Risk diversification and vote decisions in mixed-member electoral systems

Shikano, Susumu; He on , E ik S.
A icle — Published Ve sion
Risk di e si ica ion and o e decisions in mixed-membe
elec o al sys ems
Public Choice
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Shikano, Susumu; He on , E ik S. (2025) : Risk di e si ica ion and o e decisions
in mixed-membe elec o al sys ems, Public Choice, ISSN 1573-7101, Sp inge US, New Yo k, NY, Vol.
204, Iss. 1-2, pp. 203-219,
h ps://doi.o g/10.1007/s11127-025-01301-5
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Public Choice (2025) 204:203–219
h ps://doi.o g/10.1007/s11127-025-01301-5
Risk di e si ica ion and o e decisions inmixed‑membe
elec o al sys ems
SusumuShikano1 · E ikS.He on2
Recei ed: 30 Sep embe 2024 / Accep ed: 31 May 2025 / Published online: 1 July 2025
© The Au ho (s) 2025
Abs ac
This pape builds on he li e a u e abou mixed-membe elec o al sys ems, explo ing how
ballo design in e ac s wi h o e beha io . We p esen a heo e ical model o o e deci-
sion-making in mixed-membe sys ems ha akes in o accoun he in e ac ion be ween
bo h ie s. The model is g ounded in a spa ial model o o e decision-making unde isk
and inspi ed by he logic o po olio di e si ica ion unde isk. Acco dingly, o e s a e
modeled as isk-a e se decision-make s who may p e e di e si ied o e packages (i.e.
spli - icke ) when pa y and candida e unce ain ies a e highly co ela ed. The isk di e -
si ica ion s a egy aba es when o e s cas hei o es sequen ially. This inding p o ides a
po en ial explana ion o he impac o o e sequence in mixed-membe sys ems, an unde -
in es iga ed opic in he li e a u e. I hus links he es ablished li e a u e on mixed-membe
sys ems wi h schola ship on ballo design and i s e ec s. Addi ionally, he pape ’s analy-
sis explo es he implica ions o combining he p oposed model wi h he well-es ablished
was ed o e model.
Keywo ds Mixed-membe sys ems· Vo e decision unde unce ain y· Risk
di e si ica ion· Vo e sequence.
1 In oduc ion
Mixed-membe elec o al sys ems ypically allow o e s o cas wo ballo s, one o plu ali y
and he o he o p opo ional ep esen a ion. While many o ing models exis o each ie ,
heo e ical models ha inco po a e bo h ie s a e sca ce. Fo example, models based on
o e s’ expec ed u ili ies p edic s a egic o ing in he o m o spli - icke o ing, bu hey
model o ing beha io in each ie independen ly and dis ega d any possible in e ac ions
be ween hem.
* Susumu Shikano
susumu.shikano@uni-kons anz.de
E ik S. He on
eshe [email p o ec ed]
1 Uni e si y o Kons anz, Uni e si ä ss . 10, 78464Kons anz, Ge many
2 Wes Vi ginia Uni e si y, Mo gan own, USA
204
Public Choice (2025) 204:203–219
This pape in oduces a heo e ical model ha examines o e decisions in mixed-mem-
be sys ems, aking in o accoun di e en in e ac ions based on he sys em ype. A he
indi idual o e le el, he model expands on con en ional app oaches by inco po a ing
join p e e ences o bo h ballo s. By conside ing isk a e sion, ce ain o e s choose o
educe isk by spli ing hei icke , e en i hei p e e ence would be o cas a s aigh icke
i hei decisions we e independen . The model also explo es he impac o ballo cas ing
me hods, dis inguishing be ween simul aneous and sequen ial o ing, and p edic s a ying
le els o spli - icke o ing. This inding o e s a po en ial explana ion o he in luence o
o e sequence in mixed-membe sys ems, which emains an unde -in es iga ed opic. Addi-
ionally, while he p oposed model is based on since e o ing, he analysis also explo es he
implica ions o inco po a ing s a egic o ing models, such as was ed o ing.
2 Rela ed li e a u e
The e is a subs an ial body o li e a u e explo ing he impac o mixed-membe sys ems
on o ing beha io and elec ion ou comes ( o an o e iew, see He on e al. 2018). These
s udies can gene ally be ca ego ized in o wo app oaches: con olled compa ison and con-
amina ion. The con olled compa ison app oach ea s mixed-membe sys ems as con-
olled expe imen al se ings, whe e wo dis inc elec o al sys ems ope a e wi hin he same
social, economic, and cul u al con ex s (e.g. Mose and Scheine 2012). This app oach
assumes ha he mic o-le el o ing decision p ocesses in each ie a e independen o each
o he . One s aigh o wa d implica ion o his app oach is ha o ing beha io in he p o-
po ional ep esen a ion (PR) ie ends o be since e, while s a egic conside a ions, such
as a oiding was ed o es, may in luence o ing beha io in he i s -pas - he-pos (FPTP)
ie (see e.g. Bawn 1999). Consequen ly, his could lead o a speci ic ype o spli - icke
o ing pa e n. Howe e , i is impo an o no e ha his is jus one possible implica ion.
Al e na i ely, i can also be pos ula ed ha o ing beha io in he PR ie is s a egic wi h
he aim o achie ing ce ain coali ion go e nmen s (Pappi and Thu ne 2002; Shikano e al.
2009). Rega dless o how o ing beha io is modeled in each ie , models based on his
app oach commonly ea he o ing decisions in each ie as independen o each o he .
In con as o he con olled compa ison app oach, he con amina ion app oach empha-
sizes he in e dependencies be ween he wo ie s o mixed-membe sys ems (Fe a a e al.
2005). Ea ly s udies in his app oach ha e demons a ed ha unning a candida e in a FPTP
dis ic can boos he pa y’s lis o es in he same dis ic , indica ing a con amina ion
e ec (He on and Nishikawa 2001). This inding sugges s ha o e s’ decision-making
p ocesses in bo h ie s a e no independen bu a he in e connec ed. Subsequen s udies
ha e p o ided e idence o a ious ypes o con amina ion e ec s (e.g. K auss e al. 2012),
while o he s ha e exp essed skep icism abou he exis ence o such e ec s (e.g. Maeda
2008; Ku ella 2016). I is impo an o no e ha he empi ical e idence on con amina ion
e ec s is o en de i ed om agg ega e-le el elec ion ou comes ac oss di e en coun ies.
This may con ibu e o he mixed indings. In his ega d, Rheaul e al. (2020) ad anced
ou unde s anding o con amina ion e ec s a he indi idual o e le el. Thei s udy shows
ha he decision o de be ween candida e and pa y o es, as well as he o e ’s le el o
poli ical in o ma ion, plays a c ucial ole, which possibly explains he mixed empi ical
indings in he li e a u e. Howe e , while hei con ibu ion is p ima ily empi ical, he he-
o e ical ounda ions o con amina ion e ec s a he indi idual le el emain less de eloped
in he li e a u e, pa icula ly in compa ison o he mo e es ablished con olled compa ison
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Public Choice (2025) 204:203–219
app oach. Recen ly, B äuninge and Pappi (2023) ad anced a o mal heo e ical model
add essing con amina ion e ec s by explici ly in oducing non-sepa able p e e ences in
mixed-membe o ing. Ye , hei wo k lea es open he subs an i e ques ion o why and
unde wha condi ions hese non-sepa able p e e ences a ise. We s ill need a clea mic o-
ounda ion o his non-sepa abili y.
While s udies o mixed-membe sys ems ha e p oli e a ed a leas since he 1990 s,
he e a e s ill se e al unde -in es iga ed opics wi hin his ield. One such opic is he e ec
o he ballo s uc u e (Ba nes e  al. 2017). Mixed-membe sys ems can adop di e en
ypes o ballo s uc u es. Fo ins ance, in Ge many, o e s ha e a single ballo o bo h
ie s, whe eas in Japan, o e s cas sepa a e ballo s o each ie . This di e ence in bal-
lo s uc u e leads o dis inc o e sequences, which can po en ially impac o ing beha -
io . When o e s ha e a single ballo , hey can simul aneously choose a candida e and a
pa y. In con as , sepa a e ballo s equi e o e s o make sequen ial decisions. These di -
e en o e sequences can in luence o ing beha io in a ious ways. In mul iple su ey
expe imen s, Shikano e al. (2023); He on and Shikano (2025) ound ha he Ge man-
s yle simul aneous ballo s acili a e mo e spli - icke o ing compa ed o sepa a e ballo s
o each ie . Thei esea ch also p o ides u he insigh s in o he mechanisms unde lying
o e decisions:
• Simul aneous ballo s equi e less o e all decision ime compa ed o sepa a e ballo s;
• when aced wi h sepa a e ballo s, o e s end o spend signi ican ly less ime on he sec-
ond ballo compa ed o he i s one;
• wi h simul aneous ballo s, o e s alloca e hei a en ion o bo h ie s be o e making
hei ini ial decision.
While hese esul s sugges he exis ence o an in e ac ion be ween bo h ie s, hey canno
be explained by ei he o he a o emen ioned app oaches. The e o e, in he ollowing sec-
ions, a new model will be p oposed ha akes hese indings in o accoun and p o ides a
amewo k o unde s anding o e decision-making in mixed-membe sys ems.
3 Theo e ical model
The p oposed model inco po a es he basic ideas ou lined below. In his model, o e s a e
aced wi h wo decisions: selec ing a dis ic candida e in he FPTP ie and choosing a
pa y lis in he PR ie . Thei choices a e d i en by he goal o maximizing hei u ili y
based on expec ed ou comes. The e a e wo ypes o ou comes: dis ic -speci ic po k and
na ional-le el poli ics. Dis ic candida es a e esponsible o he o me , while pa ies a e
esponsible o he la e .
Candida es and pa ies make announcemen s ha align wi h ce ain ou comes in he
e en o winning he dis ic ace o gaining go e nmen al powe . Vo e s a e assumed o
ha e knowledge o hei own ideals ega ding each ype o ou come, enabling hem o e al-
ua e he p oximi y o dis ance be ween hei posi ions and hose announced by candida es
and pa ies. The close he alignmen be ween o e s and a candida e o pa y posi ion, he
highe he u ili y o he o e .
Vo e s in his model a e p ima ily conce ned wi h he ac ual ou comes ha may di -
e om he announced posi ions. This is because he ealiza ion o announced posi ions
elies on he legisla i e p ocess ollowing he elec ion. Consequen ly, he e is inhe en
206
Public Choice (2025) 204:203–219
unce ain y associa ed wi h he announced posi ions and hei dis ance om he o -
e s’ own posi ions. In his ega d, o e s a e assumed o be isk-a e se, meaning hey
discoun u ili ies o ou comes wi h highe le els o unce ain y. Addi ionally, he unce -
ain y o candida es’ and pa ies’ posi ions may no be independen om each o he . Fo
ins ance, i is easonable o assume ha a pa y and i s candida es ha e a posi i e co e-
la ion in hei unce ain y, while a pa y and candida es om ano he pa y a e less co -
ela ed o independen . Vo e s also ake in o accoun his co a iance when cas ing bo h
candida e and pa y lis o es. Speci ically, a o e combina ion wi h a highe co a iance
will be discoun ed o educe he o e all isk in he package o ou comes. As a esul ,
some o e s may choose o spli hei icke , while hey would ha e p e e ed o cas a
s aigh icke i he o es we e cas independen ly.
The basic idea abo e is inspi ed by he logic o po olio di e si ica ion unde isk,
as o iginally o mula ed by Ma kowi z (1952). In his seminal wo k, Ma kowi z demon-
s a ed ha isk-a e se in es o s do no e alua e asse s in isola ion, bu ins ead conside
he expec ed e u n and isk o an en i e po olio. He e, he o al isk is shaped no only
by he a iance o indi idual asse s bu also by he co a iance be ween hem. Anal-
ogously, he model p esen ed he e ea s o e combina ions - candida e and pa y lis
o es - as a join decision unde unce ain y, whe e he o al u ili y is in luenced by bo h
he expec ed alignmen wi h a o e ’s ideal poin s and he isk associa ed wi h de ia-
ions in ealized policy ou comes. In line wi h he in es men logic, o e s a e modeled
as isk-a e se decision-make s who may p e e di e si ied o e packages when pa y
and candida e unce ain ies a e highly co ela ed.
He e, we ha e o make some poin s abou ou model clea : Fi s , ega ding ins i u-
ional di e ences be ween co ec i e mixed-membe sys ems (e.g., Ge many) and pa al-
lel sys ems (e.g., Japan) (Massico e and Blais 1999), ou basic se up is mo e consis en
wi h he la e , whe e dis ic -le el po k and na ional-le el go e nmen a e de e mined
sepa a ely by he FPTP and PR o es, espec i ely. Ne e heless, we a gue ha ou
model can also be applied o co ec i e sys ems, p o ided we assume ha o e s ocus
on he pe cei ed consequences o hei o es, a he han on he echnical de ails o
how o es a e agg ega ed ac oss ie s. C ucially, ou model depa s om he common
assump ion ha he wo o es a e decided independen ly. Ins ead, we ea o e choice as
a join decision unde unce ain y, d i en by he o e ’s conce n o cohe ence and isk
in he o e all policy ou come. He e, i is impo an o dis inguish be ween wo ypes o
independence: ins i u ional independence in how o es a e allied, and pe cep ual inde-
pendence in how o e s e alua e he isk ha elec ed ep esen a i es may di e ge om
hei announced posi ions a e he elec ion. Ou model is agnos ic abou he o me , bu
undamen ally g ounded in he la e . Second, ou model is no a game- heo e ic, bu a
decision- heo e ic model. This means ha indi idual o e s a e ea ed independen ly,
and hey make hei own decisions based on hei pe sonal ci cums ances which a e
assumed o be gi en. Las , bu no leas , ou model deals wi h one o many di e en
possible ypes o mo i a ion. O he ypes o mo i a ion, such as exp essi e mo i a ion
(Schuessle 2000), may also play a ole in spli - icke o ing. Howe e , he goal o his
pape is no o model all po en ial mo i es, bu a he o isola e and explo e one heo-
e ical mechanism: isk-a e se u ili y maximiza ion unde unce ain y. By ocusing on
ins umen al mo i es, we aim o p o ide a clea and o mal accoun o how o e s migh
e alua e o e combina ions as a package when pa y and candida e beha io s a e unce -
ain. This will ne e exclude he possibili y o ano he model in which exp essi e and
ins umen al mo i es may in e ac o compe e in such decisions.

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Public Choice (2025) 204:203–219
In he ollowing discussion, we will i s in oduce a simple model based on since e o ing
wi hou any s a egic conside a ions. Then, we will inco po a e he was ed o e model in he
FPTP ie .
3.1 Basic se up
Conside h ee-pa y compe i ion unde a mixed-membe sys em wi h
P={p1,p2,p3}
.
Each pa y can un one candida e in each dis ic and deno e he se o candida es in dis ic
d by
Cd⊆{cd,1,cd,2,cd,3}
o
d∈{1, …,D}
. A s a egy o o e i esiding in d is a pai
i=(c,p)∈P×Cd
.
Assume ha o e i has an ideal poin on he gene al policy dimension and he po k dimen-
sion. Since each o e can be eligible o o e only in one dis ic , we d op he index d in he
ollowing so ha :
cd
,
j=cj
. Vo e i’s bene i om
pj
and
cj
is he unc ion o he dis ance
be ween hei announced posi ion and i’s ideal poin , which we deno e by
di(cj)
and
di(pj)
.
Thei announced posi ions ha e some unce ain y
𝜖cj
and
𝜖pj
, which conce ns e.g. whe he he
announced posi ions will be implemen ed in legisla ion a e he elec ion. By using he
squa ed loss unc ion, we de ine i’s u ili y om candida e j’s winning he dis ic ace (
w
c
j
) as
ollows:
Analogously, we also de ine i’s u ili y om pa y j’s winning he go e nmen powe (
g
p
j
)
as ollows:
The expec ed alue o
𝜖cj
and
𝜖pj
is assumed o be ze o and hei a iances a e deno ed by
𝜎cj
and
𝜎pj
, espec i ely. Fu he , we assume hei independence om he o he componen s
in he u ili y unc ion:
The equi alen se -up o spa ial o ing model wi h unce ain y can be ound in he li e a-
u e (e.g. Enelow and Hinich 1984). Dis inc om he p e ious models, howe e , we ea
bo h decisions in FPTP and PR ie s simul aneously. And mo e c ucially, we allow ha
andom e ms o bo h ie s can be co ela ed wi h co a iance deno ed by
𝜎cj
,
pj
.
3.2 Independen choices inFPTP andPR
Be o e we discuss simul aneous choice unde mixed membe sys ems, we discuss independen
choices in FPTP and PR, espec i ely. In he ollowing, we deal wi h each o e ’s choice inde-
penden ly om each o he in a decision- heo e ic model. The e o e, we d op he index i in he
ollowing so ha
Ui(
⋅
)=U(
⋅
)
,
i=
,
di(cj)=dcj
and
di(pj)=dpj
. Fu he mo e, wi hou loss
o w.l.o.g we assume o FPTP and PR, espec i ely:
(1)
U
i(wc
j
)=−(di(cj)+𝜖cj)
2.
(2)
U
i(gp
j
)=−(di(pj)+𝜖pj)
2.
𝜖
c
j
⟂⟂ di(cj)
,
𝜖p
j
⟂⟂ di(pj)
,
𝜖c
j
⟂⟂ di(pj)
,
𝜖p
j
⟂⟂ di(cj)
,
(3)
dc1<dc2<dc3
(4)
dp1<dp2<dp3
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Public Choice (2025) 204:203–219
Fo bo h FPTP and PR, ou s a ing poin is he decision- heo e ic model o Black (1978)
abou s a egic o ing unde mul ipa ism. We ex end i by using he abo e se -up wi h o e
decision unde unce ain y. This model’s mos impo an elemen is he pi o p obabili y
pjj′
ha a single o e can de e mine who o candida e
cj
and
cj′
o
j
≠
j′
wins he mos
o es in he elec ion.
Acco ding o his model,1 a o e would cas he o e in a o o he second closes
c2
i :
•
p23
is la ge ela i e o
p12
and
p13
.
•
dc3−dc2
is la ge ela i e o
dc2−dc1
.
Fo he la e condi ion, Black did no explici ly conside unce ain y o he dis ance o can-
dida e j:
𝜎2
cj
. By aking i in o accoun , we can add he ollowing condi ions:
•
𝜎2
c3
−𝜎
2
c2
is la ge ela i e o
𝜎2
c2
−𝜎
2
c1
.
The Black model was o iginally de eloped o FPTP elec ions bu can also be ex ended o
o e decision-making unde PR. The key dis inc ion lies in he assump ion ha he pi o
p obabili y be ween any pai s o pa ies is cons an (
pp
). Howe e , e en in PR sys ems,
he e may a ise si ua ions whe e a o e de ia es om hei closes pa y in a o o he
candida e om hei second closes pa y. This can occu i :
In o he wo ds, we expec a PR o e in a o o he second closes pa y i he di e ence
be ween he dis ance o he announced posi ions is compensa ed by he second closes pa -
y’s mo e ce ain posi ion in compa ison o he closes one’s mo e unce ain posi ion. Fo
mo e de ails see Appendix B.
3.3 Simul aneous choice
I we ake he con olled compa ison app oach, we can employ he abo e models o a
single ie . In his subsec ion, in con as , we will model simul aneous choice in bo h ie s
wi h in e ac ion.
We assume ha he o e seeks o ha e he minimum squa e o he sum o dis ances o
he candida e and pa ies. This co esponds o he ollowing:
This unc ion ep esen s a join e alua ion o he o e all policy package a bo h he local
and na ional le els. The in ui ion behind summing he dis ances o he candida e and pa y
be o e squa ing is ha o e s o m a holis ic pe cep ion o isk ac oss bo h ie s. They hen
penalize la ge o e all de ia ions mo e, consis en wi h isk a e sion. This s uc u e also
cap u es o e s’ conce ns abou cohe ence o “alignmen ” be ween he candida e and he
pa y. Fo example, he g ea e he o al misalignmen ac oss bo h dimensions, he la ge
he po en ial o disappoin men o eg e in he policy ou comes – especially when o e s
(5)
d
p2
−dp1
<𝜎
2
p
1
−𝜎
2
p
2
(6)
U
(wc
j
,gp
k
) = −(dc
j
+𝜖c
j
+dp
k
+𝜖p
k
)
2
1 See o mo e de ails Appendix A.
209
Public Choice (2025) 204:203–219
a e unce ain abou how closely elec ed ep esen a i es will adhe e o hei announced
posi ions.
We s a wi h a special si ua ion whe e
C={c1}
. Tha is, he e is only one dis ic
candida e. The e o e, o e i has only h ee possible op ions
(c1,p1)
,
(c1,p2)
and
(c1,p3)
.
Appa en ly, he o e has no need o s a egically cas he o e in he FPTP ie .
Now conside whe he he e exis s a si ua ion in which a o e would spli he icke in
a o o
p2
. Fi s , we ob ain he expec ed u ili y o cas ing he s aigh icke (
c1,p1
) :
Analogously, o a spli icke (
c1,p2
)
The o e would spli i :
F om he e, we lea n ha he likelihood o aspli icke is highe i :
• he dis ances o he announced posi ions o Pa y 1 and 2 a e simila .
• he announced posi ions o bo h pa ies and Candida e 1 a e close o he o e .
Since
dp
2>
dp1
and all d’s a e non-nega i e, he le side o he las inequali y is la ge han
ze o. The e o e we ob ain a u he necessa y condi ion o
c1,p2
:
F om his inequali y, we can ob ain some u he insigh s conce ning unce ain y o he
announced posi ion:
• The la ge he unce ain y o Pa y 1’s posi ion and/o he smalle he unce ain y o
Pa y 2’s posi ion, he mo e likely is a spli icke .
• The mo e consis en he pa y and i s candida e in he s ochas ic e m, he mo e likely
he e will be a spli icke . In o he wo ds, i he candida es ha e highe po en ials o
beha e di e en ly om hei pa ies, o e s a e mo e likely o cas a s aigh icke .2
(7)
EU
( c1,p1)=−E[(dc1+𝜖c1+dp1+𝜖p1)
2
]
=−d2
c1−d2
p1−2dc1dp1−E[(𝜖c1+𝜖p1)2]
=−d2
c1
−d2
p1
−2dc1dp1−𝜎2
c1
−𝜎2
p1
−2𝜎c1,p
1
(8)
EU
( c1,p2)=−E[(dc1+𝜖c1+dp2+𝜖p2)
2
]
=−d2
c1
−d2
p2
−2dc1dp2−𝜎2
c1
−𝜎2
p2
−2𝜎c1,p
2
(9)
EU(
c1,p1
)<EU(
c1,p2
)
−
d2
c1−d2
p1−2dc1dp1−𝜎2
c1−𝜎2
p1−2𝜎c1,p1<−d2
c1−d2
p2−2dc1dp2−𝜎2
c1−𝜎2
p2−2𝜎c1,p
2
(dp2−dp1)(dc1+dp1+dp2)<𝜎2
p1
+2𝜎c1,p1−𝜎2
p2
−2𝜎c1,p2
(10)
𝜎2
p1
+2𝜎c1,p1−𝜎
2
p2−2𝜎c1,p2>0
𝜎2
p2
−𝜎2
p1
<2(𝜎c1,p1−𝜎c1,p2
)
2 No e ha his is one o he consequences o using he quad a ic loss o e he sum o dis ance om ideal
poin s. I we ins ead use he Euclidean dis ance, ha is
U
(c1,p1) = −(x
c
1+𝜖
c
1−𝜃
ci
)
2
+(x
p
1+𝜖
p
1−𝜃
pi
)
2
,
nei he
𝜎c
1,
p1
no
𝜎c
1,
p2
plays any ole.
210
Public Choice (2025) 204:203–219
This esul can be ex ended o he case whe e we ha e no only one candida e, bu
wo o h ee candida es. Fo illus a i e pu poses, we posi ion pa ies and candida es
along a one-dimensional spa ial axis (e.g.,
{1, 2, 4}
), espec i ely. This se up is no
mean o ep esen any speci ic pa y sys em, bu a he o cap u e a s ylized case whe e
wo op ions a e ideologically close and one is mo e dis an . This is a simpli ied, bu
ealis ic con igu a ion whe e pa ies a e posi ioned in an asymme ical way. The sub-
s an i e esul s o ou model, in pa icula hose conce ning isk di e si ica ion and o e
sequence, do no depend on his speci ic con igu a ion e en i we change he ela i e
dis ance be ween pa ies and/o he numbe o pa ies. While mo e complex placemen s
such as wo-dimensional s uc u es could yield u he insigh s, ou goal he e is o iso-
la e and demons a e he e ec o co ela ed unce ain y in he simples ac able se ing.
Bo h panels in Fig.1 p esen p edic ed o es o di e en o e posi ions in a ce -
ain candida e/pa y cons ella ion on he po k and na ional policy dimensions. The g ey
do ed g id lines di ide he elec o a e o di e en o ing pa e ns based on dimension-
by-dimension dis ance-based decisions. Tha is, bo h ie s a e ea ed as independen .
By con as , he solid lines di ide he elec o a e based on he model abo e which akes
in o accoun co a iance be ween candida es’ and pa y’s unce ain y. The le and igh
panels di e in co a iance, he le one is based on a highe and he igh one is based
on a lowe co a iance. I we ocus on he le panel, some o e s who a e p edic ed o
cas s aigh icke s in he dimension-by-dimension model would spli hei icke s in
ou model. These o e s a e e y close o he g ey g id lines, which means he closes
and second closes candida es/pa ies a e almos equidis an o hem. Fu he mo e, his
a ea becomes smalle i he dis ance on he o he dimension inc eases. Fo example,
i we look a he bo de a ea a
c1
,
p1
and
c1
,
p2
, i s a ea is he la ges i o e s a e e y
close o
C1
’s announced posi ion. This a ea becomes smalle i o e s a e mo e dis an
om
C1
. This co esponds o he second p edic ion equi ing o e closeness o pa ies’
and candida es’ posi ions. I we u n o he igh panel, he size o he bo de a eas is
much smalle han in he le panel. Tha is, i o e s expec he dis ic candida e would
beha e di e en ly in he legisla u e, hey a e mo e likely o s ick o he s aigh icke .
Fig. 1 Expec ed o ing beha io depending on announced posi ions unce ain y. The le panel is based on a
highe co ela ion be ween pa y and candida e announced posi ion (
𝜌
=
0.8
). The igh panel is based on a
lowe co ela ion (
𝜌
=
0.2
). The g ey do ed lines di ide o e s depending on he closes o e combina ions
217
Public Choice (2025) 204:203–219
Based on his inequali y, Black iden i ies wo condi ions o o e i’s de ec ion om he
closes candida e in a o o he second-closes one:
•
p23
is la ge ela i e o
p12
and
p13
.
•
EU(c2)−EU(c3)
is la ge ela i e o
EU(c1)−EU(c2)
.
Fo he la e condi ion, Black did no explici ly conside unce ain y
𝜎2
. The e o e, he second
condi ion in his e sion is as ollows:
•
dc3
−dc2
is la ge ela i e o
d
c
2
−d
c
1
.
By aking unce ain y in o accoun , we can add he ollowing condi ions:
•
𝜎2
c3
−𝜎
2
c2
is la ge ela i e o
𝜎2
c
2
−𝜎
2
c
1
.
B Ex ending Black’s model op opo ional ep esen a ion
The se -up is analogous o he abo e plu ali y case, bu he pi o -p obabili y will be subs an-
i ely di e en .
Assume a p opo ional ep esen a ion wi h a d’Hond sys em. The e a e N o e s who cas
alid o es and M sea s o dis ibu e.
A o e is decisi e on whe he one addi ional sea goes o Pa y j o
j′
i he expec ed esul
o j and
j′
wi hou he o e a s ake is
m×N
M
−
1
and
m�
×N
M
−
1
wi h
m,m�∈{1, …M}
. In o he
wo ds, bo h pa ies need one addi ional o e o each he nex quo a o ano he sea .
He e, he pi o -p obabili y unde plu ali y and p opo ional ep esen a ion is appa en ly di -
e en . Unde plu ali y, bo h candida es should be expec ed o ha e a simila amoun o o es
so ha he o e ’s pi o p obabili y among hem becomes la ge . This does no ha e o be he
case unde p opo ional ep esen a ion since a o e can ha e a high pi o p obabili y among
a pa y being expec ed o ob ain a la ge amoun o o es and ano he pa y being expec ed o
ob ain a small amoun o o es.
Gi en a la ge numbe o M, we can easonably assume
p
p
1
p
2
=p
p
1
p
3
=p
p
2
p
3
=p
p . I we
apply his o he abo e esul unde plu ali y, we ob ain:
EU(c
1
)−EU(
0
)<EU(c
2
)−EU(
0
)
2
p12(EU(c1)−EU(c2))+p13(EU(c1)−EU(c3))
−p
23(
EU(c
2
)−EU(c
3
)
)
<0
EU(
p1
)−EU(
0
)<EU(
p2
)−EU(
0
)
2
pp (EU(gp1)−EU(gp2))+pp (EU(gp1)−EU(gp3))
−pp (EU(gp2)−EU(gp3))<0
EU(gp1)−EU(gp2)<0
−d2
p1
−𝜎2
p1
+d2
p2
+𝜎2
p2
<0
d2
p2
−d2
p1
<𝜎
2
p1
−𝜎2
p2

218
Public Choice (2025) 204:203–219
Acknowledgemen s This esea ch was suppo ed by a g an om he Ge man Resea ch Founda ion
(Deu sche Fo schungsgemeinscha , DFG; No.659600). An ea lie e sion o he manusc ip was p esen ed
a he annual mee ing o he Wo king G oup on Analy ical Poli ical Theo y o he Ge man Poli ical Science
Associa ion in Hambu g in June 2023. We hank he pa icipan s o hei cons uc i e eedback. We a e
also g a e ul o F anz U ban Pappi and he h ee anonymous e iewe s o hei aluable commen s, which
signi ican ly imp o ed he manusc ip . Special hanks go o Lilli Becke and Maja S ahl o hei excellen
esea ch assis ance.
Funding Open Access unding enabled and o ganized by P ojek DEAL.
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