# Summary

This chapter examines how an extreme candidate wins against a moderate candidate to provide one reason to have political polarization, based on the model of partially binding platforms with asymmetric information. There are two main equilibria, namely, semi-separating and pooling, and voters cannot determine a candidate’s political preferences in any equilibrium. In a pooling equilibrium, an extreme candidate imitates a moderate candidate by announcing the same platform as the moderate candidate. In a semi-separating equilibrium, an extreme candidate imitates a moderate candidate with some probability, and with the remaining probability reveals his own preferences by approaching the median policy. An extreme candidate who reveals his preference type will defeat an unknown candidate who may be moderate or extreme, even though this extreme candidate will implement a more extreme policy than the moderate candidate. This is because voters wish to avoid electing an extreme type who imitates a moderate candidate and will implement the most extreme policy. As a result, an extreme candidate has a higher expected probability of winning than a moderate candidate. The important reason for this result is that an extreme candidate has a stronger incentive to prevent an opponent from winning because his ideal policy is further from the opponent's policy than is the ideal policy of a moderate candidate.

Combining the implications from Chapters 2 and 3, the following result is obtained. When voters know little about candidates and are uncertain about their policy preferences, an extreme candidate tends to win an election. By contrast, when voters have sufficient information about candidates, a more moderate candidate tends to win. Therefore, political polarization tends to occur when voters have insufficient information about candidates.

# Appendix: proofs

**3.A.1 Proposition 3.3**

First, a candidate has no incentive to deviate to a more extreme platform than * zf^{1}** or r, e because off-path beliefs are

*p*(M|z,) = 0.

_{t}Hence, this candidate will be certain to lose and the expected utility will decrease for an extreme type (from Lemma 3.4) and will remain unchanged for a moderate type (from the definition of * zf^{1}*).* A moderate type does not have an incentive to approach the median policy by more than z-, because her utility from winning will become lower than her utility if her opponent wins (from the definition of

*zf*). An extreme type also does not have an incentive to deviate by approaching the median policy by more than

^{1}*z*when

*p*>

^{M}*p*, from the discussion in Subsection 3.4.2. As a result, this is an equilibrium in which both candidates announce z/

^{w}.

З.А.2 **Lemma 3.4**

Consider a case of Л, without loss of generality. Lq.x'r ^{=} Xr (^{г}д (-/,)) denote the situation where the utility when type *t R* wins is the same as the expected utility for type *t R* when *L* wins, given =l. This means that

where ='r(x'r) represents the platform in which the candidate implements Xr. First, I prove the second statement.

(ii) Differentiate both sides of (3.10) with respect to x^, given the opponent’s strategies = Xl (=l) ^{an}d Xl = Xl (*=L*))• Then,

From Lemma 3.1, w'(v^ *- x' _{R}) = c’[x'_{R}*-^(ЛГд))- Moreover, fix Xr

and differentiate w'(a« - *x' _{R}) = c'(x'_{R} ~-_{R}* (^)) with respect to

*x'*Then,

_{R}.Substitute these values into (3.11), which then becomes

If (3.12) is positive, an extreme type will implement a more extreme policy than will a moderate type. In the same way as ^='r(x'r) ^Xr was derived,

so the denominator of (3.12) is positive. To prove that (3.12) is positive, it is sufficient to show that the numerator of (3.12) is positive. In other words,

Note that, from (3.10) and Lemma 3.1,

Since 1 assume that *c'(d)lc(d)* strictly decreases as *d* increases (Assumption 1 *),c’[x' _{R}* -

*='*'

_{R}(x_{R})) /

*Cfa ~*4 (x'r )) > c(x'r ~ 4

*(x'*

_{R}))'* ^{c}* (

*X*(4)). The right-hand side of (3.13) is greater than the right- hand side of (3.14). Therefore, if the left-hand side of (3.13) is less than the left-hand side of (3.14), (3.13) holds. This means

_{R}~^{:}'_{R}

Since 1 assume that *u’(d)l u(d)* strictly decreases as *d* increases (Assumption 3), the right-hand side is positive. If Xl = Xl = *X*_{R}, both sides are the same. If *x[* and Xl become further from 4 than *X*_{R}, the left-hand side decreases. The reason is as follows. I differentiate «'(4 *~xt)l ^{U}'(^{X}'_{R} -Xr)~ *

^{u}(

^{x}r-Xt)

*'[*

^{1 u}^{x}r ~ Xr) with respect to

^{x}r - xl then M'(4

*-xt)*u"(

^{1 }^{x}'r -x'r)- "'(

^{x}r -xt)/ «'(4 - x'r )• This value is negative because I assume that

*u’[d)l u'(d)*strictly decreases as

*d*increases. As a result, the left-hand side of (3.13) is less than the left-hand side of (3.14), so (3.13) holds, and (3.12) is positive.

(i) The first statement is true because

82 *Electoral promises as a signal ***3.A.3 Corollary 3.5**

When *p ^{M}* =0, z/ = r/

^{w}*, from (3.4). Thus, the left-hand side of (3.5) is

*~*

^{и}1х?

^{л}/|)~^{с}(=^*- X? [-i^{4}*)|)

^{and is}strictly greater than

the right-hand side, since *-u^xf* <*-ul^xF [=!“*)-*a/|)

(c/^{w} |j. That is, (3.5) does not hold when *p ^{M}* = 0.

From (3.4), when *p ^{M}* =1,

*z =*(=/

^{W}*))- Here, if an extreme

type announces the platform *zfi^xt ^{1}* (

^{:}^*)), *

^{ie wd}* implement the policy of the moderate type,

*Х‘У*The left-hand side of (3.5) is

*-4х?(=Г)-^Ус( _{:}г(_{х}Г(;Г))-хУ(=Г% ***** '»*

strictly less than its right-hand side, since - а (уК | r ' j - ,v,^{£} |) a ^{_}

“(К(^{г}"‘)-^|)-ф^{£}(*"(="’))-^(^{г}"')). ^^{rom} Lemma 3.4,

and -»(| * _{X}F(-r*) - Д-,

^{£}|) - 4-Г -

*а*(=Г Ц) > (--/“•) - ,v,

^{£}|)

*-Л^(х"[-Т))-х"[--Г*)That is, (3.5) is satisfied when *p ^{M}* = I.

Both sides of (3.5) change continuously with *p*^{M}, so there always exists *p ^{M} =pe* (0,1) in which both sides of (3.5) are equal.

**3.A.4 Proposition 3.7**

The precise definition of a semi-separating equilibrium is as follows. Denote the expected utility of an extreme type who announces r,- as

*У, ^{Е}Ы-*

**Definition 3.9 ***A continuous semi-separating equilibrium is a collection *(z*,cr ^{w},F(.),n) *and a two-policy semi-separating equilibrium is a collection* (z,-,C7^{M},--,n), *where* z*, *is a platform chosen by a moderate type, <7 ^{M} is the probability of an extreme type choosing in a mixed strategy,* F(.)

*is a distribution function with the support ofzi,zj]for L and*[сд,Тд ]

*for R, and П is a scalar, such that: (а) П = Vf*(r,) =

*V*(z* j,

_{t}^{E}*for all*r

*j in support of*/-'(.)

*in a continuous semi-separating equilibrium; and (b) П =*K

^{£}(r* j =

*Vf (=i) ht a two-policy semi-separating equilibrium.*

**Define ***<У‘ ^{Х1}*

**and**

*П*

First, I discuss a continuous semi-separating equilibrium.^{12 }When an extreme type announces the expected utility is

*P ^{M})]^xf{=j)-^dF{=j).* When an extreme type announces

*z*the expected utility is

_{h }*V*(=)■)=(

_{t}^{E}*р*+

^{м}^*О*(l-

^{M}*p*))[-m(=;)-a/|) -

^{M}^{c}(|5/ ~ if (^)|)]

^{-}(l

^{_a W}) (1 “^) J

^{M}(|^/ (~y)“

^{A}/|)

^{dF}(=/')• The

value of *a ^{M}* is decided at the point at which the extreme type’s expected utilities under

*z,*and r) are the same:

When cr^{v/} = 1, the left-hand side of (3.15) is less than the right-hand side because (3.5) does not hold. If the left-hand side is greater than the right-hand side when cr^{v/} = 0, the value of *a ^{M}* e (0,1) under which an extreme type is indifferent between

*=,*and 5) exists. The following condition means that the left-hand side is greater than the right-hand side of (3.15) when = 0:

First, *—u*(|*,^{£} (=;)-*xf* |) *-c*(|z* -*xf* (.-*)|) > -m(*xf* (=i-)■-*xf* I) *-C*(|r, -. ^{A}/(-')|) because *z _{t}* is more moderate than

*z*.*Second,

*z,*is the platform with which an extreme type can defeat a moderate type who announces

**z,-.**When

*o*becomes zero, voters guess that a candidate announcing

^{M}**z**is a moderate type. From the definition of

**z),**an extreme type’s implemented policy, needs to be more moderate than a moderate type’s implemented policy, £-

^{w}(z,*j. From Lemma 3.4, a moderate type has a greater incentive to choose a more moderate

*implemented policy*than an extreme type, and a moderate type is indifferent to winning or losing at z,-. This means that

*)>~*

^{u}(xf(=t )-^{x}f)-^{c}(=t-xf*(=i*)|j- As a result, (3.16) holds, so a value of

*e (0,1) under which an extreme type is indifferent between*M

**5}**and

**z***exists.

**The other bound of support for ***F(.)*

The distribution function, F(.), satisfies the following lemma.

**Lemma 3.10**

*Suppose that a continuous semi-separating equilibrium exists. In such an equilibrium,* /-'(.) *is continuous with connected support.*

**Proof: **If *F(.)* has a discontinuity at some policy, say **z- **(i.e., *F* (*z]* +) > *F(z* -)) there is a strictly positive probability that an opponent also chooses *z'j* (the probability density function is / (z'-j > 0). If this candidate approaches the median policy by an infinitesimal degree, it increases the probability of winning by / (z'y) / 2 > 0. On the other hand, because this approach is minor, the expected utility changes by slightly less than

0^{12})/(^{z}/) -ф,^{£} (z')-A_{V}|) —c(|*A-xf* И)|)-{-ф/ (^{z}y)-*|} and is positive (or negative). This implies that if F(.) has a discontinuity, it cannot be part of a continuous semi-separating equilibrium. Assume that F(.) is constant in some region, [zi,z2], in the convex hull of the support. If a candidate chooses *z,* he has an incentive to deviate to Z2 because the probability of winning does not change. However, the implemented policy will approach the candidate’s own ideal policy, so the expected utility increases. Thus, the support of F(.) must be connected.

At z,-, the expected utility is V^{е} (z,) = *-u* (| *,^{£} (z*,)-xf* |) + *с* (| *z,* - *xf* (=,■ )|) because, from Lemma 3.10, *F(zy) =* 0, and the probability of winning

is one. If (3.9) does not hold, *Vj ^{E}* (z,) is higher than

*V,*(z)) when z,- = z), so

^{E}*Zj Zj*in equilibrium. Therefore, a continuous semi-separating equilibrium exists. If (3.9) holds, the extreme bound and the moderate bound are equivalent (a two-policy semi-separating equilibrium). Suppose that (3.9) does not hold. In equilibrium,

*(=,)*and

*V,*should be the same, so г,- and F(.) should satisfy the following equation. Further, suppose

^{E}(=,)*R,*without loss of generality, then

I assume that the two candidates’ positions are symmetric, so when гд decreases, *zj_* increases. Then, *vf(z^)* increases because

* ^{U}(^{X}R ~ (^{=}L))dF(^{=}L)* decreases while

*Vf (z#)*decreases. In

*-L* I*

addition, *F{.)* adjusts the value of **J **_ *ufxf* - Xl {^{=}l))^F(zl)- Thus,

there exist combinations of r) and F(.) that satisfy (3.17).

I denote *=f* such that -m *(|xf {=j)~ ^{x}f* |) =

*-«(|%f {^f)~*|) ~

^{x}f*cj^zf -%f{zfj.*The moderate bound,

*z*should be more extreme than

*zf.*If

*Zj*is more moderate than

*zf,*it means -

_{М}(|^(,

_{у}.)-^|)>-

_{М}^(->л/|)-ф.-^(

_{5})|). Thus, an extreme type with

*Zj*has an incentive to deviate and lose to an extreme opponent with a platform close to

*zj.*Any platform in the support of

*F*(.), say z ■,

needs to satisfy -wj*/ (zj) - л/|) > -м(|*,^{£} *(zf)* - *xf* | j - cflzf - *xf* (z,')|) to avoid deviating to lose. Therefore, *xf (в)* is more extreme than *X'f ^{1}* (z*J, because

*xf (=f)*is more extreme than

*xf*(=;*)•

^{1}**Define ***F(.)*

Suppose *R,* without loss of generality. Let

For any z'r g (£я,:Гд), the expected utility should be the same as Я.^{1? }This means that

The distribution function, *Fy* (.), is defined by the above equation for any platform in support of F(.), given *X*(z£). When *Fy* (z«) = 0, it is

*П + и*(a£ - Xr {=r)) + * (Xr (-a ) - *=R*) *X {z*'_{L}) = 0. This equation holds if and only if r_{R} = zr and X{-l) = 0, such that * П *= (zr). Then,

*X* (*z*_{L}) = 0 if and only if 2/, = *zj^.* Therefore, when *z' _{R}* and

*z*become

_{L}*z*and £/,, respectively,

_{R }*F(z'*becomes zero.

_{R})When F(z'* _{r}) =* 1, it is Я = (p

^{;V/}+ <7

^{;W}(l-

*р*))[-м(л| -*f (

^{м}*z*'

_{R}))-

^{c}{Xr{^{:}'r)~^{z}'r)[}~(z'i)]. This equation holds if and only if

^{z}-h

z'r **=z _{R}andX(z'_{L})=**

**J**m(a£ ~Xl (-£.)) (-z.), such that Я =

**V**_{R}(z_{R}).This means that w'hen *z _{R}* and

*z'i*become

*z*and

_{R}*z*respectively,

_{R},*F(z'*becomes one.

_{R})When *z'i* satisfies |л„, -2^|=|.v_{w} — -J? |, that is, *F(.)* is symmetric for both candidates, the value of *X(z _{R})* increases continuously as

*z'*(

_{R}*z*

_{R}) becomes more extreme. Therefore, if the platform moves from zr to z’r, F(z

*increases from zero to one. Thus, if F(.) is symmetric for both candidates,*

_{r})*F,*(.) can be defined for

*i = L,R.*

An extreme type does not deviate

An extreme type does not deviate to a more moderate platform than *zj, *as the probability of winning is still one. However, the cost of betrayal and the disutility following a win will increase.

If an extreme type deviates to a platform that is more extreme than *Zj*, or between *z,* and *z,-,* this candidate is certain to lose because voters believe that such a candidate is an extreme type based on the off-path belief. Therefore, the expected utility is:

Subtracting this equation from *V) ^{E}* fz*J yields

A moderate type is indifferent to wanning and losing at z*. That is, (3.7) holds. Thus, from Lemma 3.4, the value of the above is positive, and this deviation decreases the expected utility. Note that (3.10) in the proof of Lemma 3.4 uses *p*^{M}, but the same result holds when *p ^{M}* is

replaced by *р ^{м} I*[

*p*(l -

^{M}+o^{M}*p*

^{M})].

A Moderate type does not deviate

Suppose *R,* without loss of generality. As a moderate type is indifferent between winning and losing at zr, she is indifferent on whether to deviate to a platform that is more extreme than zr or between zr and zr. The second possibility involves deviating to any platform in z'r g For an extreme type, the candidate is indifferent between zr and zr. This means that

A moderate type has no incentive to deviate to *z* if

I disregard *(-a ^{M}^(-p^{M}* j and differentiate the right- hand side of the above equations with respect to

*x*to obtain

_{R}*^ _{L}^{u}X^{x,}R-xt{=L))dF{:_{L})-{-F(z'_{L}))u'^X^{,}R{z'R)-x^{,}R* |). This is

positive because the opponent’s implemented policy is further from the ideal policy than z'r, so the right-hand side of (3.18) is greater than the right-hand side of (3.19). From (3.15), at z'r = zr, the left- hand side of (3.18) is zero. From (3.7), the left-hand side of (3.19) is

P^{M}^{u}(^{x}r - Zr {-«)) + ‘-(лГд' (-л) —-д)] . SO is positive because :’r is

smaller than zr. 1 differentiate the left-hand side with respect to z'r. Note that *o ^{M}* and zr are already decided, so only z'r changes. Then,

I ignore *p ^{M} +a^{M} p^{M}*)■ From Lemma 3.1, this is negative. That is, -u'^x'r (-^) — |) < 0- This implies that if zr becomes smaller, then

the left-hand sides of both equations increase. The next problem is the degree of increase. Differentiating (-л)^{-А}'л|) "'it*^{1} respect to

a' yields

I differentiate (3.15) with respect to x^, then

Thus, the value of (3.20) is negative. This implies that if x^ is more extreme, the increase of the left-hand side is lower when zr becomes smaller. At zr = zr, the left-hand side of (3.18) is lower than the right- hand side of (3.19). If *z' _{R}* becomes more moderate, both left-hand sides increase, but an increase in (3.19) is greater than an increase in (3.18). As a result, for all z'r, the left-hand side of (3.18) is lower than the right- hand side of (3.19), and (3.19) is satisfied.

Finally, since a moderate type has no incentive to deviate to zr, she does not deviate to any policy that is more moderate than zr.

*Electoral promises as a signal* 89 A two-policy semi-separating equilibrium

When (3.9) holds, a two-policy semi-separating equilibrium exists. When an extreme type chooses the expected utility is

=(i- 4 *-° ^{u}* (■ - />(№)

*-*4

-{^_{+<}T*(i-,")}{«(|^(4-)-4_{+c}(|=;-^(=;|}]-(i-

* ^{u}^xf{?j)* - x/|j.The expected utility when the candidate chooses

- 3 is ^(5)=[р"
_{+}а"(1-_{7}«)][«(|^(?,)-^|)-(ф--^(5)|) - -(l/2)(l-cr
^{J}')(l-p^{M})[»(|jf(:_{/})-AV^{£}|) + »(|£p(5)-if|) + r(|r_{i}-

*XF* (=/)|) • When cr ^{u} =1, *Vj ^{E} (~j)* is greater than V

^{е}because we

assume that (3.5) does not hold. Assume *a ^{M},* which satisfies (3.15). If (3.9) holds, then V

^{е}[:Л is less than V

^{е}(r* j at

*■*When

*o*increases continuously from сг , V

^{M}^{е}(=, ) increases and V

^{е}decreases continuously, so there exists a

*o™*under which V

^{е}(г, ) = V

^{е}(r,*), and such

*G*should be higher than

^{M}*o*

^{M}.Platform -• should be such that *xf* (-; ) is between *xf ^{4}* (-/*) and

*xF(*>0 and >0

^{=}t) P^{M}_{5}because in this region, there exists a policy that voters prefer to the expected implemented policy of a candidate with Thus,

*xf (-i)*is more extreme than

*xf*(-Л- An extreme type does not deviate for the reason explained in the previous discussions. If an extreme type deviates to a platform that is more moderate than the expected utility changes

^{1}by (l/2)(l-cr ^{v/})(l-*p ^{M} )[u(xf (=j*)-a/|) - u(|

*x*(a)-

^{E}*x*-ф -

^{E})*xf* (3})|j . This is negative because (3.9) holds.

A moderate type does not deviate to a more extreme policy than *=,■ *for the reason explained in the previous subsection. A moderate type does not deviate to z) if

As *Zj* is more moderate than *ulW* ^{4}* (r,)-x/

^{w}|) -hc(|

*-xf*(-,)| -

^{4}^{M}(|*/

^{W}(=»*) -

*|)" ‘•(|=/ -*

^{x}f*xf**(-/ )|) is positive. For an extreme type,

*is negative because (3.9) holds. From Lemma 3.4, its value for a moderate type is lower than for an extreme type, so*

^{u}(xf ^j)-^{x}f)~^{u}(x^(=i)-^{x}f)~^{c}(=t ~Xf (=t))*(*

^{u}(xf^{r}y')

^{_A}/

^{W}|)

^{_w}(|^/

^{W}(-») _

_{А}-Л/||_

_{с}/|= _ is also negative for a moderate type. As a result, (3.21) is satisfied.

**Asymmetric equilibrium**

There does not exist an asymmetric equilibrium in which candidates choose asymmetric platforms or different values of *a ^{M}* or

*F(.).*First, suppose that the support of

*F(.)*is asymmetric. Then, the probability of winning is constant in some regions of the support for at least one candidate, and it cannot be an equilibrium for the reason explained in Lemma 3.10. This means that

*should also be symmetric. Second, suppose that moderate types’ platforms are asymmetric. This means that the probability of winning for a moderate type is also asymmetric. Suppose moderate type*M

*R*announces a more extreme platform than moderate type

*L,*and so loses to

*L.*In this case, extreme type

*R*has no incentive to imitate moderate type

*R.*The values of

*o*should be symmetric, so it cannot be a semi-separating equilibrium.

^{M}**3.A.5 Proposition 3.8**

In such a separating equilibrium, the utility of candidate *i* of type *t *when he/she wins is (-/)—л/1)^{—c}(|-/ *~x!* (-<’)|)' Then, the utility

of candidate *i* of type *t* when a *same-type* opponent (type *t)* wins is —,w(|ЛГу 1 denote *zj* as the cut-off platform under which both

of these utilities are the same for a type *t* candidate, and are symmetric ||.v„, *-=j=x _{m}*-£y|). Figure 3.4 shows the positions of

*:'*and Then, the following lemma holds.

_{R}*Figure 3.4* Separating Equilibrium.

An extreme type has an incentive to pretend to be moderate by choosing the moderate type’s platform because the probability of winning increases, and the implemented policy approaches the ideal policy.

Lemma 3.11

*The extreme type's cut-off platform is more moderate than that of the moderate type, but the extreme type’s implemented policy given the cutoffplatform is more extreme than that of the moderate type.*

Proof: To prove this, I differentiate (3.10) with respect to *x _{R}-x:'_{L}* rather than .v^. Equation (3.10) can be rewritten as

*u[x'*where

_{R}+x^{,}R-2x_{m})-u[x'_{R}-x'_{R}) = c[xR--_{R}(x^{,}R%^{X}R ~X'r = (

^{Л}л -

*=*

^{x}m )^{+}{x'r-^{x}m)*+ Xr ~ 2a,„ because the platforms are symmetric. Differentiating both sides of (3.10) with respect to*

^{X}'_{R}*x'*is the same as differentiating both sides of the rewritten equation with respect to

_{R}-x'_{L}*x'*Then,

_{R}.For the same reason given in Lemma 3.4, this is positive. Note that (x'r)^^{x}r ^{=-m,}(^{a}'« ~Xr)I^{c}'(x'r ~-r (Яд))^ > ^{ancl}

!*Xr = ^{u}'{x'r-X'r)^{1} c’(x'r--r(x'r))_>®- Moreover,

The second term of the right-hand side is negative. The first term is also negative since *u’[x‘ _{R} - Xr) *

^{= c}'(Xr

^{-}-д(/£д)) fr°

^{m}Lemma 3.1. Thus, this value is negative.

Then, a symmetric separating equilibrium in which a moderate type wins does not exist. First, if a separating equilibrium in which a moderate type wins against an extreme type exists, regardless of off- path beliefs, an extreme type should announce *zf.* If the utility when an extreme candidate wins is higher than the utility when an extreme opponent wins, the extreme candidate has an incentive to win with certainty against the extreme opponent. This is made possible by approaching the median policy, regardless of off-path beliefs.

Second, a moderate type never announces a more moderate platform than *zf.* On such a platform, the utility when this moderate type candidate wins is lower than the utility when a moderate opponent wins. Therefore, the moderate type has an incentive to deviate and lose to the moderate opponent. If this moderate type also has an incentive to deviate and lose against an extreme opponent, she will deviate to lose with certainty. If this moderate type has an incentive to win against an extreme opponent, she will deviate to approach *zf,* because she can win against an extreme opponent and lose against a moderate opponent.

Finally, suppose that a moderate type announces a more extreme platform than *zf* If an extreme type deviates to a moderate type’s platform, the extreme type can improve his chance of winning and can implement a policy closer to his ideal. As a result, the extreme type can increase his expected utility from this deviation.

An asymmetric separating equilibrium in which a moderate type wins *(z^* and *zff* are asymmetric) also does not exist. Suppose that *zf ^{1}* and

*zff*are asymmetric, so one moderate candidate defeats a moderate opponent with certainty. Without loss of generality, suppose that moderate type

*R*defeats moderate type

*L.*That is, Xr (

^{:}r )

^{- x}

*m*<

^{x}m*~*X'l [

^{:}l )’

^{anc}* moderate type

*R*defeats extreme type

*L.*Note that an extreme type announces

*zf.*If

*zff*is more extreme than

*zf {zff*an extreme type

*R*will deviate to pretend

to be a moderate type *R.* Therefore, assume that *zff* is more moderate than *zf (zf* < zfj.* There are three cases.

The first case is that moderate type *L* loses to or has the same probability of winning as extreme type *R [xf [=f)-x„,* < *x,„ - xff (*^{z}l))-

Regardless of off-path beliefs, if moderate type *R*'s platform approaches *zf,* she can win against both moderate and extreme types of L, and the disutility following a win and the cost of betrayal decrease as the platform approaches her ideal policy.

The second case is that moderate type *L* defeats extreme type *R*

[Xr [^r )~* ^{x}m* >

*)) when moderate type*

^{x}m ~Xl [~l*L*announces a more moderate platform than

^{:}l . From Lemma 3.4, moderate type

*R*has an incentive to^deviate and lose to moderate type

*L.*If moderate type

*R*approaches

^{=}r by more than she can lose to moderate type

*L*and still win against extreme type

*L.*

The final case is that moderate type *L* defeats extreme type *R *(/ff (^л)^{-А}'н ^{>x}»>~Xl [^{:}l)) when moderate type *L* announces a platform that is the same as, or closer to her own ideal policy than *zjf.* If extreme type *L* deviates to moderate type *V*s platform extreme type *L* can win against extreme type *R* with certainty and so gain a higher probability of winning. With this deviation, an extreme type can implement a policy closer to his ideal policy, so he will deviate.