Spyche , Sa ah
Wo king Pape
Elec ions and poli ical pola isa ion: Challenges o
en i onmen al ag eemen s
Quade ni - Wo king Pape DSE, No. 1196
P o ided in Coope a ion wi h:
Uni e si y o Bologna, Depa men o Economics
Sugges ed Ci a ion: Spyche , Sa ah (2024) : Elec ions and poli ical pola isa ion: Challenges o
en i onmen al ag eemen s, Quade ni - Wo king Pape DSE, No. 1196, Alma Ma e S udio um -
Uni e si à di Bologna, Dipa imen o di Scienze Economiche (DSE), Bologna,
h ps://doi.o g/10.6092/unibo/amsac a/8001
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ISSN 2282-6483
Elec ions and Poli ical Pola isa ion:
Challenges o
En i onmen al Ag eemen s
Sa ah Spyche
Quade ni - Wo king Pape DSE N° 1196
Elec ions and Poli ical Pola isa ion: Challenges o
En i onmen al Ag eemen s
Sa ah Spyche
Depa men o Economics, Uni e si y o Bologna
spyc[email p o ec ed]
This e sion: No embe 2024
Abs ac
This pape examines he ole o domes ic elec ions and poli ical pola isa ion
in shaping in e na ional en i onmen al ag eemen s and how elec o al dynamics
may explain he limi ed success o cu en clima e coope a ion. I ocus on wo
key ac o s: he impac o domes ic elec o al p essu e on in e na ional policy
decisions and he misma ch be ween sho elec ion cycles and long- e m ea y
commi men s. Using a 4-s age game modelling a bila e al en i onmen al ag ee-
men , I analyse how incumben s s a egically balance policy p e e ences wi h
eelec ion p ospec s. Resul s show ha while a g een incumben is o en o ced
o empe hei ambi ions, a b own incumben aces ewe elec o al cons ain s,
explaining why s ingen policies a e ha de o achie e. None heless, elec o al
p essu e can mode a e policies, p oducing ou comes mo e aligned wi h he p e -
e ences o he median o e . Finally, I discuss how poli ical pola isa ion, pa -
icula ly in wo-pa y sys ems, adds complexi y o in e na ional coope a ion on
global public goods.
Keywo ds: in e na ional clima e policy, poli ical economy, elec ions, poli ical pola isa ion,
en i onmen al policy making, public goods, ex e nali ies
I would like o hank Jean-Michel Benke , Philipp B unne , Nadia Ceschi, Simon Die z, Niko Jaakkola, Igo
Le ina, Ralph Winkle and pa icipan s a he AURÖ Nachwuchswo kshop 2021 (Oldenbu g), SAEE Wo k-
shop 2021 (Zu ich), SURED 2022 (Ascona), EAERE 2022 (Rimini), EPS Fo um 2022 (G az), En i onmen al
and Ene gy Economics Wo kshop 2023 (Be n), SSES Annual Cong ess 2024 (Luce ne), and IIPF Annual
Cong ess 2024 (P ague), as well as semina pa icipan s a he Uni e si y o Be n, he London School o
Economics and he Uni e si y o Bologna o aluable commen s. This esea ch was suppo ed by a SNSF
Doc.Mobili y Fellowship (P1BEP1_199981) and a SNSF Pos doc.Mobili y Fellowship (P500PS_214320).
Non- echnical summa y
This pape examines how elec ions and poli ical pola isa ion impac in e na ional en i-
onmen al ag eemen s, o e ing a po en ial explana ion o he modes success o global
clima e coope a ion. The s udy ocuses on wo main ea u es o in e na ional en i onmen al
policy: i s , policymake s in ansna ional nego ia ions o en ace domes ic elec o al p es-
su es, shaping decisions such as o inc ease hei e-elec ion chances. Second, en i onmen al
ea ies ypically ex end beyond a single go e nmen ’s e m, c ea ing a iming misma ch,
as ea ies a e long- e m while elec ion cycles a e sho . This also means ha ea y ne-
go ia ion and a i ica ion may all o di e en adminis a ions, po en ially wi h con lic ing
policy goals depending on he le el o poli ical pola isa ion.
I use a ou -s age game o model wo coun ies conside ing se ing up an emissions educ ion
ea y, concen a ing on he poli ical dynamics in one coun y wi h elec o al compe i ion.
The cu en go e nmen nego ia es he ea y while weighing i s impac on he upcoming
domes ic elec ion. A e he elec ion, he newly elec ed go e nmen – po en ially om a
di e en pa y – decides on a i ica ion, ollowed by emissions decisions based on ha ou -
come. I p o ide a de ailed analysis o each s age in he game, ul ima ely cha ac e ising he
esul ing equilib ium ea ies. In he model, he wo poli ical pa ies in he i s coun y ha e
di e ing en i onmen al p e e ences: a g een pa y ha p io i ises clima e goals and a b own
pa y wi h less willingness o engage in clima e ac ion. Poli ical pola isa ion is de ined by
he dis ance be ween each pa y’s en i onmen al p e e ences and hose o he median o e .
The pape explo es di e en scena ios based on which pa y is ini ially in powe .
The esul s sugges ha an incumben go e nmen may in en ionally p opose a ea y ha
di e s om wha i would sugges wi hou elec ion p essu es, in o de o imp o e i s e-
elec ion chances. This is due o a ade-o aced be ween a pa y’s own policy p e e ences
and i s desi e o win elec ions, esul ing in policies ha appeal o he median o e . The
s udy iden i ies wo possible ea y ypes: consensus ea ies, designed o secu e a i ica ion
ega dless o he succeeding pa y, and di e en ia ion ea ies, whe e only one pa y would
a i y, o e ing o e s a clea con as in policy app oaches.
The indings e eal ha b own incumben s can achie e consensus ac oss a wide ange o
pola isa ion le els. G een incumben s, howe e , may need o mode a e hei ambi ions o
emain elec able, especially in highly pola ised se ings. This asymme y hin s a a eason
o limi ed clima e coope a ion: g een pa ies ace elec o al p essu e o empe ambi ious
plans, while b own pa ies a e no simila ly p essu ed o adop s ic e policies. Ne e heless,
elec o al p essu e o en nudges policies owa ds he middle, aligning hem mo e closely wi h
he median o e ’s p e e ences han policies c ea ed wi hou elec ion conce ns.
1 In oduc ion
An h opogenic clima e change is widely ecognised as one o he majo global en i onmen al
issues o ou imes. In he pas ew decades, he in e na ional communi y has add essed he
subjec by nego ia ing many in e na ional en i onmen al ag eemen s (IEAs), mos ecen ly
esul ing in he Pa is Ag eemen in 2015. Howe e , li le p og ess on clima e change mi -
iga ion can be obse ed: he cu en pledges as ag eed upon in he Pa is Ag eemen a e
no ambi ious enough o mee he ecognised policy goal o keeping he inc ease in a e age
su ace empe a u e well below 2◦C compa ed o p e-indus ialised le els. In addi ion, in
almos all coun ies, cu en g eenhouse gas (GHG) emissions a e abo e he pledged pa h
(UNEP 2023).
T ansna ional coope a ion on clima e change mi iga ion poses a undamen al challenge o
he in e na ional communi y: bo h he Pa is Ag eemen and i s p edecesso he Kyo o P o-
ocol indispu ably demons a e he di icul ies o achie ing ambi ious en i onmen al ag ee-
men s as well as he eluc ance o pa icipa ing coun ies o comply wi h emission a ge s
ag eed upon. This lack o success is no su p ising om an economic poin o iew: on
he one hand, mi iga ion o an h opogenic clima e change is impeded by he public goods
p ope y o GHG emission educ ions. Each coun y’s e o s o educe emissions bene i s
all coun ies in a non-exclusi e and non- i al manne , while cos s a e bo ne domes ically. A
he same ime, no sup ana ional au ho i y exis s ha migh en o ce an e icien ou come.
We he e o e obse e a global unde p o ision o emission educ ions.
In his pape I in es iga e he ole ha domes ic elec ions and poli ical pola isa ion play
o IEAs and o wha ex en hey migh be an explana o y ac o o he modes success
o cu en in e na ional coope a ion on clima e change mi iga ion. I ocus on wo key cha -
ac e is ics o in e na ional policy: i s , agen s in ol ed in in e na ional nego ia ions a e
o en subjec o domes ic elec o al conce ns and he e o e, policy decisions migh a ec
hei chances o eelec ion in upcoming elec ions. Second, in e na ional ea ies usually las
beyond a go e nmen ’s incumbency. This, on he one hand, leads o a empo al dispa i y in
he sense ha en i onmen al ea ies a e gene ally de ised o las o e a long pe iod o ime,
while elec ion cycles a e compa ably sho . On he o he hand, his implies ha he nego i-
a ion and he a i ica ion decision migh be made by wo di e en en i ies, which depending
on he le el o pola isa ion, migh pu sue e y dis inc en i onmen al policy goals.
A good example o such delibe a ions is he beha iou o he US du ing he nego ia ions
o he Kyo o P o ocol. Al Go e, se ing as ice p esiden in he Clin on adminis a ion,
pa icipa ed in nego ia ing wha was conside ed an ambi ious a ge om he US pe spec-
i e. Howe e , he adminis a ion was ully awa e ha he Sena e would likely ejec he
1
a i ica ion o such a ea y. One could a gue ha Go e, an icipa ing his p esiden ial un in
he upcoming elec ion, s a egically posi ioned himsel on en i onmen al policy o bols e
his elec o al p ospec s. His campaign was unsuccess ul, and he newly elec ed p esiden ,
Geo ge W. Bush, chose no o a i y he Kyo o P o ocol. This example unde sco es he
a ionale o sepa a ing nego ia ion and a i ica ion decisions in he model, as hey may be
unde aken by di e en ac o s.
Wi h hese conside a ions in mind, I o mula e a ou s age game o model a bila e al en i-
onmen al ag eemen , ocussing on he s a egic incen i es a ising om domes ic elec o al
p essu e. Two coun ies conside es ablishing a bila e al ag eemen on emission educ ions,
whe e he ocus lies on poli ical compe i ion wi hin coun y 1. In he i s s age, he incum-
ben in coun y 1 nego ia es a ea y, aking in o accoun how i s s ingency migh a ec
hei chances o eelec ion in he upcoming domes ic elec ion. Following he elec ion, which is
s ochas ic and depends on he median o e ’s wel a e, he new go e nmen decides whe he
o a i y he nego ia ed ea y. Finally, emission choices a e made based on he a i ica ion
decision. I p o ide a de ailed analysis o each s age in he game, ul ima ely cha ac e ising
he esul ing subgame-pe ec equilib ium ea ies.
Poli ical compe i ion wi hin coun y 1 in ol es wo i al pa ies: a g een pa y and a b own
pa y. The pa ies di e in hei willingness o pay o en i onmen al damage educ ion,
wi h one being mo e en i onmen ally ocused and he o he less so, ela i e o he median
o e . The p e e ence dis ance o ei he pa y o he median o e is wha is e e ed o
as he le el o poli ical pola isa ion in he model. Th oughou he pape , I will explo e
wo dis inc scena ios based on which pa y is in powe a he s a o he game. The
en i onmen al ea y is modelled as a coope a i e ag eemen be ween he wo coun ies o
educe emissions p opo ionally om he s a us quo. I he ea y is no a i ied ei he by
coun y 1 o 2, bo h coun ies de aul o he non-coope a i e emissions le el o he elec ed
pa y.
I ind ha incumben go e nmen s migh indeed op o a “subop imal” ea y – ela i e
o a scena io wi hou elec ion – o enhance hei chances o eelec ion. This is in luenced
by a ious poli ical economy ac o s and he deg ee o poli ical pola isa ion. Incumben s
ace a undamen al ade-o be ween hei policy p e e ences and hei eelec ion p ospec s:
choosing a ea y ha aligns wi h hei p e e ences migh nega i ely impac hei elec o al
chances, which can make i a p o i able s a egy o appeal o he median o e ins ead.
This dynamic is e lec ed in he op imal ea y choices, which can be classi ied as ei he
consensus o di e en ia ion ea ies. In a consensus ea y he incumben an icipa es he
possibili y o being eplaced in he upcoming elec ion and designs he ag eemen in a way
ha ensu es i s a i ica ion by hei po en ial successo , which is usually o he bene i o
2
he median o e . In con as , in a di e en ia ion ea y, only one o he pa ies a i ies he
ea y, leading o a si ua ion whe e he wo pa ies p esen di e ing en i onmen al policies
o o e s in he elec ion. This app oach in ol es s ee ing he ea y’s ambi ion away om
he challenging pa y’s p e e ences, and consequen ly, om wha he median o e would
a ou .
Equilib ium ea y ou comes e eal dis inc p essu es aced by g een and b own incumben s
in shaping clima e policy. A b own incumben can achie e consensus ac oss a wide ange o
pola isa ion le els, s aying ue o hei policy p e e ences wi h limi ed elec o al concessions.
By con as , a g een incumben aces a much na owe se o condi ions unde which high-
ambi ion policies can gain consensus. This asymme y a ises because he g een pa y is
gene ally willing o a i y mos p oposals om a b own incumben , whe eas a b own pa y
will only ag ee o a i y a g een incumben ’s ea y unde low le els o pola isa ion. These
dynamics c ea e a s a egic bind o he g een incumben : while ambi ious policies may
e lec i s pla o m, scaling back becomes essen ial o main ain elec o al iabili y, especially
when pola isa ion is high, and knowing ha a loss could lead o a non-coope a i e ou come
unde a b own successo – po en ially mo e ha m ul han he s a us quo. These dynamics
can help explain he pe sis ence o modes clima e coope a ion success: he pa y pushing
o ambi ious policies is es ained by elec o al dynamics, while he opposing pa y aces
li le p essu e o engage in ambi ious clima e ac ion. On he posi i e side, elec o al p essu e
ends o mode a e policies in a classic con e gence o he middle manne , o en esul ing in
wel a e ou comes mo e closely aligned wi h he p e e ences o he median o e han would
be achie ed wi hou elec ions.
Conside ing domes ic poli ical pola isa ion in he discussion o in e na ional coope a ion is
no el, and I demons a e ha his aspec c ucially a ec s ou comes. This goes beyond he
opic o en i onmen al policy: I can illus a e how in a wo-pa y sys em, he p o ision o a
sha ed public good in gene al becomes mo e complex wi h poli ical pola isa ion, an inc eas-
ingly widesp ead phenomenon ac oss di e se na ional con ex s (Ca o he s and O’Donohue
2019). Fu he mo e, gi en he ac ha he US, a p ominen example o a wo-pa y democ-
acy, is one o he majo global playe s when i comes o in e na ional coope a ion, his
connec ion will become inc easingly ele an .
2 Rela ed Li e a u e
The ques ion o how elec ions a ec policy choices, and ice e sa, has been widely discussed
in con ex s ou side o in e na ional (en i onmen al) coope a ion. Pe sson and Tabellini
3
(1992) show ha poli ical p ocesses such as elec ions may dis o ax a e choices com-
pa ed o wha a social planne would do, while Besley and Coa e (1998) highligh how
iscal policy in es men s can be used o in luence u u e elec ions. Robinson and To ik
(2005) show how ine icien in es men s in local in as uc u es migh s em om a emp s
o in luence elec ions. Howe e , in Pe sson and Tabellini (1994), con a y o a majo i y
o he public choice li e a u e, he au ho s ind ha poli ical incen i es may also imp o e
he equilib ium ou come h ough mo e c edible commi men . Add essing how incumben
go e nmen s can in luence policies o hei successo s, Alesina and Tabellini (1990) discuss
he ole o public deb as a means o limi ing expendi u es. My pape con ibu es o his
li e a u e by adding insigh s in o he nexus o economic policy and poli ical compe i ion in
he con ex o c oss-bo de public goods p o ision, speci ically in an en i onmen al con ex .
The phenomenon o poli ical pola isa ion has ga ne ed signi ican a en ion ac oss a ious
disciplines due o i s impac on policymaking in democ a ic sys ems. Es eban and Schneide
(2008) a gue ha domes ic pola isa ion diminishes coun ies’ willingness o con ibu e o
global public goods, such as in e na ional secu i y. A he domes ic le el, pola ised poli ical
sys ems and declining poli ical di e si y weaken go e nmen s’ abili y o p o ide public goods
(Le in e al. 2021). Mo eo e , Bake e al. (2020) ind ha in ense poli ical pola isa ion in
he US, pa icula ly du ing closely con es ed elec ions, leads o spikes in economic policy
unce ain y. And eo ola and Li (2024) analyse how mass pola isa ion, ha is pola isa ion
among he elec o a e, in luences e o m design. They a gue ha he incumben has wo
iable s a egies: ei he o inc ease hei eelec ion chances o o design a e o m ha ap-
peals o he opposi ion, he eby educing he chance o epeal in case o elec ion loss. This
las esul p o ides an in e es ing pa allel o my inding o consensus and di e en ia ion
ea ies, depending on he deg ee o pola isa ion and he size o he o ice en . Few s ud-
ies ha e examined he ole o poli ical pola isa ion in he con ex o en i onmen al issues.
Aus en-Smi h e al. (2019) show ha pola ised se ings encou age ine icien en i onmen al
policies, as hese a e easie o e e se. In es iga ing na ional esponses o changes in IEAs,
Pe ings e al. (2021) show ha pa y pola isa ion weakens a coun y’s commi men o hese
ag eemen s. Howe e , s akeholde s who penalise pa ies o adop ing ex eme posi ions can
mode a e hese e ec s, akin o he mode a ing in luence o elec o al p essu e in my model.
In en i onmen al economics, he heo e ical analysis o sel -en o cing IEAs o en elies on
non-coope a i e game heo y, pa icula ly coali ion o ma ion games since he ea ly 1990s
(e.g., Ca a o and Siniscalco 1993; Hoel 1992; Ba e 1994). These models ypically in-
ol e wo s ages: coun ies i s decide whe he o join he ag eemen , and hen signa o-
ies in e nalise emission ex e nali ies, while non-signa o ies ac non-coope a i ely (Wagne
2001). Such models gene ally yield pessimis ic conclusions, p edic ing small coali ions and
widesp ead ee- iding, pa icula ly when coope a ion would yield la ge gains. This con-
4
adic ion, known as he pa adox o in e na ional en i onmen al ag eemen s (Kols ad and
Toman 2005), a ises because nume ous la ge IEAs exis in p ac ice. Finus and Maus (2008)
add ess his by in oducing he concep o modes IEAs, whe e only a ac ion o emission
ex e nali ies is in e nalised. They show ha less ambi ious ag eemen s lead o la ge coali-
ion sizes, wi h he bene i s o b oade membe ship ou weighing he highe emissions by
indi idual membe s. Simila explana ions o la ge bu less ambi ious coali ions include
Ba e (2002), Aldy e al. (2003), and Ha s ad (2022).
Mos en i onmen al economics models, including wide- anging ex ensions, ea coun ies
as homogeneous en i ies, ep esen ed by a single bene olen decision-make , o e looking he
po en ial in e play be ween domes ic and in e na ional en i onmen al policy (Finus 2008).
To add ess his, a g owing body o li e a u e in oduces hie a chical s uc u es in o models
o in e na ional coope a ion, inco po a ing insigh s om he poli ical science li e a u e, in
which he ela ionship be ween domes ic and in e na ional policy is desc ibed as a wo-le el
game (Pu nam 1988). This app oach dis inguishes be ween di e en go e nmen al bodies
wi hin coun ies and emphasises he need o accoun o poli ical economy ac o s like in-
e es g oups (Ma chio i e al. 2017; Hagen e al. 2021), elec o al conce ns (Buchholz e al.
2005; Siquei a 2003), and domes ic poli ical s uc u es (Loepe 2017). Fo ins ance, Spyche
and Winkle (2022) show ha by accoun ing o he hie a chical s uc u e o in e na ional
clima e policy ia he in oduc ion o a s a egic delega ion s age, ”b oad-and-deep“ ag ee-
men s can be s abilised.
So a , only ew pape s combine in e na ional en i onmen al coope a ion wi h na ional po-
li ical compe i ion. Köke and Lange (2017) analyse he impac o a i ica ion cons ain s
in a s a egic o ing model, whe e hey dis inguish be ween ” ep esen a i es“ nego ia ing
ag eemen s and ”pi o al agen s“ deciding on a i ica ion. They o mula e a coali ion o ma-
ion game and show how poli ical dynamics wi hin coun ies in luence he size and scope o
clima e ag eemen s, o e ing a public choice mo i a ion o he indings o Finus and Maus
(2008) by emphasising he ole o poli ical economy aspec s in shaping in e na ional nego ia-
ions. Ba aglini and Ha s ad (2020) model elec o al conside a ions in clima e nego ia ions,
ocussing on ea y design a he han pa icipa ion. In a simple wo-coun y amewo k,
hey show ha incumben go e nmen s may sign ”weak ea ies“, ha is ea ies wi h oo
low a le el o sanc ions o gua an ee compliance, in o de o in luence u u e elec ions in
hei a ou . Fo ins ance, a go e nmen wi h low en i onmen al p e e ences signs a ea y
in ol ing sanc ions small enough o he median o e o a ou non-compliance and hus
eelec ion o he incumben . This incen i e is pa icula ly s ong when he bene i s o s aying
in o ice a e signi ican .
Coming mo e om a poli ical science angle, Buisse e and Be nha d (2018) explo e how
5
ˆeBˆeG˜eG
(a) G een incumben
ˆeBˆeG˜eB
(b) B own incumben
Figu e 2: Emission Le el Scena ios
illus a ed in Figu e 2b. In his case, he g een challenge would, unde a a i ied ea y,
educe emissions by less han wha hey would do non-coope a i ely. A possible a gumen
in a ou o his assump ion could be ha he ea y nego ia ion essen ially cap u es he
se ing up o an in e na ional pe mi ma ke , whe e he nego ia ing pa y decides on he
pe mi supply, which o some ime ho izon a e he elec ion is ixed. Ano he a gumen
could be ha i he g een challenge we e o olun a ily unde cu ea y commi men s, hey
would po en ially o ego he oppo uni y o nego ia e a new and mo e ambi ious ea y in
he nea u u e. No wi hs anding he unde lying causes, his line o a gumen alls ou side
he scope o his model amewo k. Howe e , in Sec ion 5.1 I explo e an ex ension in which
he assump ion o binding ea y emission is elaxed, allowing he ea y o se e me ely as
an uppe bound, which he uling pa y can unde cu a hei disc e ion.
Wi hou loss o gene ali y, we can no malise he p oblem by se ing α== 1. Also,
h oughou he pape , he ange β∈[0,0.15] o he ma ginal damage pa ame e will be
assumed. This is a e y non- es ic i e assump ion, since β= 0.15, om a median o e
pe spec i e, would imply he wo s case scena io o unmi iga ed clima e change co espond-
ing o app oxima ely a 45% dec ease in global GDP. While his is ce ainly much highe
han wha is commonly ound o be ealis ic wi h espec o clima e change in a global
con ex (see, e.g., Hänsel e al. 2020), allowing o such high alues migh make sense om
a mo e local pe spec i e, o example o coun ies in Sou heas Asia, whe e local impac s
a e expec ed o be signi ican ly highe han he global a e age (see, e.g., Swiss Re 2021).
In any case, allowing o such high alues o βensu es ha esul s a e no d i en by a oo
op imis ic e alua ion o en i onmen al damages.
4.2 Ra i ica ion S age
Whoe e is in cha ge a he a i ica ion s age, ha is, he elec ion winne in S age 2, will
decide whe he o a i y he ea y on he able. As p e iously s a ed, we assume ha
enego ia ion o he ea y is no a ailable o he go e ning pa y a e he elec ion. This is
12
a easonable assump ion in he con ex o his model, since ea y nego ia ions usually ake
place o e a long ime ho izon and ha i is no possible o a newly elec ed go e nmen o
one coun y o immedia ely enego ia e an in e na ional ag eemen , as seen, o example,
in he case o he US and he Kyo o P o ocol. This ea u e o he model con as s he se up
o Buisse e and Be nha d (2018) as well as Melnick and Smi h (2022), whe e ag eemen s
made be o e an elec ion se e as a s a ing poin o any subsequen enego ia ion.
4.2.1 Ra i ica ion In e als
The incumben io he challenge j, when elec ed, a i y he ag eemen whene e :
∆Wh=Wh(˜e1(δi, θi),˜e)−Wh(ˆe1(θh),ˆe)≥0, h =i, j . (9)
This leads o wo in e als o which elec ed pa y will a i y he ag eemen , each de ined
by he wo h eshold alues o δ:
[δi,¯
δi] = max0,1 + βθi(β(2 + θi)−4)
(βθi−1)2,1,(10)
[δj,¯
δj] = "max0,1−β[θi−θj(β+βθi−2)] −√M
(βθi−1)2,
min1−β[θi−θj(β+βθi−2)] + √M
(βθi−1)2,1#,(11a)
whe e M=β2θj(β−1) [θj(β+ 2βθi−3) −2θi(βθi−1)] (11b)
No e ha (11b) has o be non-nega i e o (11a) o be well de ined. Also, he incumben ’s
uppe h eshold alue co esponds o no emission educ ions, ha is, o he non-coope a i e
ou come.
We will see in he ollowing ha i depends on which pa y is he incumben and which is
he challenge in o de o s a e how hese h esholds ela e o each o he . We de ine:
∆δ≡δi−δj,(12a)
∆¯
δ≡¯
δi−¯
δj,(12b)
which de ine he espec i e o de o he wo pa ies’ lowe and uppe a i ica ion h esholds.
13
While he incumben o coun y 1 sugges s a ea y pa ame e δi, coun y 2 is assumed o
no ha e any nego ia ion powe . S ill, coun y 1’s ea y sugges ion is limi ed by coun y 2’s
pa icipa ion cons ain , ha is, coun y 2 mus no be wo se o han unde no ag eemen , in
which case hey expec he incumben ’s non-coope a i e emission choice. The co esponding
a i ica ion h esholds o coun y 2 a e hen gi en as ollows:
∆W2=W2(˜e2(δi, θi, θ2),˜e)−W2(ˆe2(θ2),ˆe)≥0(13)
⇔[δ2(θi),¯
δ2(θi)] = 1 + β(2βθi+β−4)
(β−1)2,1.(14)
Coun y 2’s a i ica ion in e al only depends on he incumben s’ p e e ence pa ame e .
This is due o he ac ha hei ea y pa ne in he ag eemen s age is he incumben
and e en i he challenge we e o win he elec ion and a i y he ea y, coun y 2’s emission
educ ion commi men only ela es o he ea y signed wi h he incumben . Implic ly, we
assume coun y 2 o no be sophis ica ed enough o an icipa e he possibili y o acing he
challenge ’s non-coope a i e ou come i hey we e elec ed and did no a i y.
The h eshold alues (10) and (11a) can be o de ed esul ing in a pa i ion o anges o he
ea y pa ame e . The o de ing will depend on which pa y is he incumben and he size
o he pola isa ion pa ame e φ. Fou di e en cases can eme ge:
(A) δ∈[δi,¯
δi], δ 6∈ [δj,¯
δj]: only he incumben a i ies,
(B) δ6∈ [δi,¯
δi], δ ∈[δj,¯
δj]: only he challenge a i ies,
(C) δ∈[δi,¯
δi], δ ∈[δj,¯
δj]: bo h a i y,
(D) δ6∈ [δi,¯
δi], δ 6∈ [δj,¯
δj]: none a i y.
No e ha echnically, he e is an addi ional case in which a ea y ha would be a i ied by
a leas one o he pa ies in coun y 1 bu no by coun y 2. Howe e , om a heo e ical
poin o iew he e is no di e ence o case D ega ding he esul ing emission choices. Thus,
hence o h, his scena io will be cap u ed by case D.
In he ollowing, we will discuss speci ic a i ica ion in e als in he con ex o a g een and
a b own incumben .
G een incumben
In a i s s ep, le us assume he g een pa y se es as he incumben . We can he e o e
assign p e e ence pa ame e s θi= 1 + φand θj= 1 −φ. No e ha i he pa ies a e oo
pola ised, he challenge would ne e a i y a ea y sugges ed by he incumben . This is
14
he case i no eal alue δisa is ies (9), ha is, whene e (11b) is nega i e. This holds ue
o pola isa ion alues φ≤¯
φG(β), whe e d¯
φG
dβ >0, as shown in Lemma A.12. This implies
ha o highe en i onmen al damages, a a i ica ion in e al o he b own challenge exis s
o highe deg ees o pola isa ion.
We can now speci y a i ica ion in e als (10) and (11a) in case o a g een incumben as
de ailed in P oposi ion A.1, wi h co esponding compa a i e s a ics in P oposi ion A.2. We
ind ha while he incumben ’s uppe h eshold is independen o φ, hei lowe a i ica ion
h eshold dec eases wi h he dis ance om he median o e . In ui i ely his means ha a
g eene incumben will be mo e willing o a i y s ic ea ies. Fo he b own challenge ,
he uppe h eshold dec eases and he lowe h eshold inc eases in φ, meaning ha hei
a i ica ion in e al becomes mo e na ow wi h inc easing pola isa ion. Consequen ly, highe
pola isa ion leads o a mo e na ow ange o δi ha would allow o a i ica ion by bo h
pa ies.
P oposi ion 1 (O de ing o Ra i ica ion Th esholds wi h i=G)
1. In he case o he g een incumben and whene e φ≤¯
φG, he a i ica ion h esholds
(10) and (11a) ela e o each o he as gi en in he ollowing:
∆δi=G≤0,(15a)
∆¯
δi=G≥0.(15b)
2. The lowe a i ica ion h eshold o pa ies in coun y 1 ela e o ha o coun y 2 as
ollows:
δi=G−δ2≤0,(16a)
δj=B−δ2≥0.(16b)
The i s pa o P oposi ion 1 s a es ha he g een incumben will always sign s ic e
ea ies han he b own challenge 3. While no ea y is unambi ious enough o he in-
cumben ( echnically hey can nego ia e a ea y wi h δi= 1, which co esponds o hei
non-coope a i e ou come), he challenge will no always sign such ea ies. The eason o
his is ha i emission educ ions a e negligible, he posi i e e ec o damage educ ions
(also ia less ex e nali ies om coun y 2) does no ou weigh he nega i e e ec o no being
able o choose emissions eely acco ding o hei own op imal non-coope a i e ou come.
2Hence o h, in e media e esul s a e elega ed o Appendix A.
3The p oo s o all p oposi ions p esen ed in he main ex a e p o ided in Appendix B.
15
The second pa hen s a es ha coun y 1 wi h a g een incumben is willing o a i y mo e
s ic ea ies han allowed o by coun y 2’s pa icipa ion cons ain . This implies ha in
some cases, coun y 2’s lowe a i ica ion h eshold is binding, a he han ha o he g een
incumben . The b own challenge , con e sely, a i ies less ambi ious ea ies han coun y
2.
This leads us o a discussion o which cases a ise along he spec um o possible ea y
pa ame e s, ollowing P oposi ion 1:
0
φ≤¯
φG1
δiδ2δj¯
δj¯
δi=¯
δ2
D A C A
0
φ > ¯
φG1
δiδ2¯
δi=¯
δ2
D A
Figu e 3: G een incumben a i ica ion h esholds
The wo scena ios a e dis inguished by whe he a i ica ion h esholds o he challenge
exis , ollowing he h eshold alue ¯
φGas de ined by Lemma A.1. I he h esholds o he
challenge do no exis , he easible ange o he ea y pa ame e is limi ed by coun y 2.
No e ha he colou s in Figu e 3 will hence o h indica e which pa ies a i y he ea y:
g een and b own o he wo pa ies espec i ely, yellow o bo h pa ies and ed o none.
B own incumben
Nex , we conside he scena io in which he b own pa y is he incumben and he e o e
en i onmen al p e e ence pa ame e s a e gi en by θi= 1 −φand θj= 1 + φ. Ra i ica ion
in e als ollowing (10) and (11a) a e hen s a ed in P oposi ion A.3.
We ind ha he lowe a i ica ion h eshold o he b own incumben inc eases in he
dis ance om he median o e , whe eas he uppe h eshold is independen o φ, as de ailed
in P oposi ion A.4. In ui i ely, he b owne he incumben , he less s ic he ea y can be
o hem o a i y. Fo he challenge , highe pola isa ion dec eases bo h uppe and lowe
h esholds. The g eene he challenge , he mo e ambi ious he ea y can be on he lowe
end and has o be on he uppe end, o hem o a i y. I a ea y is oo weak, he damage
educ ion does no compensa e o insu icien ly low emission le els. This essen ially means
ha hei a i ica ion in e al shi s downwa ds.
16
P oposi ion 2 (O de ing o Ra i ica ion Th esholds wi h i=B)
1. In he case o he b own incumben , he a i ica ion h esholds (10) and (11a) ela e
o each o he as gi en in he ollowing:
∆δi=B≥0,(17a)
∆¯
δi=B≥0.(17b)
2. The incumben ’s lowe a i ica ion h eshold ela es o ha o coun y 2 as ollows:
δi=B−δ2≥0,(18a)
δj=G−δ2≤0.(18b)
The i s pa o P oposi ion 2 s a es ha he g een challenge will sign mo e ambi ious
ea ies han he b own incumben . Howe e , a he uppe end o he spec um, he e a e
e y unambi ious ea ies ha he incumben will sign and he challenge no . In ui i ely,
hese a e con ac s ha a e so unambi ious in e ms o damage educ ion ha he g een
challenge is be e o wi h hei non-coope a i e ou come.
Secondly, P oposi ion 2 s a es ha coun y 1 wi h a b own incumben is willing o a i y
less ambi ious ea ies han would be allowed o by coun y 2’s pa icipa ion cons ain ,
and ice e sa o he g een challenge . Consequen ly, he ollowing cases a ise:
0
φ≤¯
φB1
δjδ2δi¯
δj¯
δi=¯
δ2
D B C A
0
φ > ¯
φB1
δjδ2¯
δjδi¯
δi=¯
δ2
D B D A
Figu e 4: B own incumben a i ica ion h esholds
The wo scena ios a e sepa a ed by whe he a i ica ion in e als o he incumben and
challenge o e lap o no , ha is, depending on δi≶¯
δj. This condi ion yields a h eshold
alue o pola isa ion ¯
φBas de ined in Lemma A.2. In case o no o e lap, his means ha no
ea y pa ame e leads o a i ica ion by bo h pa ies, as depic ed in he second scena io (no
a ea C) in Figu e 4. Hence, i pola isa ion is e y high, he e exis s an in e al o he ea y
pa ame e , which will no be signed by any o he wo pa ies, since i is oo ambi ious o
he b own incumben while being no ambi ious enough o he g een challenge .
17
4.3 Elec ion S age
The domes ic elec ion is modelled as de ised by Ba aglini and Ha s ad (2020). The median
o e aces he choice be ween he incumben iand he challenge jand conside s how each
will a ec hei wel a e: depending on he elec ion ou come, coun y 1 can ei he (i) be pa
o he ea y and choose emissions as nego ia ed by io (ii) li e in a wo ld wi hou a ea y
whe e bo h coun ies choose he non-coope a i e ou come. In he o me case, bo h pa ies
will ac iden ically whe eas in he la e , he ou side op ion di e s be ween he wo. The
median o e can, as de ined by he a i ica ion in e als de i ed in Sec ion 4.2, an icipa e
wha he consequence o elec ing ei he o he wo pa ies is.
The median o e ’s wel a e di e ence be ween a go e nmen iand jis deno ed by ∆WM
and he incumben is consequen ly e-elec ed whene e :
∆WM≡Wi
M−Wj
M≥Ω,whe e Ω∼U−z
σ,1−z
σ.(19)
The pa ame e z≥0.5quan i ies an incumbency ad an age, ha is, he eelec ion p obabil-
i y o he incumben in he absence o any policy di e ences be ween he wo pa ies. The
pa ame e σcap u es he densi y o a popula i y shock. A high alue o σ(low a iance)
means ha policy di e ences a e mo e likely o dic a e he ou come o he elec ion, whe eas
low alues o σ(high a iance) inc ease noise and hus make andom popula i y shocks mo e
impo an . The pa ame e can he e o e also be in e p e ed as a alue o policy salience.
An example o such a shock could be an exogenous change in he poli ical clima e wi h e-
spec o en i onmen al issues, as o example, he Fukushima nuclea disas e in 2011. Fo
eelec ion p obabili ies o be in e io in (0,1), he a iance in he popula i y shock is limi ed
o be σ < ¯σas de ined in Lemma A.3, which howe e does no es ic he p esen ed esul s
in la e sec ions. P oposi ion 3 gi es a ull cha ac e isa ion o he eelec ion p obabili y o
he incumben pa y in S age 2, as a consequence o (19).
P oposi ion 3 (S age 2: Reelec ion P obabili ies)
Gi en ha σ < ¯σand z≥0.5, eelec ion p obabili ies o cases A – D a e de ined by:
pl(δi) = σ∆Wl
M+z l ∈ {A, B, C, D},(20)
18
wi h
∆WA
M=WM(˜e1(θi, δi),˜e)−WM(ˆe1(θj),ˆe)(only he incumben pa y a i ies)
∆WB
M=WM(ˆe1(θi),˜e)−WM(˜e1(θi, δi),ˆe)(only he challenging pa y a i ies)
∆WC
M= 0 (bo h pa ies a i y)
∆WD
M=WM(ˆe1(θi),ˆe)−WM(ˆe1(θj),ˆe)(none o he pa ies a i y).
This eelec ion p obabili y is a unc ion o he ea y pa ame e δi ha de e mines which
case A–Deme ges. The e o e, he eelec ion p obabili y be ween cases di e s, since Wi
M
and Wj
Mdepend on whe he a i ica ion occu s in S age 3. S aigh o wa dly, he median
o e ’s wel a e le el is a ec ed in he cases whe e he incumben and he challenge will ake
di e en a i ica ion decision (Aand B). In he case whe e bo h pa ies will a i y C, his
is no he case because he challenge is ied o he ea y nego ia ed by he incumben . In
he las case D, again he median o e ’s wel a e le els a e di e en since he wo pa ies
will choose di e ing non-coope a i e emission le els. No e ha he eelec ion p obabili y is
a unc ion o he ea y pa ame e in cases Aand Bbu no in cases Cand D.
4.4 Ag eemen S age
The incumben go e nmen nego ia es an ag eemen such ha hei expec ed wel a e is
maximised:
max
δi
p(δi)hWi(’i in powe ’) + Ri+ (1 −p(δi))hWi(’j in powe ’)i,(21)
and Wi(·)and p(δi)depend on cases A–D, as de ailed in he p e ious sec ions.
Rdeno es he en om s aying in o ice, which can cap u e any inhe en bene i s om
s aying in powe . Ba aglini and Ha s ad (2020) e e o Ras an indi ec measu e o
poli ical pola isa ion: he u he apa he wo pa ies, he mo e impo an holding he
o ice is, o example, o in luence domes ic policy un ela ed o emission choice. Fu he mo e,
he le el o o ice en s migh di e be ween poli ical sys ems: p esiden ial sys ems would
hen be associa ed wi h highe alues o R, as opposed o pa liamen a y sys ems, in which
he su plus om being in o ice is mo e sp ead ou ac oss poli ical ac o s and being in o ice
comes wi h less powe o push one’s own agenda.
The incumben ’s objec i e unc ion illus a es he undamen al ade-o ha hey ace:
choosing he ea y which maximises hei wel a e unc ion when in powe , ha is ˆ
δi, migh
no be op imal when conside ing he e ec his choice has on he eelec ion p obabili y. I
19
can he e o e be a p o i able s a egy o adjus he ea y pa ame e such as o in luence
eelec ion p ospec s as well as he wel a e le el in case o elec ion loss in a a ou able way.
The incen i e o inc ease he eelec ion p obabili y is pa icula ly s ong when he o ice
en is high: in ha case, he ela i e weigh o he ac ual policy choice is educed and
s aying in powe becomes mo e impo an .
When sol ing his ade-o , he incumben go e nmen chooses δi, pe ec ly an icipa ing
which case will ma e ialise acco ding o he de i ed a i ica ion in e als. Due o he ac
ha (21) is a non-smoo h unc ion, hey compu e expec ed maximum wel a e le els o
each case A-Dand hen op o he case yielding he highes expec ed wel a e and he
co esponding op imal ea y in ha ange.
No e ha in he ange in which bo h pa ies would a i y he ag eemen , ha is, case C,
i is wel a e-maximising o he incumben o se δ∗
i=ˆ
δi. This is due o he ac ha he
objec i e unc ion quali a i ely co esponds o he maximisa ion p oblem in he absence o
an elec ion, ha is, (5):
WC
i=pChWi(˜e1(θi, δi)) + Ri+ (1 −pC)hWi(˜e1(θi, δi))i
=Wi(˜e1(θi, δi)) + zR. (22)
In ui i ely, since he eelec ion p obabili y is no in luenced by he choice o ag eemen ,
no dis o ion o policy choice is necessa y, he e o e allowing o he i s -bes ou come o
ma e ialise. Howe e , we will see ha he ea y pa ame e ˆ
δidoes no lie in he ange o
case Cwhen pola isa ion becomes oo la ge.
Op imal ea y choices can be ca ego ised in o wo g oups: consensus ea ies, which a e
a i ied independen o he elec ion ou come and di e en ia ion ea ies, which a e only
a i ied by ei he he incumben o he challenge . Wi hin he wo g oups, ea y ypes
di e in e ms o he incumben ’s unde lying a ionale, as de ailed in he ollowing:
•Consensus ea y: a i ied by bo h (case C)
–Fi s -bes (FB): op imal ea y is equi alen o no-elec ion ea y
–Comp omise (COMP): ea y ambi ion is shi ed owa ds challenging pa y’s
p e e ences o ensu e a i ica ion independen o elec ion ou come
•Di e en ia ion ea y: a i ied by ei he incumben (case A) o challenge (case B)
–Dis inc ion (DIST): ea y ambi ion is shi ed away om challenging pa y’s
p e e ences o s ess policy di e ences owa ds median o e
20
–Assimila ion (ASSIM): ea y ambi ion is shi ed owa ds median o e p e e -
ences o imp o e elec o al p ospec s
–Insu ance (INS): ea y ambi ion adapa ed such as o ensu e an accep able ou -
come in case o elec ion loss
The espec i e a ailabili y o hese ea y ypes depends on pola isa ion le els and will be
de ined in de ail la e in his sec ion. Fo illus a i e pu poses, he op imal ea y choice
o he incumben is hence o h p esen ed in nume ical examples. The ollowing pa ame e s
will be assumed h oughou :
z= 0.55, β = 0.05, σ = 0.8.
All o hese pa ame e alues a e unexcep ional, in ha hey do no d i e any o he esul s
p esen ed, and pos ula e (i) a 5 pe cen age poin incumbency ad an age, which ollowing, o
example, Gelman and King (1990) and Le i and Wol am (1997) a e a middle-o - he- oad
es ima e, (ii) en i onmen al damages in he absence o any policies pu suing clima e change
mi iga ion would cons i u e app oxima ely an 18% educ ion o GDP, and (iii) a shock
densi y pa ame e o mi o en i onmen al policy being ela i ely salien such ha 80% o
policy di e ences be ween he con ende s ansmi in o eelec ion p obabili ies, mi o ing
he cu en ly high isibili y o he clima e c isis in policy deba es in many coun ies.
In Figu e 5 he shaded a eas indica e which pa y a i ies he ea y: yellow o bo h, g een
and b own o he espec i e pa ies. The o ange do s e e o he speci ic nume ical scena ios
which will be discussed in de ail in Sec ions 4.4.1 and 4.4.2. Figu e 5a illus a es op imal
ea y choices by a g een incumben depending on he le el o pola isa ion and he o ice
en .4A common a i ica ion in e al only exis s o ela i ely low le els o pola isa ion. In
his a ea, he ea y can be o he ype i s -bes o comp omise o su icien ly low le els
o he o ice en , depending on he a ailabili y. Fo a highe o ice en , he “gamble” o a
dis inc ion ea y is wo hwhile because o he inc eased impo ance o eelec ion. Fo highe
le els o pola isa ion, no ea y will lead o a i ica ion by he challenge and he e o e he
sole ocus lies on appealing o he median o e by choosing an assimila ion ea y.
Analogously, Figu e 5b shows he op imal ea y choices o a b own incumben . Common
a i ica ion is op imal o a la ge ange o pola isa ion le els, as is he ea y ype i s -bes
o a su icien ly low o ice en . The le el o he o ice en which sepa a es dis inc ion om
i s -bes o comp omise ea ies dec eases in pola isa ion, since consensus becomes mo e
cos ly he mo e dis inc pa y p e e ences a e and he e o e eelec ion becomes ela i ely
4No e ha his is an illus a i e example including a nume ical app oxima ion o ¯
R(φ)in he ange
φ∈(φG
A, φG
F B). A mo e de ailed discussion ollows la e in his sec ion.
21
he b own incumben ades-o he ela i e impo ance o policy ou come and chances o
eelec ion and chooses o pu mo e weigh on he la e i he o ice en is su icien ly high,
he h eshold alue o which is de ined in he ollowing:
¯
RB(φ) =
a g min |WC(ˆ
δi)−WA(¯
δj)| o φ≤¯
φB
F B
a g min |WC(¯
δj)−WA(¯
δj)| o ¯
φB
F B < φ ≤¯
φB
a g min |WB(¯
δj)−WA(δi)| o φ > ¯
φB.
(27)
The incumben will choose a consensus ea y i he o ice en lies below (27), and a dis inc-
ion ea y o he wise. When he i s -bes ea y is a ailable, he esul ing ea ies di e in
ambi ion. O he wise he sugges ed ea y pa ame e s a e only ma ginally di e en , bu he
dis inc ion ea y is only a i ied by he incumben . When pola isa ion is high, he incum-
ben chooses an insu ance ea y o low le els o he o ice en , and a dis inc ion ea y
o he wise.
P oposi ion 5 (T ea y Ou comes o i=B)
(i) Fo low & medium le els o pola isa ion, ha is o φ≤¯
φB, i holds ha :
1. i is ne e op imal o he b own incumben o choose a ea y o ype B,
2. he b own incumben chooses a consensus ea y i R≤¯
RB(φ)and a dis inc ion
ea y wi h δ∗
i=¯
δj+o he wise.
3. The chosen consensus ea y is i s -bes o φ≤¯
φG
F B, ha is δ∗
i=ˆ
δi, o he wise
i is a comp omise ea y wi h δ∗
i=¯
δj−.
(ii) Fo high le els o pola isa ion, ha is o φ > ¯
φB, he b own incumben chooses an
insu ance ea y wi h δ∗
i=¯
δji R≤¯
RB(φ)and a dis inc ion ea y wi h δ∗
i=δi
o he wise.
Figu e 9 illus a es exempla y ea y ou comes wi h a b own incumben . The i s -bes
ea y is a ailable o qui e a la ge ange o pola isa ion le els, an example o which is
depic ed in 9a. The choice be ween a comp omise and a dis inc ion ea y is illus a ed in
Figu es 9b and 9c, which shows dynamics analogous o he case o a g een incumben , albei
mo ing in he opposi e di ec ion.
28
0.88 0.92 0.96 1.
δi
E[W]
D B C A
(a) Fi s -bes ea y wi h φ= 0.3, R = 1
0.88 0.92 0.96 1.
δi
E[W]
D B C A
(b) Comp omise ea y wi h φ= 0.7, R = 1
0.88 0.92 0.96 1.
δi
E[W]
D B C A
(c) Dis inc ion ea y wi h φ= 0.7, R = 1.25
0.88 0.92 0.96 1.
δi
E[W]
D B D A
(d) Insu ance ea y wi h φ= 0.9, R = 0.5
Figu e 9: B own incumben ea y ou comes
By choosing a dis inc ion ea y, he median o e is o ced o compa e he weak ea y
o he non-coope a i e ou come o he highly g een challenge , whe e ∆WA
M>0and hus
pA(δ∗
i)> pC. The p ospec o a high o ice en makes i wo h o he incumben o exploi
his di e ence as opposed o choosing a be e policy.
Finally, he case o e y p onounced pola isa ion is shown in Figu e 9d. In his case, he e
e en exis s an in e al (¯
δj, δi)which is no a i ied by any o he wo pa ies, as he ea y
would be oo ambi ious o he incumben and no ambi ious enough o he challenge . The
choice be ween an insu ance ea y and dis inc ion ea y again depends on he size o he
o ice en : i Ris su icien ly small, i can be op imal o he incumben o sugges a ea y
ha hey hemsel es would no a i y bu hei g een challenge would. This s a egy is
bene icial and “low- isk” o he incumben o wo easons: on he one hand, hey a e pu -
suing a “cheap” policy in case o eelec ion by no commi ing o any emission educ ions.
Gi en hei low en i onmen al p e e ences, he cos o coun y 2 no educing emissions is
ela i ely low. On he o he hand, in case hey a e eplaced, he nego ia ed ea y gi es he
incumben highe wel a e le els han he challenge ’s non-coope a i e policy choice would.
In e es ingly, ∆WB
M(δ∗
i)<0, meaning ha he incumben knowingly educes hei eelec-
29
ion p obabili y. The e o e, as opposed o he di e en ia ion ea y and due o he s ong
pola isa ion, hey p io i ise policy ou come o e eelec ion. One can he e o e in e p e his
choice as an insu ance agains a po en ial successo .
Fo he nume ical example he pola isa ion h eshold alues a e gi en by ¯
φF B
B= 0.668 and
¯
φB= 0.79. Table 3 quan i ies he di e ences be ween a comp omise and a dis inc ion ea y.
Comp omise is p e e able i he ela i e impo ance o clima e policy is a leas 30%. Choos-
ing a dis inc ion ea y leads o a eelec ion p obabili y abo e he incumbency ad an age,
because he challenge ’s non-coope a i e emission choice is cos ly o he median o e , com-
bined wi h compa ably li le emission educ ions by coun y 2. Fo he median o e , due
o he ac ha he ea y ambi ion only ma ginally di e s, he ea y i is iden ical and
a sligh imp o emen o e he no-elec ion ea y. Howe e , because he comp omise ea y
a oids he cos ly non-coope a i e ou come o he g een challenge , in e ms o poli ical i
i is supe io o he dis inc ion ea y.
Table 3: B own incumben and pola isa ion φ= 0.7,ω(¯
R= 1.12) ≈0.296
T ea y Type Emissions (ei, ej) Reelec. P . R, ω(R) ∆WT F
M∆WP F
M
Comp omise
δ∗
i= 0.983 −
˜e1= 0.968
˜e2= 0.936
˜e1= 0.968
˜e2= 0.936
pC= 0.55 R= 1
ω(R) = 0.32
+0.044% +0.166%
Dis inc ion
δ∗
i= 0.983 +
˜e1= 0.968
˜e2= 0.936
ˆe1= 0.915
ˆe2= 0.95
pA= 0.5507 R= 1.25
ω(R) = 0.27
+0.044% +0.024%
A nume ical example o he insu ance ea y is gi en in Table 4. This ea y ype only
eme ges when clima e policy has a high ela i e impo ance o he incumben : in his case,
hey wan o p io i ise policy ou come o e eelec ion. This is con as ed wi h a dis inc ion
ea y a he same pola isa ion le el. In compa ison, median o e wel a e is much imp o ed
unde an insu ance ea y: concep ually, his is a way o comp omising when no common
a i ica ion is possible.
Table 4: B own incumben and pola isa ion φ= 0.9,ω(¯
R= 0.78) ≈0.39
T ea y Type Emissions (ei, ej) Reelec. P . R, ω(¯
R) ∆WT F
M∆WP F
M
Insu ance
δ∗
i= 0.9759
ˆe1= 0.995
ˆe2= 0.95
˜e1= 0.971
˜e2= 0.927
pB= 0.5485 R= 0.7
ω(R) = 0.41
+0.374% +0.158%
Dis inc ion
δ∗
i= 0.9904
˜e1= 0.971
˜e2= 0.927
ˆe1= 0.995
ˆe2= 0.95
pB= 0.5507 R= 1
ω(R) = 0.33
+0.101% +0.056%
B oadly speaking, om he pe spec i e o he median o e , elec ion p essu e has a mode -
a ing in luence on ea y ou comes ac oss he pola isa ion spec um. This mode a ing e ec
is somewha dampened by highe le els o o ice en , which incen i ises incumben s o seek
30
di e en ia ion. Howe e , his should no obscu e he ac ha inc eased pola isa ion esul s
in less a ou able policy ou comes o he median o e in absolu e e ms – as pola isa ion
ises, policy p e e ences become inc easingly dis inc by de ini ion.
Fu he mo e, his illus a ion poin s ou wo di e ences be ween he g een and he b own
incumben om he poin o iew o he median o e : i s ly, due o he lowe en i onmen al
p e e ences o he b own incumben , a gi en ea y pa ame e ansla es in o lowe absolu e
emission educ ions in coun y 1 compa ed o a g een incumben wi h he same ea y.
Secondly, he ac ha he b own incumben sugges s less s ic ea ies also implies ha
coun y 2 educes emissions by less, esul ing in highe damage ex e nali ies han unde a
g een incumben .
This highligh s he undamen al di e ence be ween he wo go e nmen s, bes illus a ed
in Figu e 5: he s a k con as in a ailable ea ies. Fo he g een incumben , compa ed o
a no-elec ion scena io, mode a ion esul s in lowe emission educ ions ab oad bu highe
domes ic cos s due o lowe s a us quo emissions. Con e sely, a comp omise by he b own
incumben leads o highe ea y ambi ion han in he absence o an elec ion, esul ing in
g ea e emission educ ions ab oad.
5 Ex ensions
While he basic model amewo k e ec i ely cap u es he co e dynamics o he issue, i
lends i sel o a numbe o ex ensions ha could u he e ine he esul s. Each ex ension
in oduces addi ional nuances, o e ing a mo e de ailed and comp ehensi e unde s anding
o he model dynamics.
5.1 T ea y Emissions as an Uppe Bound
While I assume ha ins an enego ia ion o a ea y is no possible, i could be a gued
ha a go e nmen in powe can always go beyond he p omises made in an in e na ional
ea y. P esumably, coun y 2 would no oppose o coun y 1 educing emissions by mo e
han wha was ag eed upon. While I would a gue ha his case is ha dly seen empi ically,
illus a ed by he lack o coun ies which o e shoo hei emission pledges, allowing o he
elec ed go e nmen o go beyond ea y a ge s a ec s some ou comes o he model in an
in e es ing ashion. In his ex ension, we will he e o e in e p e ea y emissions ˜e1as an
uppe bound which he elec ed go e nmen can olun a ily unde cu .
Fi s , no e ha his change in assump ion does no a ec he equilib ium ou comes in he
case o a b own challenge since he b own pa y’s non-coope a i e emission choice is ne e
31
lowe han ea y emissions nego ia ed by he g een incumben :
1−βθj=B
| {z }
ˆej=B
< δi=G(1 −βθi=G)
| {z }
˜ei=G
,since θi=G≥θj=Band δi∈[0,1].
The e o e, gi en ha he b own challenge op imally wan s o se highe emissions han he
ea y emissions and due o he conca i y o hei wel a e unc ion, unde a a i ied ea y
hey canno do be e by choosing e < ˜ei=G.
Ye , i is possible o he g een challenge o ha e lowe non-coope a i e emissions compa ed
o ea y emissions nego ia ed by he b own incumben , as p e iously illus a ed by Figu e
2. Mo e p ecisely, his is he case whene e he ea y pa ame e is abo e a lowe bound
le el δLB:
1−βθj=G
| {z }
ˆej=G
≤δi=B(1 −βθi=B)
| {z }
˜ei=B
⇒δi≥δLB ≡1−β(1 + φ)
1−β(1 −φ).(28)
No e ha dδLB
dφ <0, ha is he mo e pola ised he pa ies a e, he lowe is his lowe bound.
The e o e, i he non-coope a i e emission le el is easible (ˆej≤˜e1,i), which is he case
whene e δi=B≥δLB, he elec ed challenge will a i y he ea y and hen choose ˆej. This
makes sense in ui i ely: hey canno do be e han o choose hei indi idually op imal
emission le el, while a he same ime ge ing he ea y bene i o coun y 2 educing hei
emissions below hei non-coope a i e le el, esul ing in lowe damage cos s. Due o he
assump ion o a linea damage unc ion, emission le els a e dominan s a egies and a lowe
han ag eed emission le el o coun y 1 does no a ec he emission choice in coun y 2.
This now gi es ise o wo new cases C0and B0, which eme ge depending on he le el o
pola isa ion as illus a ed in Figu e 10 and a e sepa a ed by he pola isa ion h eshold le el
¯
φLB, ha is, whe e δiand δLB a e equal:
δi=δLB
⇒¯
φLB =β−1 + √1−β
β(29)
I pola isa ion is below ¯
φLB i holds ha δi< δLB. This means ha he e exis s a common
a i ica ion in e al ( o me ly case C), howe e , in which he g een challenge chooses non-
coope a i e emissions a e a i ica ion. This new case C0 hus indica es a ange whe e,
32
depending on who wins he elec ion, iwould a i y and se ea y emissions and jwould
a i y and choose ˆej.
0 1
φ≤¯
φLB
δjδ2δiδLB ¯
δi=¯
δ2
D B C C’
0 1
φ > ¯
φLB
δjδ2δi
δLB ¯
δi=¯
δ2
D B B’ C’
Figu e 10: New cases depending on pola isa ion le els
Pola isa ion abo e ¯
φLB leads o he ac ha δi> δLB and he e o e he e exis s a ange
whe e only he g een challenge a i ies ( o me ly case B) and hen op s o choose non-
coope a i e emissions. In his new case B0, depending on who wins he elec ion, iwould
hus no a i y and se non-coope a i e emissions ˆeiand jwould a i y and choose ˆej. No e
ha his ou come di e s om case Din ha coun y 2 will se ea y emissions.
Independen o pola isa ion le els, we ind ha classic dis inc ion ea ies no longe exis .
This is in ui i e: he only way o a b own incumben o di e en ia e om a g een challenge
in he basic model was o nego ia e a ea y oo weak o he challenge o a i y. Now,
howe e , he challenge a i ies any ea y δ∈[δj,1]. The comp omise and insu ance ea y
ypes s ill exis , albei esul ing om sligh ly di e en mo i es.
0.88 0.92 0.96 1.
δi
E[W]
D B C C'
(a) Fi s -bes ea y wi h φ= 0.3, R = 1
0.88 0.92 0.96 1.
δi
E[W]
D B C C'
(b) “Comp omise” ea y wi h φ= 0.45, R = 1
Figu e 11: T ea y ou comes wi h low pola isa ion
Wi h pola isa ion le els below ¯
φLB, which he e co esponds o ¯
φLB(0.05) = 0.494, i s -
bes and a a ia ion o a comp omise ea y a e possible as seen in Figu e 11. The o me
occu s when pola isa ion is su icien ly low. The la e di e s om he o iginal comp omise
ea y in he sense ha we ind ou sel es in he ange o case C0, whe e e en hough
33
he challenge a i ies, hey choose non-coope a i e emissions. S ill, coun y 2 engages in
emission educ ions as de ined by he ea y. No e ha e en hough ˆ
δiis a ailable wi hin case
C0, he incumben op imally chooses a sligh ly mo e ambi ious ea y. This is due o he ac
ha in case o elec ion loss, he challenge will in any case choose non-coope a i e emissions,
howe e , coun y 2 will engage in mo e emission educ ions i he ea y pa ame e is lowe ,
which compensa es o a sligh ly lowe eelec ion p obabili y.
In he case o high pola isa ion, ha is o φ > ¯
φLB, ano he di e ence o he main model
ma e ialises. Now he g een challenge canno be “locked” in wi h a weak ea y as be o e,
since hey can go beyond ea y emission educ ions and hus he classic insu ance mo i e
is no longe a ailable o he b own incumben .
Analogously o he low pola isa ion case and o s anda d o ice en le els, we obse e a
ype o comp omise ea y, an example o which is shown in Figu e 12a. We a e in a ea C0,
whe e bo h pa ies a i y he ea y, and whe e he incumben op s o a ea y which is
s ic e han he no-elec ion ea y in o de o achie e highe emission educ ions by coun y
2. Now in e es ingly, as he o ice en dec eases and hus less impo ance is pu on secu ing
an elec ion ic o y, a new a ia ion o an insu ance ea y eme ges, as illus a ed in Figu e
12b.
0.88 0.92 0.96 1.
δi
E[W]
D B B' C'
(a) “Comp omise” ea y wi h φ= 0.7, R = 1
0.88 0.92 0.96 1.
δi
E[W]
D B B' C'
(b) “Insu ance” ea y wi h φ= 0.7, R = 0.05
Figu e 12: T ea y ou comes wi h high pola isa ion
The da ke g een ange indica es case B0, whe e only he challenge a i ies he ag eemen
bu o any ea y pa ame e chooses non-coope a i e emissions. In he basic model, he
incumben chose he uppe limi o his ange o lock in a cheap ea y because he challenge
was bound o he ea y emissions, while now, hey op imally p opose he o he end o
he ange δLB. A his poin , ea y emissions exac ly equal he g een challenge ’s non-
coope a i e emissions. In ui i ely, his is op imal because a any o he poin o he da k
g een ange, he challenge would also se non-coope a i e emissions, while his is he poin
a which emission educ ions by coun y 2 a e maximised. Unchanged o he basic model,
34
in case o an elec ion win, he incumben does no a i y he ea y. So concep ually, his
ea y choice s ill esembles an insu ance ea y in he sense ha he incumben aims a
op imising he ou come in case o elec ion loss.
5.2 P e e ence Asymme y
The basic model assumes he pa ies’ p e e ence pa ame e s o be symme ic a ound he
median o e . Allowing o p e e ence asymme y, ha is o p e e ence pa ame e s ha
di e in e ms o dis ance o he median o e , no only ep oduces all o he p esen ed
esul s, bu p oduces e en mo e dis o ed ou comes. In pa icula , we de ine p e e ence
pa ame e s o be:
θ1,G = 1 + µ, θ1,B = 1 −λ.
The deg ee o p e e ence asymme y is cap u ed by he a io o µand λ, he symme ic
cases being de ined by µ=λ. Emission choices s ill ollow as desc ibed in Sec ion 4.1.
The e a e a ew changes in he a i ica ion s age om he esul s desc ibed in Sec ion 4.2.
Fi s ly, in he case o a g een incumben , he h eshold alue in Lemma A.1 now becomes
wo-dimensional, as pic u ed in Figu e 13a: Any combina ion o µand λwi hin he shaded
a ea ensu es ha he b own challenge ’s a i ica ion in e al exis s, he bounda y o which
is he unc ion ¯
λ(µ). Simila ly o he b own incumben , he h eshold alue o sepa a e
he wo scena ios in Figu e 4, as de ined by Lemma A.2, also becomes wo-dimensional as
illus a ed in Figu e 13b. The shaded a ea hen indica es he pa ame e combina ions o
which an o e lap in a i ica ion in e als exis s, he bounda y being he unc ion ˜µ(λ).
(a) Th eshold alue analogy o ¯
φG(b) Th eshold alue analogy o ¯
φB
Figu e 13: Two-dimensional analogy o h eshold alues ¯
φGand ¯
φB o β= 0.05
35
Fo a mo e de ailed and o mal discussion o he changes in he a i ica ion s age, e e o
Appendix C. The elec ion s age is quali a i ely una ec ed by he in oduc ion o p e e ence
asymme y. In he ag eemen s age, all o he ou comes p esen ed in he symme y case can
also be eplica ed wi h asymme y.
5.3 Mo e sophis ica ed coun y 2
I could be a gued ha in he assump ions o he basic model, coun y 2 ac s in an o e ly
nai e ashion, igno ing he possibili y o a po en ial go e nmen change in coun y 1 while
aking hei pa icipa ion decision. He e I will elax his assump ion by allowing o a sligh
sophis ica ion in coun y 2: ha ing an unde s anding o he poli ical en i onmen in coun y
1, coun y 2 can an icipa e eelec ion p obabili ies p io o nego ia ions cap u ed by he
incumbency ad an age z. We will see ha his does no a ec ea y ou comes o he g een
incumben as discussed in Sec ion 4.4, while i educes he op ions o a b own incumben
in some ins ances.
This elaxa ion o assump ion means ha coun y 2 will only accep he sugges ed ea y i
hei expec ed wel a e change is non-nega i e, he e o e a ec ing he pa icipa ion condi ion
(13). Gi en a ea y pa ame e δi, coun y 2 can an icipa e coun y 1’s a i ica ion decision
condi ioned on which pa y will be elec ed, and he e o e compu e an expec ed wel a e le el
as ollows:
E[W2|δi] = zhW2(’i in powe ’)i+ (1 −z)hW2(’j in powe ’)i(30)
Thus, coun y 2 will pa icipa e in an ag eemen i hei expec ed wel a e is non-nega i e,
which depends on he case esul ing om he ea y pa ame e , as gi en by he ollowing:
∆E[WA
2] = zh˜
W2(δi, θi)−ˆ
W2(θi)i+ (1 −z)h
=0
z }| {
ˆ
W2(θj)−ˆ
W2(θj)i
=zh˜
W2(δi, θi)−ˆ
W2(θi)i(31a)
∆E[WB
2] = zh
=0
z }| {
ˆ
W2(θi)−ˆ
W2(θi)i+ (1 −z)h˜
W2(δi, θi)−ˆ
W2(θj)i
= (1 −z)h˜
W2(δi, θi)−ˆ
W2(θj)i(31b)
∆E[WC
2] = ˜
W2(δi, θi)−hzˆ
W2(θi) + (1 −z)ˆ
W2(θj)i(31c)
36
We will now analyse whe he his changed pa icipa ion cons ain will a ec any o he
equilib ium ou comes in Sec ion 4.4. Fi s no e ha (31a) is quali a i ely unchanged o he
o iginal pa icipa ion cons ain , since only he incumben e e a i ies and he e o e, any
ea y o case A, ha is all dis inc ion and assimila ion ea ies, will s ill be alid.
Figu e 14 displays all consensus and insu ance ea ies om Sec ion 4.4. The shaded a eas
indica e he ea y ype which a ises in a speci ic ange, as a consequence o a i ica ion
in e als in coun y 1. The plo ed unc ions ollow (30) o he espec i e cases and in
co esponding colou s (nume ical example wi h β= 0.05, z = 0.55). No e ha his di e s
om unc ions plo ed in Figu es 7 and 9, whe e he unc ions in black ep esen he coun y
1 incumben ’s expec ed wel a e, which de e mines he op imal ea y choice. He e, whene e
he unc ion akes a non-nega i e alue, coun y 2 pa icipa es in he ea y.
0.90 0.92 0.94 0.96 0.98 1.00
δi
E[W2]
A C A
(a) i=G,φ= 0.17: comp omise ea y
0.90 0.92 0.94 0.96 0.98 1.00
δi
E[W2]
B C A
New PC
(b) i=B,φ= 0.3: i s -bes ea y
0.90 0.92 0.94 0.96 0.98 1.00
δi
E[W2]
B C A
(c) i=B,φ= 0.7: comp omise ea y
0.90 0.92 0.94 0.96 0.98 1.00
δi
E[W2]
B D A
(d) i=B,φ= 0.9: insu ance ea y
Figu e 14: Expec ed wel a e o coun y 2 and op imal ea y choice o incumben
Fo a g een incumben , he esul s a e una ec ed: any dis inc ion and assimila ion ea y is
s ill alid and, as pic u ed in Figu e 14a, he p oposed comp omise ea y yields a posi i e
expec ed wel a e o coun y 2. Howe e , he new pa icipa ion cons ain s o coun y 2
a ec he indings o a b own incumben : Figu e 14b illus a e he case o low pola isa ion,
whe e a i s -bes ea y is p oposed. The new pa icipa ion cons ain educes he ange o
easible consensus ea ies, ye , i pola isa ion is su icien ly low (de ailed in Lemma C.7),
37
ollows:
A+pMj=B
B<1,
and hen e o mula ing:
qMj=B<B−A ≡ C ⇔ C2−Mj=B>0
⇒4β2φ2(β(1 + φ)−1)2>0which is ue.
The e o e, we ha e shown ha A+√Mj=B
B<1.
Lemma A.1 (Exis ence o Ra i ica ion In e al o B own Challenge )
The challenge ’s a i ica ion h esholds exis i i holds ha :
φ≤¯
φG(β) = 5β−5 + p(β−1)(41β−25)
8β.(A.4)
In addi ion i holds ha :
d¯
φG
dβ >0.(A.5)
P oo o Lemma A.1
(i) The challenge ’s h eshold alues exis i he e m in he squa e oo in (A.3), ha is,
Mj=Bis non-nega i e. The e o e:
β2
<0
z }| {
(β−1)
<0
z }| {
(φ−1)
| {z }
>0h1−5φ+β(4φ2+ 5φ−1)i≥0.
I hus su ices o conside he e m in squa e b acke s o de e mine he sign:
1−5φ+β(4φ2+ 5φ−1) ≥0
φ≤5−5β+p(β−1)(41β−25)
8β≡¯
φG.
(ii) P oo o (A.5):
d¯
φG
dβ =25 −33β−5p(β−1)(41β−25)
8β2p(β−1)(41β−25)
44
The e m unde he squa e oo is non-nega i e since β−1<0and 41β−25 <0∀β∈
[0,0.15]. The sign is hus de e mined by he nume a o :
25 −33β−5q(β−1)(41β−25) ≶0
(25 −33β)2≶25(β−1)(41β−25)
625 + 1089β2−1650β≶625 + 1025β2−1650β
64β2>0,
and he e o e d¯
φG
dβ >0.
P oposi ion A.2 (S age 3: Compa a i e S a ics wi h i=G)
The ollowing condi ions hold o he equilib ium a i ica ion in e als unde he condi ion
ha h esholds exis and ha hey a e wi hin he in e al [0,1]:
dδi=G
dφ <0,d¯
δi=G
dφ = 0,dδj=B
dφ >0,d¯
δj=B
dφ <0.
P oo o P oposi ion A.2
(i) Fo he g een incumben ’s h esholds:
dδi=G
dφ =−2β
<0
z }| {
(β−1)
>0
z }| {
(1 + β(1 + φ))
(β(1 + φ)−1)3
| {z }
<0
<0(A.6)
The uppe a i ica ion h eshold o he g een incumben is a cons an and he e o e no a
unc ion o φ.
(ii) Fo he b own challenge ’s h esholds:
•dδj=B
dφ >0
dδj=B
dφ =
β(β−1)hβ2(φ2−15φ−8) + β3(2φ3−5φ2−10φ+ 5) + 3pMj=B−
(β(1 + φ)−1)3pMj=B
β(−3+5pMj=B+φ(5 + qMj=B))i
(β(1 + φ)−1)3pMj=B
(A.7)
45
I will now show ha his e m is posi i e. The denomina o is nega i e. In he nume -
a o , because β(β−1) <0, I will show ha he e m in squa e b acke s is posi i e.
Rew i ing his e m:
qMj=B(3 −5β−βφ
| {z }
A
) + φ3(2β3
|{z}
>0
) + φ2(β2−5β3
| {z }
B
) + φ(15β2−10β3
| {z }
>0
)
+β(3 −5φ+ 5β2−8β
| {z }
C
)(A.8)
whe e
A= 3 −5β−βφ > 0 o β < 3
5−φ∈[0.58,0.6],depending on φ∈[0,¯
φB].
B=β2−5β3>0 o β < 1
5
C= 3 −5φ+ 5β2−8β > 0(see below)
No e ha he exp ession Cis s ic es a β= 0.15 and φ=¯
φG≈0.206, whe e
C(β= 0.15, φ = 0.206) = 0.7925 >0. Consequen ly, since all e ms in (A.8) a e
posi i e, he exp ession in squa e b acke s is also posi i e. Hence he nume a o is
nega i e, and combined wi h he nega i e denomina o , i holds ha dδj=B
dφ >0.
•d¯
δj=B
dφ <0
d¯
δj=B
dφ =
β(1 −β)hβ2(φ2+ 15φ−8) + β3(2φ3−5φ2−10φ+ 5) −3pMj=B+
(β(1 + φ)−1)3pMj=B
β(3 + 5pMj=B+φ(−5 + qMj=B))i
(β(1 + φ)−1)3pMj=B
(A.9)
I will now show ha his e m is nega i e. Again, he denomina o is nega i e, so
ha i emains o be shown ha he nume a o is posi i e. Fi s , β(1 −β)>0. Then,
ew i ing he nume a o :
A+ [−3+5β+βφ]qMj=B>0
A>[3 −5β−βφ]
| {z }
>0qMj=B
A2
[3 −5β−βφ]2> Mj=B
46
The e o e:
Mj=B−A
[−3+5β+βφ]2
<0
which expands o
1
(·)2
4β2φ(β(φ−1) −1)3
| {z }
<0
[6 −
max=1.03
z}|{
5φ−
max=1.5
z }| {
β(10 −9φ+φ2)]
| {z }
>0
<0.
Consequen ly, he e m in squa e b acke s in he nume a o is posi i e, making he
whole exp ession nega i e.
P oposi ion A.3 (S age 3: Ra i ica ion In e als wi h i=B)
In he case o a b own incumben , a i ica ion h esholds a e gi en as ollows:
[δi=B,¯
δi=B] = "1 + β(φ−1) [4 + β(φ−3)]
(1 + β(φ−1))2,1#,(A.10)
[δ2(θB),¯
δ2(θB)] = 1 + β(β(3 −2φ)−4)
(β−1)2,1.(A.11)
The challenge ’s a i ica ion h esholds always exis and a e gi en by:
[δj=G,¯
δj=G] = "max0,1−β[3 + φ+β(φ−2)(1 + φ)] −pMj=G
(1 + β(φ−1))2,
1−β[3 + φ+β(φ−2)(1 + φ)] + pMj=G
(1 + β(φ−1))2#,(A.12)
wi h Mj=G=β2(1 −β)(1 + φ) (1 + 5φ+β[φ(4φ−5) −1]).
P oo o P oposi ion A.3
(i) Ra i ica ion in e al o i=B:
Th eshold alues ollow om (9), which is a conca e unc ion and has wo oo s. The uppe
a i ica ion h eshold is equal o 1 because i co esponds o he incumben ’s non-coope a i e
emission choice and hus makes hem equally well o as wi hou a ea y.
47
The lowe a i ica ion h eshold is always posi i e, wi hin he assumed pa ame e anges:
δi=B=1 + β(φ−1)[4 + β(φ−3)]
(1 + β(φ−1))2>0
1> β(1 −φ)[4 + β(φ−3)],
whe e he igh hand side a i s la ges akes he alue 0.555, and he e o e is always smalle
han 1.
(ii) Ra i ica ion in e al o coun y 2:
Th eshold alues ollow om (13), which is a conca e unc ion and has wo oo s. The uppe
a i ica ion h eshold is equal o 1 because i co esponds o coun y 2’s non-coope a i e
emission choice and hus makes hem equally well o as wi hou a ea y.
The lowe a i ica ion h eshold is always posi i e:
δ2=1 + β[2β(1 −φ) + β−4]
(β−1)2>0
1+3β2
| {z }
∈[1,1.07]
>4β+ 2β2φ
| {z }
∈[0,0.64]
Gi en ha he wo in e als ne e o e lap, his is always ue.
(iii) Ra i ica ion in e al o j=G:
Th eshold alues ollow om (9), which is a conca e unc ion and has wo oo s. They exis
i Mj=Gis non-nega i e:
Mj=G=β(1 + φ)(1 −β)
| {z }
>0h1+5φ+β(4φ2−5φ−1)i
| {z }
A
whe e
A= 1 −β
|{z}
>0
+5φ(1 −β)
| {z }
>0
+4βφ4>0
and hus Mj=G>0, hence he h eshold alues always exis .
To show when δj=Gis non-nega i e, conside he alue o φ∈[0,1] ha ende s δj=G= 0:
δj=G(φj=G
0)=0
⇒φj=G
0=2−2β−p3−4β+β2
β
48
Knowing ha he lowe a i ica ion h eshold o he g een challenge dec eases in φ(see
P oposi ion A.4), we can show ha φj=G
0is only es ic i e i ma ginal damages a e su i-
cien ly high:
φj=G
0≤1⇒β≤0.1465
No e ha his is analogous o he condi ion in P oposi ion A.1.
Also, he challenge ’s uppe h eshold is ne e abo e 1:
¯
δj=G=1−β[3 + φ+β(φ−2)(1 + φ)] + pMj=G
(1 + β(φ−1))2≤1
Rew i ing he condi ion:
B
C≤1⇔ B −C ≤ 0
−4β2φ2(1 + β(φ−1))2≤0
which is always ue.
P oposi ion A.4 (S age 3: Compa a i e S a ics wi h i=B)
The ollowing condi ions hold o he equilib ium a i ica ion in e als unde he condi ion
ha hey a e wi hin he in e al [0,1]:
dδi=B
dφ >0,d¯
δi=B
dφ = 0,dδj=G
dφ <0,d¯
δj=G
dφ <0.
P oo o P oposi ion A.4
(i) Fo he b own incumben ’s h esholds:
dδi=B
dφ =2β(
<0
z}|{
β−1)(
<0
z }| {
β(φ−1) −1)
(1 + β(φ−1))3
| {z }
>0
>0(A.13)
The uppe a i ica ion h eshold o he b own incumben is no a unc ion o φ.
(ii) Fo he g een challenge ’s h esholds:
49
•dδj=G
dφ <0
dδj=G
dφ =
β(β−1)h3pMj=G+β(3 + β(φ2−15φ−8)+
(1 + β(φ−1))3pMj=G
β2(5 + 10φ−5φ2−2φ3)−5pMj=G+φ(5 + pMj=G)i
(1 + β(φ−1))3pMj=G
(A.14)
No e ha he denomina o is posi i e, as seen in (A.13). In he nume a o , gi en ha
β(β−1) <0, I will show ha he e m in squa e b acke s is posi i e. Rew i ing his
e m:
qMj=G
A
z }| {
(3 −5β+βφ) +β
B
z }| {
3−8β+ 5β2+φ
C
z }| {
5β−15β2+ 10β3
+φ2β2−5β3
| {z }
D
+φ3−2β3
| {z }
E
whe e
A= 3 −5β+βφ > 0 o β < 3
5−φ∈[0.6,0.71],depending on φ∈[0,¯
φB].
B= 3 −8β+ 5β2>0 o β < 0.6
C= 5β−15β2+ 10β3>0 o β < 0.5
D=β2−5β3>0 o β < 0.2
E=−2β3<0
All e ms bu Ea e posi i e. Howe e , he nega i e impac o Eis co e ed, e.g. by
e m Cas ollows:
φC −φ3E=φ(5β−15β2+ 10β3−(2β3φ2)) = φ(
>0 o β< 1
3
z }| {
5β−15β2+β3(
>0
z }| {
10 −2φ2)) >0
Consequen ly, he e m in squa e b acke s is posi i e, making he nume a o o (A.14)
nega i e and hence he whole exp ession nega i e.
50
•d¯
δj=G
dφ <0
d¯
δj=G
dφ =
β(β−1)h3pMj=G+β(−3 + β(8 + 15φ−φ2)+
(1 + β(φ−1))3pMj=G
β2(−5−10φ+ 5φ2+ 2φ3)−5pMj=G+φ(−5 + pMj=G)i
(1 + β(φ−1))3pMj=G
(A.15)
The denomina o is posi i e, as seen in (A.13). Again, gi en ha β(β−1) <0, I will
show ha he e m in squa e b acke s is posi i e. Rew i ing his e m:
A+ [3 −5β+βφ]qMj=G>0
qMj=G>−A
[3 −5β+βφ]
Mj=G−A
[3 −5β+βφ]2
>0
which simpli ies o
1
(·)2
4β2φ(1 −β(φ−1))3
| {z }
>0
[6 + 5φ−
max=3
z }| {
β(10 + 9φ+φ2)]
| {z }
>0
>0.
The e o e, he e m in squa e b acke s is nega i e, making he whole exp ession neg-
a i e.
Lemma A.2 (O de ing o Coun ies’ Ra i ica ion In e als wi h i=B)
The e exis s a h eshold alue ¯
φB o a i ica ion in e als o ouch, i.e. a which δi=¯
δj.
Then, i :
φ≤¯
φB(β)∈[0.768,0.8) o β∈[0,0.15],(A.16)
a common a i ica ion in e al exis s.
P oo o Lemma A.2
The wo scena ios a e sepa a ed a he poin whe e he a i ica ion in e als ouch, ha is,
51
whe e δi=¯
δj. Sol ing his o φyields:
¯
φB=
3
q3√3p(β−1)3β6(β(β(7β−15) + 41) −25) −2(β−1)2β3(4β−13)−
3β2
(β−1)β2(5β+1)
3
p3√3√(β−1)3β6(β(β(7β−15)+41)−25)−2(β−1)2β3(4β−13)
+ 4(β−1)β
3β2(A.17)
whe e d¯
φB
dβ <0, i.e., he highe en i onmen al damages, he smalle he ange o common
a i ica ion.
Lemma A.3 (Res ic ions on Shock Densi y)
Fo eelec ion p obabili ies o be in e io in (0,1), he a iance in he popula i y shock is
es ic ed o:
σ < ¯σ= min (1−z
∆Wl
M
,z
|∆Wl
M|),(A.18)
which is mos es ic i e o he case l∈ {A, B, C, D} o which |∆Wl
M|is highes .
P oo o Lemma A.3
The eelec ion p obabili y has o be in e io , ha is pl∈(0,1). No e ha he eelec ion
p obabili y is inc eased e sus he incumbency ad an age i ∆Wl
M>0and ice e sa o
∆Wl
M<0. In he i s case, we hus ha e o ensu e ha :
σ∆Wl
M+z < 1
σ < 1−z
∆Wl
M
,
while o ∆Wl
M<0i has o hold ha :
σ∆Wl
M+z > 0
σ < z
−∆Wl
M
=z
|∆Wl
M|.
No e ha o high alues o ∆Wl
M hese condi ions become ha de o ul il (since he uppe
limi is lowe ). The e o e, whiche e case A–D leads o he highes alue o di e ence in
median o e wel a e |∆Wl
M|will be es ic i e o he shock densi y and is hus de ined as
¯σ.
52
Lemma A.4 (Pola isa ion Th esholds wi h i=G)
In addi ion o ¯
φG, which go e ns whe he o no a common a i ica ion in e al exis s as
de ined in Lemma A.1, he h eshold alue ¯
φG
F B de e mines whe he he i s -bes ou come,
i.e. he no-elec ion ea y pa ame e , leads o a consensus ea y.
I holds ha ¯
φG
F B <¯
φGin he ele an pa ame e ange o β.
P oo o Lemma A.4
Fi s ly, he pa ame e ¯
φG
F B is de ined as he pola isa ion le el a which he no-elec ion
ea y, as de ined by (5), is jus wi hin he a ea C, ha is when ˆ
δi=δjholds. Fo a g een
incumben , (5) is gi en by:
ˆ
δi(1 + φ) = 1 + β(1 + φ)[β(2 + φ)−3]
[1 −β(1 + φ)]2,(A.19)
whe e
dˆ
δi(1 + φ)
dφ =
>0
z }| {
β[1 + βφ −β2(1 + φ)]
[β(1 + φ)−1]3
| {z }
<0
<0.
The e o e, as pola isa ion inc eases, (A.19) dec eases and hus, ¯
φG
F B is he highes le el o
pola isa ion which allows o a i s -bes ea y.
The h eshold pa ame e ¯
φGis de ined in (A.4). Fo he pa ame e ange β∈(0,0.15] he
wo h eshold pa ame e s ake he ollowing alues:
¯
φG
F B ∈[0.133,0.136] <¯
φG∈[0.2,0.206].
Lemma A.5 (Pola isa ion Th esholds wi h i=B)
In addi ion o ¯
φB, which go e ns whe he o no a common a i ica ion in e al exis s as
de ined in Lemma A.2, he h eshold alue ¯
φB
F B de e mines whe he he i s -bes ou come,
i.e. he no-elec ion ea y pa ame e , leads o a consensus ea y.
I holds ha ¯
φB
F B <¯
φBin he ele an pa ame e ange o β.
P oo o Lemma A.5
Fi s ly, he pa ame e ¯
φB
F B is de ined as he pola isa ion le el a which he no-elec ion
ea y, as de ined by (5), is jus wi hin he a ea C, ha is when ˆ
δi=¯
δjholds. Fo a b own
53
0.17 0.18 0.19 0.20 ϕ
0.2
0.4
0.6
0.8
1.0
R
ii. In a second s ep, we ha e o ind he ange o φ o which his is he ele an
compa ison. We a e hus looking o a h eshold alue ¯
φG
Abeyond which he
global maximum is a ailable wi hin a ea A. We can de ine:
¯
φG
A:dWA
dδ δ=δj,φ=¯
φG
A
= 0 (B.1)
This essen ially s a es ha a his alue ¯
φG
A, he global maximum δ∗
i,A coin-
cides wi h δj, and hus he slope o WAe alua ed a his poin is ze o. Hence,
i he de i a i e e alua ed a δj akes on a nega i e alue, his means ha
δ∗
i,A < δj, which is ue o φ > ¯
φG
A, and ice e sa o a posi i e de i a i e.
No e ha his h eshold alue depends on R. Thus, i can only be com-
pu ed o a speci ic alues o he o ice en , howe e , o R∈(0,3),¯
φG
A∈
(0.1345,0.1648) wi h d¯
φG
A
dR >0.
In combina ion, o any le el o he o ice en , hese wo s eps p o ide he ange
(¯
φG
A,¯
φG)in which he global maximum o a ea A, i.e. δ∗
i,A is a ailable, and hen
he co esponding le el o ¯
Rwhich in his ange sepa a es comp omise om
dis inc ion ea ies.
(ii) While we canno sol e explici ly o he op imal ea y pa ame e wi hin a ea A (δ∗
i,A),
i is implici ly de ined by he ollowing exp ession:
dWA
i
dδi
=dpA
dδih˜
Wi(δi)−ˆ
Wi(θj) + Ri+d˜
Wi
dδi
pA= 0.(B.2)
To see ha δ∗
i,A ∈(ˆ
δi, δ∗
M), we will e alua e (B.2) a ˆ
δiand δ∗
Mand show ha i is
inc easing in he o me and dec easing in he la e , meaning ha g aphically, δ∗
i,A is
loca ed in be ween he wo. Two p e equisi es a e necessa y o show his.
60
Fi s , no e ha he ea y pa ame e wi hin case A ha maximises he eelec ion
p obabili y coincides wi h he median o e ’s op imal ea y pa ame e :
δmax
A=δ∗
M=1 + β(β(1 + θi)−2−θi)
(1 −βθi)2whe e dpA
dδi
=
>0 o δi< δmax
A
<0 o δi> δmax
A.
Second, i holds ha ˆ
δi< δ∗
M, because:
δ∗
M−ˆ
δi=2βφ −β2φ(φ+ 2)
(β(φ+ 1) −1)2>0,
because 2> β(2 + φ)as seen in he p oo o pa (i) o his P oposi ion.
Now, no e ha ˆ
δiis de ined by d˜
Wi
dδi= 0. The de i a i e (B.2) e alua ed a ˆ
δi hus
becomes:
dWA
i
dδiδ=ˆ
δi
=dpA
dδih˜
Wi(δi)−ˆ
Wi(θj) + Ri>0.(B.3)
This is ue because:
˜
Wi(δi)−ˆ
Wi(θi) = β2
>0
z }| {
0.5−β+ 0.5β2(φ+ 1)2
(β(φ+ 1) −1)2>0,
and ˆ
Wi(θi)>ˆ
Wi(θj). Also as shown, ˆ
δi< δ∗
Mimplies ha dpA
dδi>0. The posi i e sign
o (B.3) implies ha ˆ
δi< δ∗
i,A.
In a nex s ep, no e ha δ∗
Mis de ined by dpA
dδi= 0, as shown abo e. The de i a i e
(B.2) e alua ed a δ∗
M hus becomes:
dWA
i
dδiδ=δ∗
M
=d˜
Wi
dδi
|{z}
<0
pA<0,(B.4)
since d˜
Wi
dδi= 0 and ˆ
δi< δ∗
M. The nega i e sign o (B.4) hus implies ha δ∗
M> δ∗
i,A.
61
P oo o P oposi ion 5
(i) 1. To p o e ha he i s poin is ue, we will show ha WB
i(δ=δi)≤WC
i(δ=δi),
meaning ha he e always exis s a poin in a ea C which yields a weakly highe
expec ed wel a e o he incumben han he highes le el in a ea B, hus hey
ne e choose a ea y in a ea B. Fi s , ew i ing:
WC
i−WB
i=B(˜e1)−θiD(˜
E) + pCR−
hpB(B(ˆe1,i)−θiD(ˆ
E) + R) + (1 −pB)(B(˜e1)−θiD(˜
E))i
=hB(˜e1)−θiD(˜
E)ipB+R(pC−pB)−hB(ˆe1,i)−θiD(ˆ
E)ipB
=pB
˜
Wi−ˆ
Wi
| {z }
B
+R
pC−pB
| {z }
C
.
We will now e alua e his di e ence a δ=δi, whe e cases B and C mee . Now
no e ha by de ini ion o a i ica ion h eshold alues, Bis equal o ze o: a
δi, he incumben is indi e en be ween a i ying o no , making he wo alues
exac ly equal.
The e o e, o show ha WB
i(δ=δi)≤WC
i(δ=δi)holds, Chas o be non-
nega i e:
(pC−pB)δ=δi
=4−4φ+β−2φ2+ 10φ−8+β22φ2−6φ+ 4
(1 + β(φ−1))2.
Gi en ha he denomina o is quad a ic, he nume a o de e mines he sign o
he exp ession.
4−4φ+β−2φ2+ 10φ−8
| {z }
D
+β22φ2−6φ+ 4
| {z }
≥0 o φ≤1
No e ha Ddec eases as φinc eases (wi hin he gi en ange) and equals 2β−2β2
a φ= 1, meaning ha Dis non-nega i e o all alues β∈(0,0.15]. Conse-
quen ly, a ea y in a ea B is always weakly domina ed by a ea y in a ea C.
2. The o ice en which di ides consensus and di e en ia ion ea ies is de ined
by (27) o di e en anges o he pola isa ion spec um. Con a y o he case
62
o a g een incumben , he e, we do no ha e o conside he global maximum
wi hin a ea A o compa ison. In he ollowing, I show ha his is ue o he
pola isa ion ange φ∈(¯
φB
F B,¯
φB), whe e he choice is be ween a comp omise and
a di e en ia ion ea y.
Analogously o he p oo o P oposi ion 4 pa (i), 4., we can nume ically compu e
he le el o he o ice en ¯
Rwhich sepa a es a comp omise ea y wi h ¯
δjand a
di e en ia ion ea y wi h δ∗
i,A, displayed in Figu e 15b wi h he black do s. Now
in a second s ep, we ha e o check whe he he global maximum is e en a ailable
wi hin a ea A: compu ing he de i a i e o WAand e alua ing a δ=¯
δjindica es
whe he ¯
δj≷δ∗
i,A, depic ed in Figu e 15a. He e we see ha he global maximum
is only a ailable o e y small alues o he o ice en (whe e he de i a i e is
posi i e). Nume ically sol ing o he oo o his de i a i e o he ele an φ
ange gi es ise o he highes R alues o which δ∗
i,A is in a ea A, plo ed as
black iangles in Figu e 15b.
0.1 0.2 0.3 0.4 0.5
R0
dWA
dδ
|δ=δj
(a) φ= 0.7, slope o WAa h eshold ¯
δj
0.68 0.70 0.72 0.74 0.76 0.78 ϕ
0.1
0.2
0.3
0.4
0.5
0.6
R
(b) Do s: Rwhich sepa a es WA(δ∗
i,A)
and WC(¯
δj). T iangles: h eshold R
a which δ∗
i,A ≥¯
δj
Figu e 15: Global maximum o WAis no an equilib ium in δ∈(¯
δj,1)
In his combined plo , we hus see ha he R alues o which δ∗
i,A is in a ea A a e
clea ly smalle han he R alues o which he incumben would be indi e en
be ween a dis inc ion and comp omise ea y. In o he wo ds, he Rle els which
ende he incumben indi e en co espond o δ∗
i,A alues which a e ou side o
a ea A, and hus a e no iable choices o a dis inc ion ea y. Hence, in case
o a b own incumben , o dis inc ion ea ies i holds ha he op imal ea y is
gi en by δ∗
i=¯
δj.
3. Analogously o P oposi ion 4, whe he he consensus ea y is o ype i s -bes
o comp omise simply depends on a ailabili y. By de ini ion, a i s -bes ea y is
he global maximum o he WC unc ion and he e o e will be picked i a ailable.
63
A ailabili y is de e mined by whe he ˆ
δiis con ained wi hin a ea C, which is ue
o :
δi≤ˆ
δi≤¯
δj
Fi s :
ˆ
δi−δi=1
(·)2[β(1 −φ+β(φ−1))] >0 o β < 1,which is always ue.
Second:
ˆ
δi−¯
δj=1
(·)2hA−qMj=Gi,(B.5)
whe e he e m in squa e b acke s is nega i e whene e :
Mj=G− A2>0
β2(−4φ4+ 12φ3−15φ2+ 6φ+ 1) + β(−12φ3+ 26φ2−12φ−2) −11φ2+ 6φ+ 1 >0.
We canno sol e his explici ly o φ(β), howe e , nume ically o ele an β
alues. We ind he h eshold alue o he a ailabili y o he i s -bes ea y, i.e.
¯
φB
F B, a he poin a which he abo e inequali y holds wi h equali y. Thus, o
β∈(0,0.15],¯
φB
F B ∈(0.6448,0.6793) wi h dφB
F B
dβ <0.
(ii) The o ice en which sepa a es insu ance and di e en ia ion ea ies is de ined by
(27) o φ > ¯
φB. Again, con a y o he case o a g een incumben , he e, we do
no ha e o conside he he global maximum wi hin a ea A o compa ison. In he
ollowing, I show ha his is ue o he pola isa ion ange φ∈(¯
φB,1), whe e he
choice is be ween an insu ance ea y and a di e en ia ion ea y. The a gumen a ion
is pe ec ly analogous o (i), 2.
Figu e 16a shows he slope o WAe alua ed a he h eshold alue δi, which in his
case is he bounda y alue be ween a ea Dand A. Again, o e y small le els o R
he slope is posi i e, meaning ha δ∗
i,A > δi. The h eshold alues o R, o which
δ∗
i,A is a ailable in a ea Aa e depic ed as iangles in Figu e 16b. In he same igu e,
he do s e e o he h eshold alue ¯
R, esul ing om he compa ison o an insu ance
ea y wi h ¯
δj(sepa a ing a ea Cand a ea D) and a di e en ia ion ea y wi h δ∗
i,A.
Again, we can see ha hese o ice en le els clea ly di e ge. Thus, o he necessa y
le els o Ra which he incumben would be indi e en be ween a comp omise and
64
an insu ance ea y, he global op imum o WAis no a ailable as a ea y pa ame e .
Hence, o dis inc ion ea ies, i holds ha δ∗
i=δi.
0.1 0.2 0.3 0.4 0.5
R0
dWA
dδ
|δ=δli
(a) φ= 0.9, slope o WAa h eshold δi
0.80 0.85 0.90 0.95 1.00 ϕ
0.1
0.2
0.3
0.4
0.5
0.6
R
(b) Do s: Rwhich sepa a es WA(δ∗
i,A)
and WB(¯
δj). T iangles: h eshold R
a which δ∗
i,A ≥δi
Figu e 16: Global maximum o WAis no an equilib ium in δ∈(δi,1)
65
C Ex ensions
P e e ence Asymme y
He e we p o ide he o mal backg ound o he ex ension wi h p e e ence asymme y and
how hey ela e o he main P oposi ions. Mo e de ailed o mal p oo s o he compa a i e
s a ics a e no p o ided a his poin , bu can be g aphically con i med.
Lemma C.1 (Exis ence o Ra i ica ion In e al o B own Challenge )
The challenge ’s a i ica ion h esholds exis i :
λ≤¯
λ=β−1+2µ(1 −β−βµ)
3(β−1) + 2βµ (C.1)
µ≤¯µ=1−β−p1−4β+ 3β2
2β(C.2)
and whe e i holds ha :
d¯
λ
dµ <0,d¯µ
dβ >0.(C.3)
P oo o Lemma C.1
The challenge ’s h eshold alues exis i Mj=Bis non-nega i e. The e o e:
β2
<0
z }| {
(β−1)
<0
z }| {
(λ−1)
| {z }
>0
[1 −2µ−3λ+β(2µ(1 + µ)−1 + λ(3 + 2µ))] ≥0
I hus su ices o conside he e m in squa e b acke s o de e mine he sign:
λ(3(β−1) + 2βµ)≥β−1+2µ(1 −β−βµ)
λ≤β−1+2µ(1 −β−βµ)
3(β−1) + 2βµ ≡¯
λ
No e ha he sign swi ches be ween he wo lines because he denomina o is nega i e
o he pa ame e ange conside ed. This can be shown by assuming he pa ame e s ha
maximise his exp ession, i.e. β= 0.15 and µ= 1 and hen 3(β−1) + 2βµ =−2.25 <0.
No e ha a ¯
λ(¯µ)=0, meaning ha µ≤¯µensu es ha ¯
λis non-nega i e.
66
Lemma C.2 (S age 3: Ra i ica ion In e als wi h i=G)
In he case o a g een incumben , he incumben ’s and coun y 2’s a i ica ion h esholds a e
gi en as ollows:
[δi=G,¯
δi=G] = max0,1 + β(1 + µ) [β(3 + µ)−4]
(β+βµ −1)2,1(C.4)
[δ2(θG),¯
δ2(θG)] = 1 + β[2β(1 + µ) + β−4]
(β−1)2,1(C.5)
The challenge ’s a i ica ion h esholds exis when λ≤¯
λand µ≤¯µ. In ha case, hey a e
gi en by:
[δj=B,¯
δj=B] = "1−β[3 −2λ+µ+β(λ−1)(2 + µ)] −pMj=B
(β+βµ −1)2,
min1−β[3 −2λ+µ+β(λ−1)(2 + µ)] + pMj=B
(β+βµ −1)2,1#(C.6)
whe e Mj=B=β2(β−1)(λ−1) [1 −2µ−3λ+β(2µ(1 + µ)−1 + λ(3 + 2µ)].
Lemma C.3 (S age 3: Compa a i e S a ics wi h i=G)
The ollowing condi ions hold o he equilib ium a i ica ion in e als unde he condi ion
ha h esholds exis and ha hey a e wi hin he in e al [0,1]:
dδi=G
dµ <0,d¯
δi=G
dµ = 0,dδi=G
dλ = 0,d¯
δi=G
dλ = 0
dδj=B
dµ >0,d¯
δj=B
dµ <0,dδj=B
dλ >0,d¯
δj=B
dλ <0
Analogously o he case o symme y, P oposi ion 1 holds and as a consequence, cases as
illus a ed in Figu e 3 ollow. Howe e , now he wo scena ios a es dis inguished by whe he
µand λa e below h eshold alues as de ined in Lemma C.1.
67
Lemma C.4 (S age 3: Ra i ica ion In e als wi h i=B)
In he case o a b own incumben , a i ica ion h esholds a e gi en as ollows:
[δi=B,¯
δi=B] = "1 + β(λ−1) [4 + β(λ−3)]
(1 + β(λ−1))2,1#(C.7)
[δ2(θB),¯
δ2(θB)] = 1 + β(2β(1 −λ) + β−4)
(β−1)2,1(C.8)
The challenge ’s a i ica ion h esholds always exis and a e gi en by:
[δj=G,¯
δj=G] = "max0,1 + β[λ−3−2µ−β(2 −λ)(1 + µ)] −pMj=G
((1 + β(λ−1))2,(C.9)
1 + β[λ−3−2µ−β(2 −λ)(1 + µ)] + pMj=G
((1 + β(λ−1))2#(C.10)
wi h Mj=G=β2(1 −β)(1 + µ) (1 + 3µ+ 2λ+β[2λ(µ+λ−1) −3µ−1]).
Lemma C.5 (S age 3: Compa a i e S a ics wi h i=B)
The ollowing condi ions hold o he equilib ium a i ica ion in e als unde he condi ion
ha h esholds exis and ha hey a e wi hin he in e al [0,1]:
dδi=B
dλ >0,d¯
δi=B
dλ = 0,dδi=B
dµ = 0,d¯
δi=B
dµ = 0
dδj=G
dλ Q0,d¯
δj=G
dλ <0,dδj=G
dµ <0,d¯
δj=G
dµ <0.
Lemma C.6 (O de ing o Coun ies’ Ra i ica ion In e als wi h i=B)
The wo scena ios a e sepa a ed a he poin whe e he a i ica ion in e als ouch, i.e. whe e
δi=¯
δj. Sol ing his o µyields he ollowing h eshold:
˜µ=8−9λ−β(λ−1)(6λ−16 + β[8 + λ(λ−5)])
(1 + β(λ−1))2,(C.11)
whe e d˜µ
dλ <0. The h eshold alue is non-nega i e as long as λ<λ0, whe e λ0(β)is mos
es ic i e a β= 0.15 and akes a alue o λ0(0.15) ≈0.85.
Analogous o symme y, he indings o P oposi ion 2 hold and consequen ly, cases as illus-
a ed by Figu e 4 ollow.
68
Mo e sophis ica ed coun y 2
Lemma C.7 (Equilib ium Consensus T ea ies wi h Sophis ica ed Coun y 2)
Fo su icien ly low le els o pola isa ion and a b own incumben , i holds ha :
¯
δnew
2≥δ∗
i,(C.12)
which implies ha any consensus ea y which esul ed as an equilib ium in he basic model
is s ill alid wi h a mo e sophis ica ed coun y 2. The highes pola isa ion le el which allows
o his is gi en by he ¯
φB
max, o which (C.12) holds wi h equali y.
P oo o Lemma C.7
Fi s , he new uppe a i ica ion h eshold o coun y 2 is implici ly de ined as ollows,
ollowing (31c):
∆E[WC
2]=0 ⇒hδnew
2,¯
δnew
2i
I he e is no pola isa ion, i.e. o φ= 0:
δ∗
i<1,¯
δnew
2= 1
Now wi h inc easing pola isa ion, he op imal ea y alue o he b own incumben in-
c eases, whe eas he uppe pa icipa ion h eshold dec eases:
dδ∗
i
dφ >0,d¯
δnew
2
dφ <0
The e o e, i ¯
δnew
2−δ∗
i≥0a φ=¯
φB
max, i is ue o all o he alues φ < ¯
φB
max. We
ind his uppe limi by nume ically sol ing (C.12) o β∈(0,0.15] and ind ha ¯
φB
max ∈
(0.4452,0.4456) wi h d¯
φB
max
dβ <0.
69
