich, and Ordeshook quotation
could be substitutedinto all two-partyresultswithout
presents the act of abstentionin probabilistic
changingany of the resultsobtainedhere.
1983
A Downsian Spatial Model withPartyActivism
979
definedoverthethreemutually
itydistribution
fashion,yetthetermsof themodelare determinisexclusiveand exhaustivesetsof behavior,partic. It is possible to change the formulationto a
ticipationin party e, party 1, and neither
probabilisticmodel, which also yields a more
party, respectively.These three distribution
generalmodel. RichardMcKelveydid so forthe
functionsare dependenton U,(x), 0, 6, and ci,
spatial model of electoral competition(1975).7
and all threemustsum to one at all xi forall
Here, I willparallelhis formulationforthespatial
pairs (0,O6).In addition,the two partysupport
model of partisanactivity.
functionsmust be of the same functional
Considerthe set of people who sharethe same
form,as orientedby a and b.'
ideal pointposition,e.g., xi. These people facethe
same choices of actions,the same utilityfunction
AlthoughSI definesand constrainsthesefuncand evaluationof the two parties,and yet,some
will choose to become involvedand otherswill tions, the constraintsare not verygreat. Other
not. Why? In themodel theansweris thatpeople conditionsmustbe imposed.There are fiveother
benefits conditions(offeredbelow) that theymay satisfy
evaluatethe costs and non-policy-related
That is, the net costs term,ci, varies eithersinglyor in combination.Propositionswill
differently.
evenwithall else held constant(and, by construc- be offeredto tie theseconditionsto the assumption, that variance is independentof policy- tionsof the last section.
The firstconditionon P(0) and P(06) I call the
related evaluations). This leads naturallyto a
the probabilitythat "Rational Choice Condition." This conditionis
probabilisticinterpretation,
any givencitizenwithany given ideal point and that the probabilityof supportingthe fartherpair of partypositionswill become involvedwith removed party is always zero. It is obviously
a partyor "abstain,' or the probabilitythat the closelytied to the rationalchoice assumptionof
"PB and D" termsoutweighthe variable ci. (I the last section(U6) and Prop. 1:
laterin this
willprovidea different
interpretation
S2: The two partysupportfunctions,P(0) and
section.)
There are threesuch probabilityfunctions,the
P(o), are said to meet the Rational Choice
Conditionif, forall xi and for{j,k} = {,01:
probabilitythatone is activein partye, in party
PU/xj,k) = 0 if Ixi - jl > Ix, - kI and
I, and in neither.Call these P(O), P(I), and
>0 ifIxj - jI < Ixi - ki.
P(A), respectively.These are functionsof the
ideal pointlocation(and, moregenerally,theutilityfunction),thepositionsof thetwoparties,and Figure 1 illustratesthis,and the following,conof c1.Since ci is an independentrandomvariable, ditions.
The nexttwo conditionsare used togetherand
we can writeP(e/x,, 0, @), etc. Finally,sincethese
are probabilityfunctions,and since these are so willbe definedtogether.Theyare, simply,that
supportfunctionsare unimodaland
assumed to be mutuallyexclusiveand exhaustive thetwo-party
about therespectivepartypositions,at
acts, the probabilitytermsmust sum to one at symmetric
each xi: P(O) + P(I) + P(A) = 1. Given this least up to the midpointbetweenthe two party
we can place conditionson, say,the two positions (at which point, the Rational Choice
identity,
Conditionbeginsto come intoplay). The motiva"party support functions" [P(O) and P(I)],
which implicitly constrains the abstention tion for this conditionis the alienationformof
one is fromthe
"abstention,"in whichthefarther
function.
The key here, of course, is to examinecondi- closer party,the less likelyone is to become intionson theprobabilityfunctions,indicatingthat volvedwiththatparty:
if they meet certain of these conditions (or
S3: The two partysupportfunctionsmeetthe
take on an appropriateshape or distribution),
Conditions of Unimodality and Symmetry
thencertainresultscan be proved.The definition
about the partypositionsif, forall xi and for
of them actuallycontainsenough constraintsto
{j,k} = 10,{1
be set as an assumption:
SI: There exist three probabilityfunctions,
'The last sentence of this condition is similar to
a probabil- McKelvey's "metricsymmetry"condition.His condiP(O), P(I), and P(A), representing
7I use terminology
similarto McKelvey's. Some care
must be used in comparingthe two models, because I
definesome of the termsdifferently
(and of course put
themto a different
use). A more formalderivationof
the probabilisticmodel fromU1-U7 may be found in
Aldrichand McGinnis(1983).
tionrequiresthatthetwo (candidate)supportfunctions
be of the same functionalform,but mirrorimages of
each other.For example,ifP(O) is unimodaland symmetricabout 0, thenP(I) also must be unimodal and
but about &. McKelveyshows thatif utility
symmetric,
is quadraticbased, and votingand abstentionare policy
related (which will hold here if assumptionsU1-U7
holds.
apply), thenmetricsymmetry
980
The AmericanPoliticalScienceReview
Vol. 77
Figure1. ExamplesofProbability
SupportFunctions
P(O)
la:
the
P(O) satisfies
Rational Choice, Unimodality,
and SymmetryConditions.
Abstention due to
Alienation only.
x
lb: P(?) satisfies
the
Rational Choice and Strong
Extremist Support Conditions.
Abstention due to
Indifference only.
lc:
m
the
P(G) satisfies
Rational Choice and Weak
Extremist Support Conditions.
Abstention due to
Alienation and Indifference.
X
partythan are theirmore moderatepeers. The
"strong" conditionis thatthepartysupportfunction is monotonicallyincreasingfrom the midpointbetweenthetwopartypositions,throughits
own position, to the appropriateinfinity.This
condition is related to abstentiondue to indifference.The "weaker" extremistsupportcondiThe finaltwo conditionsare twoversionsof ex- tion is that for any two pointsequidistantfrom
tremistsupport; that is, relativelymore extreme thecloserpartyposition,themoremoderatepoint
individualsare more likelyto supportthe closer has a lower(or no higher)probabilityof supportmonotonedecreasingforall
a) P(j) is strictly
xi such that Ixi - il < Ix1- kj, and
b) For any pairs of points,x and y, such that
Ix - il < Ix - kj, ly - il < ly - kj, and
Ix - il = ly - ji, thenP(//x) = Plj/y).
1983
A Downsian Spatial Model withPartyActivism
ing thatpartythanits moreextremecounterpart.
This conditionholds if abstentionis due to indifference, to alienation, or to both reasons
together:
S4: The followingtwo conditionsapplyto any
ideal point positionscloser to the positionof
partyj thank.
a) Pi), for{jk} = {G,t} are said to meetthe
StrongExtremistSupportConditionif, forall
xi, PUi) is strictlymonotone increasingas
1increases.
txi- [(K+ ,6)/2]
b) PU), for{j,k} = {O,b} are said to meetthe
Weak ExtremistSupportCondition,if for all
pairsof points,say {x,y}suchthatif Ix - il =
ly - jil and Ix - kI > Iy - kj, thenP(j/x) >
Ptj/y).
Note thatS3 and S4 can hold iftherationalchoice
conditionholds. S3 and the strongversionof S4
are not compatible,but S3 and the weak version
are (if all theweak inequalitiesare strictequalities
in S4b). Also note that if the StrongExtremist
SupportConditionholds, thenthe Weak version
but thereversemaynotbe true.
holds necessarily,
To this point, the aggregateprobabilityfuncas theprobabilitythat
tionshave been interpreted
one "joins" a partyor not, similarto the interpretation of McKelvey's probabilityof voting
functions as representingthe probability (or
relativefrequency)of votingforone candidateor
justifiable
the other.Althoughthisis a perfectly
for partyactivismunderstoodas,
interpretation
witha party,or beingactivein a
say, identifying
party without regard to how many scarce
thatis notthe
resourcesan individualcontributes,
only, nor the most general possible, interpretation.
Probabilitydensityfunctionsare basicallydistributionsthat are normalized to sum to one.
as relativefrequenThey need not be interpreted
cies of behavior. Instead, they could be interof individuals.
pretedas therelativecontributions
This interpretation
suggeststhata probabilityof
activismtermcan be less tiedto individuals,more
closely related to the amount of contributions.
ofP(O)
For example,themean of thedistribution
indicatesthe ideal
under the firstinterpretation
pointpositionof the "typical" activist.Underthe
it representsthe average
second interpretation,
contribution(be it of time, money, or effort).
Looked at in terms of individual partisans,it
represents the ideal point position of the
"typical" activist,weightedby the amount of
scarce resourceshe or she contributes.9Briefly,
is presentedin
9A formalizationof thisinterpretation
Aldrichand McGinnis(1983).
981
used hereis a special case of a
the interpretation
more general class of models. The major (and,
admittedly,it is major) restrictionis that the
are discostsforgonebycontributing
opportunity
of the ideal pointposition
tributedindependently
in X. The amount the individual activistconvia profitmaximizing
tributes,then,is determined
procedures.If R, in the calculus, is not greater
than zero, theni does not become an activist.If,
however,R is positive,then i will contributea
positivebut finiteamount. That amountis measured by the proportion of possible forgone
actuallyforgone.
opportunities
ConnectionsbetweenIndividualUtility
and PartySupportFunctions
The followingpropositionstie the assumptions
made about individualutilityfunctionsand choice
to the conditionsplaced on the aggregateprobabilityof activismfunctions.It is importantto
rememberthatutilityassumptions(those denoted
by U's) imply the aggregateconditions (those
denoted by S's), but not vice versa. That is, the
probabilityconditionsare more generaland may
hold even if the utilityconditionsdo not.
Throughout,the A-typeassumptionsdefining
the spatial model are assumedto hold. Then,
Proposition 3: If the conditionsof Prop. 1
(U1-U7) hold, thenthe Rational Choice Condition(S2) holds. In addition,if abstentionis
itslastsenpolicyrelated,thenS1 (specifically,
tence)holds.
Essentially,Proposition 3 states that if utility
functionsare quadratic-based,thenthe Rational
Choice Conditionholds,so thattheprobabilityof
supportingthespatiallymoredistantpartyis zero.
Further,if abstentionis due to eitheror both of
or alienathe policy-basedreasons (indifference
tion) or is constantin expectedvalue across the
policyspace, thenthe two supportfunctionswill
be mirrorimages of each other,orientedabout
the midpointbetweenthe two partypositions.
The other propositionsdeal with particular
forms of abstention.Both propositions,therefore,assume U1-U7 and thatabstentionis policy
related (U8). They differover what part of U8
applies.
Proposition4: If abstentionis due solely to
alienation(i.e., U8b holds),thenthepartysupportfunctionswillbe unimodaland symmetric
(S3 holds).
Proposition5: a) If abstentionis due solelyto
indifference,
and if Ui(x) is convex as well as
982
The AmericanPoliticalScienceReview
Vol. 77
quadratic based, then the Strong Extremist mean ideal point of currentactivists(but demonstratethat the median could be substitutedwithSupportCondition(S4a) holds.
probaor aliena- out changingthebasic conclusions).If the
b) If abstentionis due to indifference
as the probability
are
interpreted
functions
bility
then
the
or
c
holds),
tion or both (U8a, b,
of joining or not, thenthe mean ideal point is a
Weak ExtremistSupportConditionholds.
simple, unweightedaverage. In the alternative
the mean is the weightedaverage,
interpretation,
The basic relationships,then,are thatif utility
are based on the extentof the
weights
the
where
ConChoice
Rational
then
the
based,
is quadratic
In eithercase, the entiredismade.
contribution
ditionfollows.If, in addition,abstentionis policy
by a singleparameter.In
related,thetwo supportfunctionsare of a similar tributionis summarized
that the degreeto
implies
the
assumption
effect,
form,and theWeak ExtremistSupportCondition
containsconsensusor dissension
party
the
which
formof abstentionis
follows.If thepolicy-related
does not matter.Similarly,at thispoint,thereis
alienation,thenthesupportfunctionswillbe unino
separateaccountingof thepositionof thecaninstead
is
due
If abstention
modal and symmetric.
and the parties. Candidates are simply
didates
to indifference
(and iftheutilityfunctionsare also
activists
(although they may be weighted
party
convex), then the support functionssatisfythe
is followed).
that interpretation
if
heavily,
more
StrongExtremistSupportCondition.
or at leastmaybe
are
such
calculations
Certainly,
These propositionstie the support function
quite important,and I hope to generalizethe
approach to the more common rational choice
along theselines.For now, thisassumption
model
framework. Nonetheless, the probability apbe
takenas a firstapproximation.
must
proach is moregeneral.For example,it could be
if
that assumptionis a firstapproximaEven
the case that extremistsare more likely to be
it
up some theoreticallyinteresting
involvedin partisanactivitythan moderatesand, tion, opens
in particular,thequestionof
Consider,
questions.
thus, the extremistsupport condition applies,
an individual decides to
say
us
Let
dynamics.
even though the utility function approach is
activistin one of the partiesfor the
an
become
wrong.
firsttime. He or she does so because he or she
prefersthe positionof the averageactivistin that
Dynamicsof PartyActivism
party,in comparisonto theotherparty'sposition,
to outweighthe costs of activism.By
sufficiently
The model to this point has focused on in- becominginvolvedin theparty,he or she changes
dividualchoicesonly.I thinkit reasonableto sug- the positionof the party,even if so slightlyas to
gest that the formulationof the problemis not be all but unobservable.But thatchange(especialespeciallynovel, nor the resultsparticularlyun- lyifcombinedby similardecisionsmade by "peoexpected. In this section, I add the assumption ple like himor her") maybe big enoughto make
that providesa social aspect to the problemand it worthwhile
forstillothersto join, perhapsdrive
pose the social-dynamicquestion to which the some out of the party,and perhapscause similar
choice "machinery"of the last sectionscan pro- changesin the otherparty.Thus, the assumption
vide answers.
thatpeople becomeinvolvedin partyaffairsbased
The key assumptionfor these questionsis the on the preferences
of othersalreadyinvolvedimdeterminationof the two-partypositions that plies a dynamicto the distributionof activistsin
enter the individual utilityfunctions.What I the two parties:
assume is this:The individualevaluatesthe position of the party,as he or she considerswhether
A5: For all i, the points0 and 4, as theyenter
or not to become actively involved in party
into i's utilityfunction,are defined as the
affairs,by thepositionsof thosewho are already
ideal point position of those currently
mean
activein the party.Afterall, the goals of a party
activein party0 and A, respectively.
are what its membersmake them.Further,many
of the most importantrewards of activityare
thosethatinvolveworkingforand withothersto
The restof the articleexaminesthis dynamic,
achieve(or evento attemptto achieve)valued out- especially by focusing on such factors as the
comes. In short,the social solidaryrewardsand stabilityand location of the mean ideal point of
in a good and worth- activists (because if these two means do not
rewardsfromparticipating
while cause withotherswhom you like, admire, change, thenthe fulldistributions
will stabilize).
and respectweigh heavily for the activistcon- The questions asked are whethersuch activism
sidered here (i.e., the non-careeristand non- calculationslead to any stabilityin the distribucandidate).
tions of the two parties' activistsin the policy
Now, what positionis taken by the "typical" space, and wherein the space would we expectto
activistis less obvious. Here I assumethatit is the findthe typicalactivistin the two parties.As we
A Downsian Spatial Model withPartyActivism
1983
will see, the answers conform roughlyto the
generalview of partycleavages.
The terms such as "equilibrium," "convergence," and "divergence" appear in thismodel,
just as theydo in the spatial votingmodel. There
however.In the voting
are importantdifferences,
models, "equilibrium" refersto positions that
candidatesmightwant to take to ensurethe best
possible outcome, and such points are in
equilibriumif the candidateshave no incentiveto
change their positions, once adopted. Here
"equilibrium"refersto thebehaviorof citizensas
potentialactivists.Since activistsgetinvolvedin a
partydue to the preferencesof those already in
the party,a decisionby some citizensmay affect
thedecisionsof others.The twopartiesare said to
be in equilibriumif no new citizens want to
become involvedin eitherpoliticalparty,nor do
any currentactivistswant to change theirdecision. As a result of these decisions, the party
centersare also in equilibrium,in the sense that
thelocationof thetypicalactivistwillnot change,
since the individualswho make up the partiesdo
not changetheirbehavior.To say thattheparties
are "converging"meansthatthepartycentersare
gettingcloser together,i.e., more people in betweenthetwopartiesare becomingactiveor more
extremeactivistsare droppingout or both, and
similarlyfor "divergence."
Some EquilibriumResults
The basic approach here is to impose certain
conditionson the partysupportfunctionsand to
in the
of preferences
look at variousdistributions
electorateas a whole. This approachis useful,for
off(x,) thatare examthe kinds of distributions
ined are fairlygeneral(unimodal, bimodal, and
multimodal),and also because once theprobabilityof activismfunctionsand f(xi) are specified,
the full distributionsof activists(say, F(O) and
in the electoratecan be speciF(*), respectively
fiedby solvingequationslike:
F(e/x1, 0,
')
= lb P(E()
f(x,) dXh
The firsttwo resultsdeal witha one-partysystem. They are listedas lemmas,because the onepartysystemis a usefulfictionin thatit imposes
limitson the possiblelocationof equilibriain the
two-partycase. Let e be the one party.
LEMMA 1. If f(xj)is a continuousfunctionover
the interval(a,b) and P(O/xi, 0) is a continuous function not always equal to zero, then
983
thereexistsat least one equilibriumvalue, 0*,
interiorto theinterval[a,b].I0
LEMMA2. If, in addition, f(xj) is also strictly
unimodal,withmode at y c X, and if P(e/xi,
0) satisfiesthe conditionsof unimodalityand
symmetry
about 0, then:"
a) Any equilibria,0', willbe "near" themean
of f(xo,and hence "near" y.
O* = y, themean
b) If f(x) is also symmetric,
and mode of f(xj), is theunique equilibrium.
The two-party
case is, of course,themoreinterestingand relevant.The firstpropositionis a very
general existence result. It states that if the
Rational Choice Conditionholds (i.e., no one is
activein the furtherremovedand hence less preferredparty),thenthereis at leastone equilibrium
pair of partypositions(let m denotethemidpoint
betweenthe two partycenters(m = (0 + ,6)/2)).
PROPOSITION
6. Iff(x,) is continuousover the
interval[a,b], and if P(O) and P(I) are continuousover [a,m] and [m,b],respectively,
not
zero, and meettheRationalChoice
everywhere
Condition, thenat least one equilibriumpair
of party positions, (0*,
&*), exists.'2 Further,
theseequilibriapositionswillfindthetwopartycentersinteriorto [a,b] and separatedfrom
each other,so that0* ? 0*.
This proposition imposes only the Rational
Choice Condition, which thereforeexamines
choicesbetweenthetwo partiesonlyand does not
imposeanyconditionson abstention.In addition,
the conclusionis fairlygeneral;it statesthatthe
decisionsof theindividualcitizens,as theychoose
whetheror not to becomepartyactivistsbased on
where currentactivistsin the two parties are
located, will lead to a stable distributionof
activistsin the two parties.Further,the two parties will be located such that the centerof each
partywillnot be at an extreme,and the two parties will be distinctivefromeach other.That, by
I"Allequilibriaanalyzedin theAppendixare stable
noted.
unlessotherwise
equilibria
"These proofsassumethatthemodeis clearlyinterior
to therangeofidealpoints(andthatbothmodes
to theinterval[abJ in thebimodal
are wellinterior
For Lemma2,
case). Thisis assumedforconvenience.
ifthemodeoff(xi) is at pointa, thenitis
forexample,
to showthatthepartycenter
relatively
straightforward
willconverge
as closetopointa as ispossible,
consistent
in theAppendix.
withtheequilibrium
condition
assumesthat
theproposition
as written
'2Although
0 < & at theoutset,itobviously
holdsas wellif0 > t.
984
Vol. 77
The AmericanPoliticalScienceReview
itself,is a long way towardestablishingthe sorts
one would expectto findif party
of distributions
cleavagesexist.
The above propositionincludesthe resultthat
the two partycenterswillbe separatedfromeach
other; that is, therewill be some distinctionbetweenthe positionsof the "typical" activistsin
the two parties.In a sense, this resultis not too
surprising,as one of the major motivationsfor
activismis joining like-mindedpeers, especially
betweenthe two parwhen thereare differences
ties. Proposition 6, however,does not indicate
the two partiesmay be. If people
how distinctive
do become activistsin partto be withlike-minded
peers, then it would seem that the case of least
divergencewould be when the distributionof
ideal points in the electorateis unimodal. The
followingproposition,then,deals withthatcase,
whereit is to be understoodthatthemode off(xi)
is interiorto the interval(ab); that is, f(xi) is
strictly
increasingfroma to themode, say pointy,
decreasingfromy to b:
and thenstrictly
PROPOSITION 7. If the conditionsof Proposi-
tion6 hold, and if f(xi) is unimodalwithmode
at y; a < y < b, thenfor {j,k} = I , ,}:
a. If PO) and P(k) satisfyS3, or S4a, or S4b,
thenthe twopartymeans,at equilibrium,will
be divergentwithj* < mean of f(xi) < k*.
b. (1) If Poi) and P(k) meet theConditionsof
Unimodalityand Symmetry
(S3), the distance
separatingj* and k* willbe least.
(2) If P(j) and P(k) meet WeakExtremistSupport (S4b) and not S3 and not S4a (StrongExtremistSupport), the distance separatingj*
and k* willbe greaterthancase (1).
(3) If P(j) and P(k) meetStrongExtremistSupport (S4a), the distanceseparatingj* and k*
willbe greaterthanin case (2).
The pointof thisproposition,in general,is that
abstentionwilllead to
any formof policy-related
"divergence" of thetwo setsof partyactivists.If
abstentionis the resultof alienation,the two sets
of party activists will be the most similar
(although still divergent).If abstention is the
however,this will lead to
resultof indifference,
are more
sets of partyactivistswhose preferences
on the policydimension.
sharplydistinguished
One final propositionabout a unimodalf(xi)
as
concernsthe special case of it beingsymmetric
well as unimodal.
PROPOSITION 8. If the conditionsof Proposi-
then
tion 7 hold and if f(xi) is also symmetric,
an equilibriumpair of positions of theparty
centersexistssuch that0* and
tantabout themode of f(xi).
/*
are equidis-
Presumably,the case of a unimodal distribution of ideal preferencesin the electorateis the
most "consensual" and thereforethe mostlikely
to lead to relativelysimilarpartycenters.As it
distribuhappens,however,a bimodalpreference
tion may have at least threepairs of equilibrium
positions.
PROPOSITION 9. If the conditionsof Proposi-
tion 7 hold, exceptthatf(xi) is bimodal, with
modes,say yand z, such thata < y < z < b,
thenat least one, and possiblyall three,of the
following equilibrium pairs {j*, kV} =
{0*,1I*} exist:
a) A pair existsabout themode y,
b) A pair existsabout themode z, and/or
c) A pair existssuch thatj * is neary, k* is near
z.
Note that, like Proposition 7, this proposition
holds for any sort of policy-relatedabstention.
Note that the increaseddivergenceof the party
centers, as in Proposition 7, follows straightforwardly.That is, abstentionresultingfrominwilllead to partycentersthat
difference
ordinarily
are furtherapart than if abstentionresultsfrom
alienation.
The threepossible pairs of equilibriafoundin
Proposition 9 only indicate that thereare local
equilibriapossibilities.And thelogic of joiningor
withdrawingfrom activism implies a marginal
adjustmentprocess, so that the party activists'
decisionsmay lead to, and remainstableat, local
equilibria.
The mostinteresting
partof Proposition9 is the
thirdset of equilibriaoutcomes. Here, ratheras
we would expect, the parties are dominatedby
setsof activistsat each mode; thatis, therewould
be a dominantlyliberaland dominantlyconservativeparty,rathermoreclearlyso than in the unimodal case. Still, even in the unimodal case,
wheremostpeople are moderate,one partywould
consist mostly of moderatelyliberal to liberal
activists.Thus, the additionof the bimodal case
givesonlya sharpenedsense of partycleavages.
Finally,theanalysiscould be extendedto multiby repeatedapplicationof the
modal distributions
last proposition.Of moreinterestwould be an excontext.For
tensionof thelogic to themultiparty
example, we could redefinethe Rational Choice
Conditionto be thattheprobabilityof activismis
zero forall partiesbut theclosestone and assume
that the partysupportfunctionis unimodal and
about its own partycenter,withinthe
symmetric
1983
A Downsian Spatial Model withPartyActivism
985
setbytheredefined
constraints
choicecondition. like equilibria were found, and although that
Then,itis relatively
easyto showthatthereis an model is much less generalthan the currentone,
equilibrium
setofpartypositions,
witheachparty the principalresultsobtainedtherecan be shown
activist
distribution
beingcentered
nearoneofthe to be special cases of the decision calculus used
here. Also, Aldrichand McGinnis(1983) provide
modes.13
extensionto n-dimensionsand find
Althoughthe analysishas assumedthatthe a preliminary
citizenevaluates
theposition
ofthepartybyrefer- that the resultsdo generalizein cases studiedso
enceto themeanidealpointlocationof current far. Thus, it seems thatthe most obvious limitatherelianceon themeanper se is not tion is not a major problem.
activists,
veryrestrictive.
As the followingproposition There are other limitationsto the current
shows, the results are changed very little model. For example,thecitizen'sdecisionsreston
(specifically,
onlyinterms
oftheexactlocationof evaluatingthe partyas onlythe policypreference
ifsomeotherdefinition
equilibria)
of "theparty of itstypicalactivist.Thereis no measureof such
factorsas the importanceof dispersionof prefercenter"is used.
ences among activists,variable probabilitiesof
potentialsuccess, and the role of relativeadvanholdifthemedianidealpointlocationis sub- tagesone partymayhave in termsof "selectiveinstituted
for themeanidealpointpositionof centives." Thereis no studyof leadershipmotivation,althoughI am currently
current
inUj(x).
workingon a model
partyactivists
with a party leader seeking to attractactivists.
Summary
Further,thereis no institutionalstructurehere.
At thispoint,a partyis no more than a distribuThe threebasicconclusions
are:
tion of its activists.There are no rules,norms,or
1. Equilibria(and locallystableones at that) traditions,just as thereis as yetno incorporation
existinthismodelofpartyactivist
dis- of differing
ordinarily
officesor regionalpatterns.In short,
tributions,
based onlyon the RationalChoice thereis a greatdeal of room forfurtherresearch
Conditionwhichconcernsonlythe choicebe- to make a more completeand realisticmodel of
notthechoiceofwhether
ornotto politicalparties.
tweenparties,
be an activist.
The second topic is how this model may be
it appearsthattheparties mergedwiththe traditionalspatialmodel of elec2. At equilibrium,
arerelatively
cohesive
andrelatively
dis- toral competition.This model was designedwith
internally
fromeach other,withpartycenters,
tinctive
or that goal in mind,so its formalstructureshould
ina waythat make such a combinationrelativelyeasy. There
typical
partisan
activists,
distributed
resembles
partycleavages
of seem to me to be threemajor ways that such a
(andthedistributions
activistsassumedin, for example,Aranson& combinationmay be rewarding.First,partyacOrdeshook,1972).
tivistsmay impose constraintson the strategies
3. These conclusions
seemratherrobust,for that its candidates can follow. Aranson and
theyholdin a generally
similarwayforeitheror Ordeshook (1972) illustrateone way in which
bothtypesof spatially
relatedabstention
and for theseconstraintsmay operate. There, candidates
unimodal,
bimodal,andmultimodalf(xi).
need to win the support of activists to win
nomination,whichthen servesto constraintheir
Discussion
possible generalelectionstrategies.Activistsalso
may serveto constraincandidates,if candidates
Two topicsmeritfurther
discussion.
The first need theirsupportto make theirgeneralelection
ofthemodelat thispoint, strategieseffective.For example,adoptinga positopicis thelimitations
and the secondis theway in whichthismodel tioncloserto thecenterof thepartymaygenerate
be combined
withthespatialvotingmodel. greatercontributionsfrom those activists,conmight
of themodelpre- tributionsthatmaybe usefulforincreasingturnThe mostobviouslimitation
sentedinthisarticleis thatitis unidimensional.
A out of thecandidate'ssupporterson electionday.
unidimensional
modelis empirically
limited
and,
Second, possible candidatesfor electiveoffice
if past experience
withspatialmodelingis any generallyemergefromtheranksof partyactivists.
guide,may be quite different
fromits multi- If candidatesthemselveshave policypreferences,
dimensional
An earlyversionof this thenwe have some explanationof what sortsof
equivalent.
to the multidimensional
modelwas generalized
preferencesmost candidates are likelyto have.
case (Aldrich,in press).There,stable,cleavage- Further,of course,a candidate'sown policypreferencesmay impose importantconstraintson the
platformsthey offer the public (see Wittman,
"Theseresults
are availablefromtheauthoron 1983, for theoreticaland some empirical acrequest.
counts; Page, 1978, formore evidence).
PROPOSITION
10. All results
abovecontinue
to
986
The AmericanPoliticalScienceReview
Third,thismodel of partyactivismmaybe usefulformakingrationalchoicemodelsof elections
richer and more useful models of electoral
behavior.For example,campaignparticipationis
a fairlycommon act among the mass public, by
itself.Often, nearly200o of the respondentsto
national election surveysreport some form of
campaign activism during presidentialelection
years. Further,partisan identificationmay be
considered,at least in part, as a kind of partisan
Vol. 77
activitylike thatmodeledhere.
In sum, althoughthe model presentedhere is
limitedin manyways,myhope is thatit opens up
a wide rangeof possibilitiesforimprovingformal
modelsof electorallyrelevantbehavior,at leastin
termsof theirutilityas empiricalexplanationsof
manycommonformsof behavior,and for makingthesemodelsmoreconsistentwiththefindings
of several decades of empiricalfindingsabout
electoralbehavior.
ProofsofPropositions
Appendix.
I. Propositions
on Individual
Choice
is deletedforconvenience.
N.B.: The i subscript
j1 < jx - kIthenEU(Aj) > EU(Ak),for{j,}
1) EU(AJ) = pjjUU) + pkjU(k) + U'Cj) - aU'(k) - c by Ul-U5.
2) EU(Ak) - EU(Aj) = (pjj - Pjk)U(j) - (pkk- Pkj)U(k) + (1 - a)[U'tj) - U'(k)].
PROPOSITION 1. If U1-U7hold,andif lx -
{0'}.
3) (pjj - Pjk) = (pkk - Pkj) > 0 (callthisp) byU7.
4) EU(Aj) - EU(Ak) = p[Uj) - U(k)] + (1 - a)[U'Cj)
- U'(k)].
thesignoftheabovedependson thesignofthetwoU-differentials.
5) Sincep and(1 - a) arepositive,
in distance,
ByU1 and U4, U(x) and U'(x) arestrictly
negative
monotone
therefore
bothdifferentialsarepositive.
6) Therefore,
EU(Aj) - EU(Ak) > 0. ByU6, Ak is neverchosen.
2. Giventhatj is preferred,
PROPOSITION
i choosesAj overAo if:
1) EU(AJ) - EU(A0) > 0 byU6, or if
- aU'(k) - c - {pjoUtj) + pkoU(k)} > 0 or
pjo)U(J) + (Pkj - Pko)U(k) + [U'(j) - aU'(k)J - c >0.
2) pjjU(j) + pk1U(k) + U'(j)
3) (Pjj
-
4) (Pjj - Pjo) = (Pko- Pk) > 0 (callitp ') byU7. Therefore,
5) 1) may be writtenp'[Uj) - U(k)] + [U'(J) - aU'(k)J - c > 0.
II. Propositions
on Utility
andProbability
Functions
PROPOSITION 3.
A. If U1-U7 (and henceProposition1) hold, thenS2 holds. Since Ix - jl < Ix - kI implies
EU(AJ) > EU(Ak),byU6 no oneclosertoj joinsk. HenceP(k/xj,k) = 0.
of SI holds.Forproof,seeMcKelvey
B . If U1-U7holds,thelastsentence
(1975,prop.4.3, p. 840).
PROPOSITION
4. IfU1-U7andU8bholds,P0i) andP(k) areunimodal
andsymmetric
aboutpoints and
k, respectively.
1) For x closerto than k, U8b statesthatEU(AJ) - EU(Ao) = R = U'(i) - c.
2) ByU3, E(c) = C >O foralli, i.e., E(R) = U'(j) - C.
atj, so E(R) andP(i) aremaximized
3) ByU4, U'(j) is maximized
atj.
1983
A DownsianSpatialModelwithPartyActivism
987
aboutj, so E(R) andPUi) are,too,sinceU '(j) is theonlyvariableargu4) ByU4, U 'U) is symmetric
mentofE(R) andPU).
PROPOSITION 5.
(1975,prop.4.5, p. 841).
A. See McKelvey
Supportis met.
B. If U1-U7andanypartofU8 hold,WeakExtremist
x andj, etc.
1) Letd(xj) denotetheEuclideandistancebetween
twopoints,e.g.,x andy, suchthatd(xj) < d(x,k),
Support,consider
2) To defineWeakExtremist
d(y,j) < d(y,k),d(x,j) = d(yj), andd(x,k)>d(y,k).
3, P(k/x) = P(k/y)= 0.
3) ByProposition
whereasU(k/x)<
4) By U1 and U4, U(j/x) and U'(j/x) equal U~ily) and U'lj/y), respectively,
U(k/y)and U'(k/x) < U'(k/y).
SupportCondition
4, andtheWeakExtremist
5) If U8b holds,thenP(j/x) = P(i/y)byProposition
holds(trivially).
differenforEU(Aj) >EU(A0) to holdhaveutility
ofthetwoconditions
6) If U8a holds,theneither
arelargerforx thany,
(givenU3). Bystep4, thedifferentials
tialsas theonlyvariablearguments
Supportholds.
hencePUj/x)> Plj/y) andWeakExtremist
7) If U8a and U8b hold,thenEU(Aj) - EU(A0) is p ' [U() - U(k)] + [U'Cj) - aU '(k)J - c. The
holds.
areagaintheonlyvariable,andthecondition
differentials
utility
Proofs
Equilibrium
Somedefinitions:
in party for{j,k} = {0,4} is F(j,k) = lb p(j)f(x)dx
ofactivists
1. The density
2. Themeanidealpoint,j*, is: j*_ [flbxP(j)f(x)dx] /F(j,k).
if:fJxP(j)f(x)cx = fJbxP(j)f (x)dx,i.e.,ifj* = j.
3. j is an equilibrium
pairif3 holdsforbothsimultaneously.
4. U*,k*)arean equilibrium
5. Let m = U+k)/2, andletj < k initially.
l
6. ByProposition
3, P(j) = 0 forx > m,P(k) = 0 forx < m. Therefore,
FUj,k) b PU)f(x)dx =
forF(k,]).
a PU)f(x)dx,and similarly
TheLHS andRHS ofDef.3. arealsocon7. SincePU) andftx)arecontinuous,
FU,k)is continuous.
as
varies
tinuous j
continuously.
8. Let h(j) = xP(j)f (x)dx andh(k) = xP(k)f(x)dx.
(say,forj*).
Lemma1. A onepartyequilibrium
froma to b.
1) Letj* existandvarycontinuously
h(j) = fa h(j) = F(j).
2) Ifj* = a, Jf1h(j) = fa h(j) = 0, whilefm7
LHS = FU), whileRHS = 0.
3) Ifj* = b,,fromdef.3 of equilibrium,
values.
4) Consequently,
j* = a andj* = b cannotbe equilibrium
a
froma tob, like* above,yielded
that
varied
continuously
are
and
RHS
LHS
continuous,
Since
5)
aj
RHS wentfrom
FU) to zero.Thus,therewassome
LHS thatbeganat zero,endedatFU), whereas
value,exists.
valueofj in (a,b) suchthatLHS = RHS, hencea j*, equilibrium
988
PoliticalScienceReview
TheAmerican
Vol. 77
Lemma 2.
A. Unimodal and symmetricf(x) and P(i).
1. Let the mode of fix) be 0, so a = - b, etc.
2. Let ] = 0. Then, the questionis does l a h(x) =
b h(x)?
3. There is a one-to-onecorrespondenceforthe pairs (-xx) where -x e (aO), x e (o,b], and thusa
one-to-onemappingof each pointin the integralon LHS withthaton RHS.
4. By conditionsof lemma,P(j/x) = P(j/ - x) and f (x) = A - x). Therefore,h(x) = h- x) forall x
in rangeof integrals.Therefore,j* = 0 is an equilibrium.
5. For uniqueness,supposej < 0. Let x be any pointin LHS integral.Let y be thepointin RHS such
thatd(xj) = d(yj). For each x, thereis exactlyone suchy,whereasforsomey, thereis nox, forthe
restthereis but one x. By assumption,P(j/x) = P(j/y), butfix) < fly) forall such (xy) by definitionof unimodalityofAx). Thus, LHS < RHS. The same holds in reverseifj > 0, hencej* = 0 is
unique.
PU).
B. UnimodalJ(x),PU), symmetric
1. Let t be the mean ofJix).
I
2. By definitionof a mean, t is mean if: f xf(xo)dx
3. j* is an equilibriumif: Ji*xPU)f(x)dx =
f xf(x)dx.
'
bxP(])f(x)dx
4. Since PU) is unimodaland symmetric,
ifj = t, thenthePU) "weight" applied to theequationfort
the same value, i.e., j* "near" t.
will make the two equations hold at approximately
5. Sincey, the mode off(x), is "near" t,j* willbe "near" y.
PROPOSITION 6. Existenceof (j*,k*) whenP(U), P(k) meetSI, S2 and f(x) is unimodal.
1. Firstj* cannotequal a, forLHS(j) of the equilibriumconditionwould be 0, implyingthatRHS(j)
= 0, implyingthatm = a, thatk* = a, and thatLHS(j) = 0, but,as in Lemma 1, RHS(k) willnot
be zero. Similarly,neither nor k* can be at b. Hence, Uj*k*) ? {ab}.
2. By S2, PU}) = P(k) = 0, ifj > m, k < m forj < k. So, ifj = m (and hencek = m) RHS(j) =
LHS(k) = 0 in theequilibriumequations,so that,sincebothpartiescannothaveF(j) = F(k) = 0,
any equilibriacannot be at m, either.Steps 1 and 2 prove the last sentenceof Proposition6.
3. Existenceof an equilibriumpair forsymmetricfx).
for
a. Setj = - k throughout,so thatifone partyis at equilibrium,theotherwillbe simultaneously
fix).
symmetric
b. Let j = a, initially,and let it move continuouslytowardb. Therefore,both sides of the equilibriumconditionwill be continuous.
c. At j = a (k = -a = b), LHS(j)
= 0, RHS(j)
> 0.
d. At j = m = 0, RHS(i) = 0, LHSU) > 0.
e. Since thechangesare continuous,LHSU) = RHSU) at somea <j
the same time,a (j*, k*) exists.
< 0. Sincethisis truefork at
4. Existenceof equilibriumforasymmetric
f(x).
a. Since it has been shownthatno partof theequilibriumequationscan be zero, we can rewritethe
joint equations as:
LHS(j)/RHSU)
= RHS(k)/LHS(k) = 1.
For continuouschangesinj,k (withoutcrossingm), both ratiosare continuous.
1983
A Downsian Spatial Model withPartyActivism
989
withj = a, k = b, bothratiosequal zero.
b. By starting
c. Bythenadjustingandk (primarily
bymoving
themtowardeachother),whilekeeping
theequalityof thetworatios,as j approaches
k (and hencej and k approachm) theratioscan be madeto
increasetowardinfinity.
Atj = k = m, bothareinfinite.
d. By construction,
thetworatioswerekeptequal. Sincebothwentfromzeroto infinity,
they
i.e., a (j*,k*)exists.
equaledone together,
PROPOSITION
7.
A. PUi) is unimodaland symmetric,
fix) is unimodal.
1. Let t be themeanofAix),and recallthedefinitions
in Lemma2.
2. Suppose we considera one-partyequilibrium,withj = t. Then, fj=' xP(j)f(x)dx =
I b'_xP(j)f(x)dxifj is an equilibrium.
3. But,theremustbe a k e (Jb), withF(j,k) > 0, hencek < b.
4. BytheRationalChoiceCondition,
PUi) = 0 forall m < x 6 b.
5. Hence,forj* to be at t,theremustbe somepairsofpoints,say(eg) withd(ej) = d(gj) ande < j
< g < m withP(j/e) > P(j/g), in violationof Symmetry
ofPUi).
6. Bythesamereasoning,
t < k* < b.
B. PUi) satisfies
WeakExtremist
Support,notStrongExtremist
Supportand notSymmetery.
1. Forpairssuchas (eg) defined
in step5 above,byWeakExtremist
Support,
P(j/e) > P(j/g), and
forat leastone suchpair,theinequality
mustbe strict
(or PU) satisfies
Symmetry).
2. Consequently,
conPU) imposesmore"weight"ontheLHS andlessontheRHS oftheequilibrium
dition,whileP(k) workssimilarly.
3. Therefore,
thanin case a.
dUj*,k*)mustbe greater
C. PU) satisfies
StrongExtremist
Support.
1. Similarreasoning
applieshereas in comparing
casesa and b.
2. Hence,d(j*,k*)mustbe evengreater.
PROPOSITION
8. Ax) is unimodaland symmetric.
1. Let themodeoffx) be zero,constrain
j to equalk.
2. Beginwithj at pointa (hence,k at pointb). As before,
pair.
j = a, k = b is notan equilibrium
=
m
3. Asj continuously
approaches 0, LHS(J) oftheequilibrium
0
condition
goescontinuously
from
to F(j,k), RHS goes continuously
fromF(j,k) to zerowithbothF(j,k) > 0 (althoughpossibly
unequal).
4. Therefore,
a j* exists.Byconstruction
a k* existsat thesametimewithk* =PROPOSITION
9. AX)is bimodal.
1. Thatan equilibrium
pairexistsaboutmodey followsalmostimmediately
frompriorpropositions,
sincethemodeat z maybe very"small,"relative
to themodeaty. Thus,ftx)maybe bimodalbut
"almost"unimodal.The sameholdsforpartb.
2. Forpartc, supposeftx)
is symmetric
withmidpoint
of0 (andy = -z). Then,constrain]= - k. As
j movesfrompointa (or - b) to0, wegetthesamesortofequilibrium
result
as earlier.However,
as
m = 0, and bytheRationalChoiceCondition,
h(j) is positiveover[aOJonly,overwhich
fx) is
unimodal.
HenceviaLemma1,j*willbe "near"y,as inthe"one-party"
essentially
case.Thesame
willbe trueofk* simultaneously.
990
The AmericanPoliticalScienceReview
Vol. 77
as medians.
aredefined
centers
PROPOSITION 10. The party
ofj is] = [I b PUj)fXdxJ/Fjx
1. Thedefinition
if:
is that(j*,k*)arein equilibrium
condition
2. The newequilibrium
* a P(j)f(x)dx = Ij* P(j)f(x)dx and
k*: Ik P(k)f(x)dx= Ib*P(k)f(x)dx.
3. Notethatifi = a, thenfiJa PU)f(x)dx = 0, etc.
therestoftheaboveproofsfollow.
4. Bysimilar
reasoning,
7a,j* willbe lessthanthemedianofj(x) whichis,inturn,
5. Themajorchangeis that,inProposition
are:
lessthank*. Thatfollows,sincethecomparable
expressions
medianofJ(x) = s, such that: isJ(x)dx = i b
j(x)dx,
j* is an equilibriumif: j* PpU)f(x)dx = b P(j)f(x)dx, etc.
The same holds forLemma 2.
6. Otherthantheselocationresults,theremainderof theproofsholds for(jk) definedas themedian
insteadof the mean ideal pointlocation.
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