Candida e quali y in a Downsian Model wi h a
Con inuous Policy Space.1
En ique a A agonès2
Ins i u d’Anàlisi Econòmica, CSIC
Dimi ios Xe e is3
Uni e si y o Cyp us
Janua y 10, 2011
1A agones acknowledges inancial suppo by he Gene ali a de Ca alunya G an numbe 2009 SGR 1126, he Spanish
Minis y o Educa ion g an numbe ECO2009-08820 and he Ba celona G adua e School o Economics.
2Ins i u d’Anàlisi Econòmica, CSIC, Campus UAB, 08193 Bella e a (Spain). Email: [email p o ec ed]
3Depa men o Economics, Facul y o Economics and Managemen , Uni e si y o Cyp us. P.O. Box 20537, CY-1678
Nicosia (Cyp us). Email: xe e is.dimi [email protected]
Abs ac
This pape cha ac e izes a mixed s a egy Nash equilib ium in a one-dimensional Downsian model o wo-candida e
elec ions wi h a con inuous policy space, whe e candida es a e office mo i a ed and one candida e enjoys a non-
policy ad an age o e he o he candida e. We assume ha o e s ha e quad a ic p e e ences o e policies and ha
hei ideal poin s a e d awn om a uni o m dis ibu ion o e he uni in e al. In ou equilib ium he ad an aged
candida e chooses he expec ed median o e wi h p obabili y one and he disad an aged candida e uses a mixed
s a egy ha is symme ic a ound i . We show ha his equilib ium exis s i he numbe o o e s is la ge enough
ela i e o he size o he ad an age.
Key wo ds: spa ial compe i ion; mixed s a egies; candida e quali y
1In oduc ion
Candida e quali y is conside ed o be a c i ical a iable in elec o al compe i ion. I affec s he
decisions o poli icians ega ding whe he o un o office, campaign und- aising, o e beha io ,
elec ion ou comes, and, ul ima ely, policy ou comes. Quali y diffe ences be ween wo candida es
can a ise o many easons, including cha isma, office-holding expe ience, incumbency, ad e ising,
scandal, and any o he non-policy dimension ha may affec he ela i e a ac i eness o wo
candida es. In Poli ical Science candida e quali y is also deno ed by “ alence dimension” and i s
impo ance has been widely demons a ed o e se e al decades o ca e ul empi ical esea ch.1
All else cons an , high quali y candida es will a e be e han low quali y candida es. Fu he -
mo e, quali y diffe ences p oduce signi ican changes in he na u e o poli ical compe i ion. The
equilib ium p ope ies o spa ial compe i ion be ween wo candida es who diffe in quali y ha e
been analyzed heo e ically. Recen pape s by Ansolabehe e and Snyde (2000), A agones and
Pal ey (2002), G oseclose (2001) and Hummel (2010) epo a numbe o heo e ical esul s abou
he equilib ium p ope ies o spa ial compe i ion be ween wo candida es who diffe in quali y.2
These pape s use a amewo k o s udying he effec o candida e quali y on poli ical com-
pe i ion, based on he s anda d Downsian model compe i ion be ween wo candida es wi h an
impo an wis : any o e will s ic ly p e e he “highe quali y” candida e o he “lowe qual-
i y” candida e i he candida es loca e so ha he o e is indiffe en be ween he wo candida es
on he policy dimension.
G oseclose (2001) shows ha , o office mo i a ed candida es, exis ence o pu e s a egy equi-
lib ium is especially p oblema ic o small- o-in e media e alues o he quali y ad an age. When
he policy space is an in e al o he eal line and o e s’ p e e ences a e Euclidean he payoff
unc ions o he candida es a e discon inuous. This discon inui y implies ha he bes esponse
o he disad an aged candida e is no well de ined. In his se up A agones and Pal ey (2002)
desc ibed he equilib ium s a egies o a pa icula case: when he policy space is a ini e g id o
1See, e.g., S okes (1963), Kiewie (1983), and Kiewie and Zheng (1993).
2The e a e also some ea lie heo e ical pape s ha s udied ela ed kinds o asymme y, such as incumbency o pa isanship, e.g.,
Adams (1998), Be nha d and Ingbe man (1985), and Lond egan and Rome (1993).
1
poin s on he [01] in e al and when he ad an age is ela i ely small. Hummel (2010) s udies he
same en i onmen wi h he only diffe ence ha he conside s ha he ad an age migh be highe
han A agones and Pal ey (2002). In his case he median o e migh p e e o o e o he can-
dida e wi h supe io alence e en i he p e e s he policy o he o he candida e. Hummel (2010)
cha ac e izes he op imal ac ions o bo h candida es in a pa icula equilib ium o he game bu ,
unlike A agones and Pal ey (2002), he does no ully cha ac e ize any equilib ium o he game.
The cha ac e iza ion o he equilib ium s a egies o a con inuous policy space has no been
s udied so a and i is he main objec i e o his pape . The model we analyze ollows he s anda d
Downsian model wi h a con inuous policy space and he ollowing modi ica ions: we assume ha
o e s ha e quad a ic p e e ences ins ead o Euclidean and we assume ha he belie s o candida es’
on he dis ibu ion o he median o e ’s ideal poin a e unimodal (no necessa ily uni o m). By
changing he o e s’ p e e ences om linea o quad a ic he payoff unc ions o he candida es
become con inuous and he bes esponse o he disad an aged candida e is well de ined. Wi hin
his se up we a e able o ind condi ions o exis ence o mixed s a egy equilib ium and we a e
also able o desc ibe he equilib ium s a egies.
We ind a amily o unimodal dis ibu ions o he median o e ’s ideal poin ha gua an ees
exis ence o an equilib ium in which he ad an age candida e chooses a pu e s a egy ha con-
cen a es all he p obabili y in he expec ed loca ion o he median o e , while he disad an aged
candida e chooses a mixed s a egy ha alloca es p obabili y symme ically a ound he expec ed
loca ion o he median o e .
We ind necessa y and sufficien condi ions o his equilib ium o exis . These condi ions impose
es ic ions only on he candida es’ belie s on he o e s’ dis ibu ion o p e e ences. We ind ha
his equilib ium exis s only i he le el o unce ain y abou he loca ion o he median o e is
low enough ela i e o he size o he ad an age, ha is, when candida es belie e ha he median
o e ’s ideal poin is a ound 1/2 wi h high enough p obabili y.
Thecondi ionswe ind o exis ence do no es ic a all he size o he quali y ad an age, ha
is, he diffe ence in quali y be ween he wo candida es, as i was he case in G oseclose (2001).
2
Thus we show ha his mixed s a egy equilib ium may exis e en o small alues o he quali y
ad an age, which is when pu e s a egy equilib ium ails o exis .
As in simila models, we ind ha in equilib ium he ad an aged candida e ob ains a la ge
p obabili y o winning han he disad an aged candida e. We also ind ha as he alue o he
ad an age becomes la ge he p obabili y o wining o he ad an aged candida e inc eases, he
equilib ium s a egies o he wo candida es a e mo e diffe en ia ed, and he condi ions o exis ence
o equilib ium a e elaxed, ha is, he numbe o o e s (o he p obabili y wi h which he expec ed
median o e is a ound 12) needed o an equilib ium o exis is smalle . Finally, as he alue
o he ad an age becomes smalle , ha is, as he diffe ence be ween he wo candida es anishes,
he op imal s a egy o he disad an aged candida e mo es close o he ad an age candida e’s.
Tha is bo h playe s’ equilib ium s a egies con e ge o he expec ed median o e as candida e
’s ad an age sh inks o ze o.
The es o hepape p oceedsas ollows. Thenex sec ion desc ibes he o mal model. Sec ion
3 p esen s he de i a ion o he equilib ium s a egies and analyzes i s p ope ies. Finally, sec ion
4 con ains some concluding ema ks.
2TheModel
The policy space is he [01] in e al. The e a e  o e s, whe e is an odd and ini e numbe .
Each o e has a u ili y unc ion, wi h wo componen s, a policy componen , and a candida e
image componen . The policy componen is cha ac e ized by an ideal poin in he policy space,
wi h u ili y o al e na i es in he policy space a quad a ic unc ion o he dis ance be ween he ideal
poin and he loca ion o he policy. The image componen is cap u ed by an addi i e cons an o
he u ili y a o e ge s i he highe quali y candida e wins he elec ion.
The e a e wo candida es, and , who a e e e ed o as he ad an aged candida e and he
disad an aged candida e, espec i ely. Each candida e’s objec i e is o maximize his p obabili y
o winning he elec ion. We assume ha candida es belie e ha he ideal poin o each o e is
3

an i.i.d. d aw om a uni o m dis ibu ion in [01]
Thegame akesplacein wos ages. In he i s s age, candida es simul aneously choose posi-
ions in [01]. In he second s age, each o e o es o he candida e whose elec ion would gi e
him he highes u ili y. In case o indiffe ence, a o e is assumed o o e o each candida e wi h
p obabili y equal o 12.
Le deno e he policy posi ion chosen by candida e ,andle deno e he policy posi ion
chosen by candida e  Then, he u ili y ha a o e wi h ideal poin ob ains i wins he
elec ion is gi en by ()=−(−)2and his u ili y i candida e wins is gi en by ()=
−(−)2whe e 0deno es he size o candida e ’s ad an age.
Since he beha io o he o e s is unambiguous in his model, we de ine an equilib ium o
hegameonlyin e mso heloca ions a egieso he wocandida esin he i s ound. A
pu e s a egy equilib ium is a pai o candida e loca ions ( )such ha bo h candida es a e
maximizing he p obabili y o winning, gi en he choices o he o he candida e. A mixed s a egy
equilib ium is a pai o p obabili y dis ibu ions ()o e [01] such ha he e is no mixed
s a egy o  ha gua an ees highe p obabili y o winning han ,gi enand he e is no
mixed s a egy o  ha gua an ees highe p obabili y o winning han ,gi en.
No ice ha in his se up, i  hen all o e s wi h −(−)2−(−)p e e o o e
o candida e . The e o e, we ha e ha all o e s wi h an ideal poin +
2+
2(−)=b( )
p e e o o e o candida e  Since he ideal poin o each o e s is d awn om a uni o m
dis ibu ion, his implies ha he p obabili y ha a o e o es o he ad an aged candida e is
gi en by ( )=min{b( )1}and he p obabili y ha a o e o es o he disad an aged
candida e is gi en by ( )=max{01−b( )}
Simila ly i we ha e ha all o e s wi h an ideal poin +
2+
2(−)=b( )p e e
o o e o candida e  This implies ha he p obabili y ha a o e o es o he ad an aged
candida e is gi en by ( )=min{11−b( )}and he p obabili y ha a o e o es o he
disad an aged candida e is gi en by ( )=max{0b( )}
4
Since we assume ha he e a e  o e s, he p obabili y wi h which he ad an aged candida e
wins he elec ion is gi en by he p obabili y ha he ad an aged candida e ob ains he o es o a
leas a majo i y o he o e s. Because each o e will o e o candida e wi h p obabili y ( )
he p obabili y wi h which candida e is elec ed may be compu ed by he sum o he Be noulli
dis ibu ions co esponding o a leas a majo i y o successes o e  ials, ha is,
( )=

P
=+1
2¡
¢( )(1 −( ))−
Simila ly we could also show ha he p obabili y wi h which he disad an age candida e wins
he elec ion is gi en by ( )=

P
=+1
2¡
¢( )(1−( ))−=

P
=+1
2¡
¢(1−( ))( )−=
1−( )
Obse e ha ( )and ( )a e con inuous unc ions o ∈[01] and ∈[01]and he e o e
( )and ( )a e con inuous unc ions o ∈[01] and ∈[01] as well.
Finally, i = we ha e ha ( )=1and ( )=0, ha is, i bo h candida es choose
he same loca ion hen he ad an aged candida e wins wi h p obabili y one, because in his case
all o e s would p e e o o e o him.
No ice ha he payoff unc ions o he candida es in ou se up coincide wi h he c.d. . o a Be a
dis ibu ion wi h pa ame e s ==+1
2Such a dis ibu ion is unimodal and symme ic a ound
1
2This obse a ion allows us o offe an al e na i e in e p e a ion o ou model. Suppose ha
we ha e any numbe o o e s, e en a con inuum, and a unique median o e . Suppose ha he
candida es’ belie s abou he dis ibu ion o he median o e ’s ideal poin a e ep esen ed by his
Be a dis ibu ion. In his case he candida es’ payoff unc ions would be ep esen ed by he same
amily o c.d. . pa ame ized by he pa ame e o he Be a dis ibu ion ins ead o he numbe o
o e s o he o iginal se up. In his case ( )would ep esen he ideal poin o he o e ha
is indiffe en be ween he wo candida es. Thus in bo h cases we ha e ha he candida es’ belie s
abou he loca ion o he median o e ’s ideal poin a e mo e concen a ed a ound 1
2whene e he
numbe o o e s inc eases o , wha is he same, when he pa ame e o he Be a dis ibu ion
inc eases.
5
The payoff unc ions o he candida es in ou se up also coincide wi h hose o he Condo ce
ju y membe s (see, o example, Ki s ein and Wangenheim, 2010). This coincidence will p o e o
be help ul o ou analysis.
3 Equilib ium S a egies
When =0, nei he candida e has an ad an age, and we a e in he s anda d Downsian wo ld,
whe e in equilib ium he wo candida es loca e a 1
2and each wins wi h p obabili y 1
2. In gene al,
when 0, he e does no exis a pu e s a egy equilib ium. Diffe en e sions o his esul ha e
been s a ed and p o en in G oseclose (1999) and Be ge , Munge , and Po hoff(1999). The in u-
i ion is simple. I he disad an aged candida e’s loca ion is pe ec ly p edic able, he ad an aged
candida e can copy ha s a egy and win o su e. The e o e, a leas he disad an aged candida e
mus be mixing. The esul is ue unless is sufficien ly la ge ha canloca ea hemedian
and gua an ee a payoffo 1.
P oposi ion 1 I ≥1
4 he e is a pu e s a egy equilib ium in which wins wi h p obabili y
one.
(All p oo s may be ound in he appendix.)
In ou case, i 1
4, hen he e will be no pu e s a egy equilib ium. The aim o his pape is
o show ha i ∈¡01
4¢ he e exis s a mixed s a egy Nash equilib ium3in which he ad an aged
candida e chooses a pu e s a egy and he disad an aged candida e chooses a mixed s a egy. In
pa icula we show ha in his equilib ium he ad an aged candida e chooses a pu e s a egy
co esponding o he ideal poin o he expec ed median o e , =1
2while he disad an aged
candida e mixes be ween he wo policy loca ions =1
2−√and =1
2+√each wi h equal
p obabili y. We ind ha his equilib ium exis s as long as he numbe o o e s is la ge enough
ela i e o he size o he ad an age .Wealso ind he minimal numbe o o e s ha gua an ees
3I candida es ha e p i a e in o ma ion wi h con inuous ypes, hen his mixed equilib ium can be “pu i ied.” Tha is, he e will
exis an equilib ium in pu e s a egies, whe e he equilib ium loca ions o candida es will a y wi h his p i a e in o ma ion (A agones
and Pal ey 2005).
6
exis ence o his Nash equilib ium as a unc ion o he size o he ad an age.
We s a by demons a ing ha he s a egy p oposed o candida e ,e=(=1
2−√wi h
p obabili y 1
2and =1
2+√wi h p obabili y 1
2) is an op imal esponse o candida e choosing
e=1
2We p o e ha his holds ue o all alues o 
P oposi ion 2 Fo all 0and o all 01
4we ha e ha e=(=1
2−√wi h
p obabili y 50% and =1
2+√wi h p obabili y 50%) is a bes esponse o e=1
2
Nex we ha e o show ha he s a egy p oposed o candida e ,e=1
2is a bes esponse
o candida e choosing e=(=1
2−√wi h p obabili y 1
2and =1
2+√wi h p obabili y
1
2)No ice ha when candida e is choosing s a egy e=(
1
2−√w.p. 1
2;1
2+√w.p. 1
2)
candida e ’s p obabili y o elec ion is gi en by:
( e)=1
2( 12−√)+1
2( 12+√)
whe e
( 12−√)=

P
=+1
2¡
¢( 12−√)(1 −( 12−√))−
and
( 12+√)=

P
=+1
2¡
¢( 12+√)(1 −( 12+√))−
Mo eo e , obse e ha he unc ion ( )inc eases wi h  o all 1−p−2+2+1
emains cons an (( )=1) o ∈[1 −p−2+2+1p2+]and dec eases wi h  o
all p2+
[Figu e 1]
Tha is, ( 12−√)is inc easing in ∈[01−q1
4+2+√)cons an in ∈[1 −
q1
4+2+√ q1
4+2−√]and dec easing in ∈(q1
4+2−√ 1] and ( 12+√)is
inc easing in ∈[01−q1
4+2−√)cons an in ∈[1 −q1
4+2−√ q1
4+2+√]and
7
a pu e s a egy equilib ium in which he ad an aged candida e ob ains he o es om all o e s.
The e o e, wins wi h p obabili y one. ¨
P oo o p oposi ion2:
Gi en ha =1
2we sea ch o a alue o  ha maximizes he payoff unc ion o he disad-
an aged candida e, ha is, he ollowing exp ession.
(1
2)=

P
=+1
2¡
¢(1
2)(1 −(1
2))−
Ki s ein and Wangenheim (2010) show ha ()
 =¡−1
−1
2¢[( )(1 −( ))]−1
20 o
≥1
2.Thusweha e ha (1
2)is s ic ly inc easing in (1
2).
The e o e, in o de o ind a alue o  ha maximizes (1
2)i is enough o ind he alues
12 ha maximize (1
2)=max©0b(1
2)ª=max{01
4+
2−
1−2}; and he alues 12
ha maximize (1
2)=max{01−b( )}=maxn03
4−
2+
1−2o
No ice ha 12−√=a gmax1
4+
2−
1−2,and12−√∈£01
2¤when ∈(01
4]. Simila ly,
when 12we ind ha 12+√=a gmax3
4−
2+
1−2and 12+√∈£1
21¤when ∈(01
4]
The e o e, he p oposed mixed s a egy o candida e is a bes esponse o =12 o any
0¨
P oo o p oposi ion3:
I =1candida e ’s p obabili y o winning is gi en by
1( e)=1
2( 12−√)+1
2( 12+√)
We ha e seen ha 1( e)is inc easing in ∈[0q1
4+2−√)and ha 1( e)is
dec easing in ∈(1 −q1
4+2−√ 1] and, he e o e, he bes esponse o mus belong in
∙q14+2−√ 1−q14+2−√¸
14

Obse e ha o all ∈(01
4)we ha e ha 12−√q14+2−√121−
q14+2−√12+√and ha i =1candida e ’s p obabili y o elec ion 1( e)=
1
2( 12−√)+1
2( 12+√)can also be w i en as
1( e)=12(1 −+12−√
2+
2(−12+√))+12(+12+√
2−
2(−12−√))
o ∈[q14+2−√ 1−q14+2−√]
Thus, 1()
 =−
4(−12+√)2+
4(−12−√)20i and only i ³−12+√´2³−12−√´2
This implies ha 1( e)is dec easing o ∈[q14+2−√ 12) and i is inc easing o
∈(121−q14+2−√].
[Figu e 2]
The e o e 1( e)is inc easing o ∈[0q14+2−√],dec easing o ∈[q14+2−√ 12
)
inc easing o ∈(121−q14+2−√]and dec easing o ∈[1−q14+2−√ 1].This
implies ha when =1 he op imal esponses o candida e a e ei he =q14+2−√o
=1−q14+2−√bu no =1
2¨
P oo o p oposi ion4:
Le ’s show ha ()
0 o ∈[q14+2−√ 1
2)A simila analysis would p o e ha
()
0 o ∈(1
21−q14+2−√]
Since ( e)=1
2( 12−√)+1
2( 12+√)we ha e ha
()
 =1
2
(12−√)
 +1
2
(12+√)

whe e
( 12−√)=

P
=+1
2¡
¢( 12−√)(1 −( 12−√))−
15
and
( 12+√)=

P
=+1
2¡
¢( 12+√)(1 −( 12+√))−
We can compu e he de i a i e o ( 12−√)wi h espec o ()using he esul s in Ki s ein
and Wangenheim (2010) and ob ain (12−√)
 =¡−1
−1
2¢[( 12−√)(1−( 12−√))]−1
2
Thus we ha e ha he o al de i a i e o ( 12−√)wi h espec o by composing i wi h
i s pa ial de i a i e, ha is, (12−√)
 =(12−√)

(12−√)
 which can be w i en as
(12−√)
 =¡−1
−1
2¢[( 12−√)(1 −( 12−√))]−1
2(12−√)

and simila ly we ha e ha
(12+√)
 =¡−1
−1
2¢[( 12+√)(1 −( 12+√))]−1
2(12+√)

The e o e,
()
 =
2¡−1
−1
2¢[[( 12−√)(1 −( 12−√))]−1
2(12−√)
 +[( 12+√)(1 −
( 12+√))]−1
2(12+√)
 ]
and we need o p o e ha o la ge enough alues o we ha e ha
[( 12−√)(1−( 12−√))]−1
2(12−√)
 +[( 12+√)(1−( 12+√))]−1
2(12+√)
 ]
0
whene e ∈[q14+2−√ 1
2)
This holds i and only i
∙(12−√)(1−(12−√))
(12+√)(1−(12+√))¸−1
2(12−√)
 +(12+√)
 0
Fi s o all we will show ha ∙(12−√)(1−(12−√))
(12+√)(1−(12+√))¸−1
2
dec eases wi h and i ends o
ze o as  ends o in ini e.
No ice ha o ∈[q14+2−√ 1
2)we ha e ha ( 12−√)( 12+√)1
2
16
which implies ha ( 12−√)³1−( 12−√)´( 12+√)³1−( 12+√)´
always holds, since i does no depend on .
Thus ∙(12−√)(1−(12−√))
(12+√)(1−(12+√))¸−1
2
1and lim→∞ ∙(12−√)(1−(12−√))
(12+√)(1−(12+√))¸−1
2
=0
Sinceweha e ha (12+√)
 0we also ha e ha ∙(12−√)(1−(12−√))
(12+√)(1−(12+√))¸−1
2(12−√)
 +
(12+√)
 0will hold o la ge alues o 
Simila ly we could show ha o ∈(121−q14+2−√]we ha e ()
0which
comple es he p oo o he p oposi ion. ¨
P oo o p oposi ion5:
F om he las p oposi ion we know ha ()
 0 o ∈[q14+2−√ 1
2)i and only
i
∙(12−√)(1−(12−√))
(12+√)(1−(12+√))¸−1
2(12−√)
 +(12+√)
 0
which can also be w i en as
2
ln−
(12+√)

(12−√)
 
ln(12−√)(1−(12−√))
(12+√)(1−(12+√))+1
becauseweha e ha (12−√)
 0andwealsoha e ha (12−√)(1−(12−√))
(12+√)(1−(12+√))1implies
ln ∙(12−√)(1−(12−√))
(12+√)(1−(12+√))¸0
Fo =12 o be a local maximum we need o ha e ()
 ≥0 o =12−whe e 0
and →0
Thus we ha e o compu e lim→122
ln−
(12+√)

(12−√)
 
ln(12−√)(1−(12−√))
(12+√)(1−(12+√))+1
17
No ice ha when app oaches 1
2we ha e ha −
(12+√)

(12−√)
 →1and (12−√)(1−(12−√))
(12+√)(1−(12+√))→
1 hus we may apply l’Hopi al’s ule and we ob ain ha

lim→12
ln−
(12+√)

(12−√)
 
ln(12−√)(1−(12−√))
(12+√)(1−(12+√))= lim→12
−
(12−√)

(12+√)
 

−
(12+√)

(12−√)





(12+√)(1−(12+√))
(12−√)(1−(12−√))
(12−√)(1−(12−√))
(12+√)(1−(12+√))

Obse e ha since o ∈(12−√ 12+√)we ha e ha
( 1
2−√)=1−+1
2−√
2−
2(1
2−√−)and ( 1
2+√)=+1
2+√
2+
2(1
2+√−)
hen
( 1
2−√)
 =−1
2−
2(1
2−√−)2and (1
2+√)
 =1
2+
2(1
2+√−)2
which implies ha
µ−
(12−√)

(12+√)
 ¶=
1+ 
(1
2−√−)2
1+ 
(1
2+√−)2→1
and
−(12+√)(1−(12+√))
(12−√)
 
 =
2
(1
2+√−)3−2
(1
2−√−)3+22
(1
2+√−)2(1
2−√−)21
(1
2+√−)−1
(1
2−√−)

1+ 
(1
2−√−)2

2→2
√
Simila ly we ind ha
(12+√)(1−(12+√))
(12−√)(1−(12−√))→
1
4−
1
4−=1
and
(12−√)(1−(12−√))
(12+√)(1−(12+√))
 =
=(1−2(12−√))(12−√)

(12+√)(1−(12+√))−(12+√)(1−2(12+√))(12+√)
 (12−√)(1−(12−√))
(12+√)2(1−(12+√))2→4√
1
4−
becauseweha e ha
18
(12−√)
 →−1; (12+√)
 →+1
and
1−2( 12−√)→−2√;1−2( 12+√)→−2√
The e o e, we ob ain ha
lim→122
ln−
(12+√)

(12−√)
 
ln(12−√)(1−(12−√))
(12+√)(1−(12+√))+1=2¡1−4
8¢+1= 1
4
Fu he mo e, we compu e he sign o

ln

−
(12+√)

(12−√)




ln(12−√)(1−(12−√))
(12+√)(1−(12+√))
 o all ∈(q1
4+2−√ 1
2)
and o all ∈(01
4)using Ma hema ica and we ge ha i is posi i e (see igu e 7).
This implies ha i 
ln−
(12+√)

(12−√)
 
ln(12−√)(1−(12−√))
(12+√)(1−(12+√))+1 o →1
2 hen he inequali y should hold
o all ∈(q1
4+2−√ 1
2)as well.
Thus, =1
2is a global maximum i and only i ≥1
4¨
19

Figu e 1: The p obabili y ha a o e o es o he ad an aged candida e, p(x,y), as a
unc ion o he ad an aged candida e’s policy choice (x) gi en a policy choice o he
disad an aged candida e (y).
x
p(x,y)
½ ( y+ d/y )
½ ( (1-y)+ d/(1-y) )
y
Figu e 2: The p obabili y ha candida e A, P1(x,D), as a unc ion o he ad an aged
candida e’s policy choice (x) gi en he bes esponse o he disad an aged candida e
(D) when n=1 and d=0.05.
Figu e 3: The p obabili y ha candida e A wins, Pn(x,D), as a unc ion o he
ad an agedcandida e’s policy choice (x) gi en he bes esponse o he disad an aged
candida e (D) when n=5 and d=0.1, hus n>1/4d
Figu e 4: The p obabili y ha candida e A wins, Pn(x,D), as a unc ion o he
ad an aged candida e’s policy choice (x) gi en he bes esponse o he disad an aged
candida e (D) when n=5 and d=0.05 hus n=1/4d.