# Summary

This chapter examined the effects of partially binding platforms in elections. Although a candidate can choose any policy that differs from the promised policy before an election, it is costly to do so. I extend the standard political-competition model introduced by Downs (1957) by considering that candidates choose a platform and a policy separately, and by also adding the cost to renege on the platform. The model of partially binding platforms shows the following two notable implications, which cannot be analyzed by previous formal models. First, when candidates have different characteristics, one candidate has a higher probability of winning. That is, asymmetric electoral outcomes can be analyzed. Second, even if a candidate knows that he/she will lose an election, this loser runs to induce the opponent to approach the loser’s ideal policy. That is, it can describe the loser’s incentive more clearly.

# Appendix: proofs

**2.A.1 Corollary 2.2**

With *u"(d) =* 0, for all *cl* > 0, condition (2.1) is

w'hich does not depend on the position of.v,-. If Я < *ii* / - лу|), *Xi* (-;)>

w'hich satisfies (2.1), is further aw'ay from r, than л, . However, if a candidate chooses *Xj* rather than this ,£, (=,), the disutility is minimized at w(0) = 0, and the cost of betrayal also decreases. Thus, the winner implements л,- if Я < *U* / c'(|-- - л_{(}|).

**2.A.2 Corollary 2.3**

Suppose Candidate *L,* without loss of generality. Note that an increase in =l means that *=i* moves further from the ideal policy, лRewrite

(2.1) as *Xc'{= _{L}-Xl{=l)) = u'{Xl{=l)-Xl)*

^{and}differentiate it by

*z*Then, it becomes

_{L}.

Differentiate u(xl(=l)~^{x}l) by =l■ Then,

Differentiate *Xc(z _{L} - Xl* (=/.)) by

*z*Then,

_{L}.

■

**2.A.3 Lemma 2.4 Existence**

Suppose that both candidates commit to implementing the median policy *(Xi{ ^{:}i) = Xj(^{=}j) ^{= x}n)-* Then, the expected utility for candidate / is —m(|a

_{w}, — л-,|) + (1 / 2)|-Яс(|г,(х

_{т})-+ If

*i*deviates to lose by committing to a more extreme policy, /’s expected utility is—w(|a„, - a,|). Thus, if > Ас(|г,-(л

_{т})-a„,|), candidate / does not deviate.

**Uniqueness**

First, suppose that *Xi{=i)* ^{x}nr Xj{=j)*^{x}nt* and Ь | >

*\%j* thus, / loses. The expected utility of/' is -м||^у(су)-х,|).

If / deviates to commit to a policy, *xl,* which is slightly closer to *x _{m}* than

*Xj(zj),*and which is also closer to a, than

*x*the expected utility is

_{m},*-u(x!-*a,-|)-

*Xc(zi(xi)~*Я/|) +

*b*since / will win for sure. Thus,

^{if} “(] *Xj* (-/) — */|) - *u[x! **~* |) - Acflr,- *(xl) **- Xl) + b>* 0, candidate / deviates. First, *-Xc(zj(x!)-x!) + b>0* since Яс(|.-,(я/)- d) < Ac(|=,-(a,„)-a,„|) from Corollary 2.3, and *b >* Яс(|г,(а„,)-a„,| • Second, if *Xj[ ^{z}j)* >

^{s}further away from a,- than

*x*)

_{m}, u[xj[~j}~^{x}i-м(|я/-л/|)>0. If *Xj{=j)* is closer to a,- than a„„ м(|я_{;}(-/)-a, ) - *u(xl ~* a,|) is only slightly lower than but almost the same as zero. Thus, *u(xj* (-O-H)-" (к/ - *« |) - Ac (| *=i(x!)* - *Xl) + b>* 0.

Second, suppose that *X,* (-;) ^{[1]} * ^{x}m, Xj{=j)^^{x}»o* and

*Xi{=i)~*thus, there is a tie. The expected utility of

^{x}m = ^Xj*i*is (1

*12)[-u(x*,(-,)-

^{[1]},|)-^'(|-< -

*Xi(=i))*4)-

^{+ h}-^{u}(Xj{=j)-^{lf}

*i*deviates to commit to a policy,

*xl,*which is slightly closer to

*x*than

_{m}*Xi{=i),*and which is also closer to

*x,*than

*x*the expected utility is

_{m},*-u(jx! ~*

^{A}'/|)- Ac(|

*=i(x!) ~ Xl)*+

*b*since/' will win for sure. From the same reason as above,

*-u^-x^-Xc^ix')-x!) + b>-a(xj(=j)-*is further away from

^{x}i)■ l^ Xi{=t)^{[1]}, than

*x„„*-

_{M}(|^,(- )-x

_{1}|)-Ac(|r,--

*Xt (=i*)|) <

*~*Ac(|--,

^{U}{X!-^{X}i)-*(xi)~ X*!)

^{If}

*Xt(=i*) is closer to л, than

*x*-n(|^/- x,-1) - Ac(|z,-(

_{m},*Xj)~ Xi*) is °niy slightly lower than but almost the same as -м(|

^{[1]}/-|)-Ae(|r,-

*- Xi{=i))-*Thus, (1 / 2)(-w(|^, (r, ) -А-,|)-Яс(|=

_{/}-^(^)|) + 6-п(|^(гу)-л-,|) is lower than -m(|^-x,|)- - Ае(|г,-

*(x!)~*A/|)

*and as such,*

^{+}b,*i*deviates in this way.

Third, suppose that *Xi*(-,■) * ^{[1]}^{x}m* and

*Xj(=j) =*thus,

^{x}m-*i*loses. The expected utility of

*i*is -и(|х

_{т}~

^{л}'/|). If

*i*deviates to commit a policy

*x*the expected utility is

_{m},*-u(jx*- x,|) + (1 / 2)|-Ac(|г,- (л,„) -

_{m}*x*

_{m}|) +

*b*j since they tie. Because

*b*> Ac(|r,(x

_{m})-/ deviates in this way. Thus, when at least one candidate does not choose, here is no equilibrium in the case of

*b>*Ae(|r,- (x,„)- x,„|).

2.A.4 Lemma 2.5

First, suppose that *IS*s policy is more extreme than his/her ideal policy, *XL {=L* ) < * ^{X}L-* That is, Xl{=l)<

*+ (*

^{x}L <^{x}m <^{X}R <^{x}m^{[1]}m ~ Xl{=L ))• There are five possible positions of Xr{=rY (1) Xr(=r)^ Xl{=lY (2)

*x,„*+ (x,„ -Xl{

^{:}l))~Xr(

^{:}r); (3) Xl(=l)< Xr(=r)<

^{x}r^ (

^{4})

^{x}r < Xr(=r)<

*))l and (5) xr(=r) =*

^{x}m + {^{x}m ~Xl(=l^{x}r• In case (5),

*R*wins with certainty. Here,

*L*has an incentive to deviate to choose

*:*= Xl (=l ) and so has a 50% chance of winning with the maximized expected utility from winning.

_{L}= x_{L}For the same reasons, if *x _{R}* < ;^

_{л}(;:

_{л}), there is no equilibrium.

Next, suppose that *V*s policy encroaches on *R's* side of the policy space, *x _{m} *< Xl{

^{:}l)- Here, there are three possible positions Of Xl(=lY- (A)

*X*-{xl{

_{m}^{:}l)~

^{X}m )<^{X}L <^{x}m <

^{X}R*<*Xl{=L )l (B)

*X*—(xl (

_{m}*) —*

^{z}L

^{x}m )^{= X}L <^{x}m <

^{X}R^{=}Xl{

^{=}l )> and (C)

*X*(Xl(=l)~ xr- In each case, there are five possible positions of Xr(=rY (!) Xr(=r) <

_{L}*)*

^{x}m^{[7]}#»

*(*

^{x}n, ~{Xl(=l)~^{x}m},**2**) Xl(=l)< Xr(=rY (

**3**) Xr{

^{:}r) =

*(*

^{х}т~{Х1-{^{:}1.)-^{x}mp**4**)

Xl(=l) = Xr(=rY and (**5**) *x _{m} - {xl (=l)~ ^{x}m* )< Xr{=r)< Xl {=l)-

- • Suppose (A):
- • In cases (1) and (2),
*L*wins with certainty. Here,*L*has an incentive to deviate to choose =l =^{x}l = Xl(^{=}l) and still definitely wins with the maximized expected utility from winning. - • In cases (3) and (4), they tie. Both candidates have an incentive to deviate to choose r, = лу =
*Xi*(-/) and definitely win with the maximized expected utility from winning. - • In case (5),
*R*wins with certainty. If Xr{^{:}r)^{[7]}^{Л}'л- ^ l^{ias}an

incentive to deviate to choose =x_{r} = and still definitely wins with the maximized expected utility from winning. If Xr{=r) ^{= X}R’ L has an incentive to deviate to choose *z _{L} = xi = Xl{=l)* and has a 50% chance of winning with the maximized expected utility from winning.

- • Suppose (B):
- • In cases (1) and (2),
*L*wins with certainty. Here,*L*has an incentive to deviate to choose*=*_{L}= x_{L}= Xl(^{:}l) and still definitely wins with the maximized expected utility from winning. - • In cases (3) and (4), they tie. Here,
*L*has an incentive to deviate to choose*zi = xi*= Xl (-/.) and has a 50% chance of winning with the maximized expected utility from winning. - • In case (5),
*R*wins with certainty. Here,*R*has an incentive to deviate to choose Xr{^{:}'r) such that Xr{^{:}'r)^{[7]}^{s}closer to*x*than Xr (=r) but is still closer to x_{R}_{m}than Xl (=l)• In cases (3) and (4), they tie. Here, if*L*deviates to choose

*x _{m}*

^{A}'m)> the expected utility increases.

• In case (5). *R* wins with certainty. Here, *R* has an incentive to deviate to choose Xr{^{:}'r) such that Xr{^{:}'r) is closer to *x _{R}* than Xr{=r) but is still closer to

*x*than Xl (-/.)•

_{m}For the same reasons, if ^(cr) < *x _{m},* there is no equilibrium.

**2.A.5 Proposition 2.6**

Sufficient condition

If the pair of platforms satisfies condition (2.2) and is symmetric, it is in equilibrium. If no one deviates, the payoff for candidate *i* is

(1 / ^{2})[-M (I*; (=/) ■- *Xf* |) ■- Ac (|r, - *Xi (=i* )|)'+ * ^{b} ~* «

*iXj {=j) ~*■

If candidate *i* deviates to any policy that diverges from *x _{m}*, he/she is certain to lose, and the payoff becomes -м(|^у(-у)-av|). The change in payoff from this deviation is

- (1/2)
*—u*(|*Xi*(=,-)'- */|) -(Ь -^{Лс}*Xi*(-/)|) +(|^{ft}-ч*Xj**/1)_*■*^{From} - (2.2) , it is zero, and therefore, there is no profitable deviation that diverges from
*x*_{m}.

If the candidate deviates to a more moderate platform, say he/she is certain to win. Suppose that the candidate deviates from to After this deviation, the payoff becomes-г/^, (-/) - л,| j - - *Xi* (-/)|) + ^{b}-

The change in the payoff from this deviation is—*/ (|(^,-)—лу|)—Ac (| — *Xi* (=/ )|) ■+ *b* - (1/2)[-w (| *Xi* (-/)-*Xi* |) ■- Ac (| *=i**-Xi(=i))+b-ui^Xj(=j)-**Xi* |)] ■ Since *-u(xi* (-,) - *Xj*|) - Acflr, - *Xi* (-/ *)) + ^{b} = ~"(Xj {=j)~* *.]). from

- (2.2) , this can be rewritten as
*-u xty-Ac ^z'*_{{}- ft - 6 -[-«(|
*Xi (*^{:}i)~^{A}V |) “ Ac (I"/ -*Xi*(a )|)+/>]. From Corollary 2.3, - -и(Ы=/)-*/|) < -»(|^(=/)-^|)andAc(|r/-^(=/)|) > Acflr, -
*Xi{=i)*Thus, the change in the payoff from this deviation is negative. Therefore, there is no profitable deviation approaching*x*As a result, the two platforms satisfy (2.2) and are symmetric. Therefore, this is a state of equilibrium._{m}.

**Necessary condition**

To show the necessary condition, I use a contradiction; that is, if this pair does not satisfy equation (2.2) or is not symmetric, it is not in equilibrium.

First, if the pair of platforms is asymmetric, one candidate loses and the other wins. The winning candidate prefers another platform that has a higher utility, that is, one that approaches his/her own ideal point, a-,-, but still wins. Thus, the asymmetric position is not in equilibrium. In what follows, 1 assume that the candidates’ platform positions (and policies they would implement) are symmetric.

Second, if equation (2.2) is not satisfied with a pure strategy, it is not in

equilibrium. If *-u[x,* (с,-)- л,-|) - Ac(jz,- *- XiЫ*)|) + *b <* -n(*Xj(=j*)~ */|) and there is a tie, the candidate has an incentive to deviate to lose. Then, he/she can choose any platform that is worse for the median voter and lose. Before this deviation, the expected utility is

(1/2) -«d*,■(=,-)—А-,-|)-Аф,--*,-(г,■)!)+/>—*u(xj(=j)-*л-,|) • After the deviation, it is — w АГу (- у)^{—} |) • Thus, this candidate can increase

his/her utility by (1/2)[ф (=,)-*!+*- _{Xi}Щ-Ъ-и1_{Х}}*

-a,I) from this deviation. Since x,|)-Ac(|z,-*-XiW))+b*

< *Xj( ^{:}j)~^{x}if)* >

^{ап}У candidate will deviate.

If -“(I *Xi* (-/) - */|) - Щ *=i - Xi* (-/ )|) + *> -и(| *Xj* (-./)- *,-|) and there

is a tie, then the candidate has an incentive to deviate to be certain of winning. The candidate can move slightly to a platform that is better for the median voter and be certain to win. Assume that the deviation to approach *x _{m}* is minor. Before this deviation, the utility is

(1/2) -w (| (-/) - лу |) - Ac (| (r,- )|)+- «(| ^ / (-y) - Ay I)] - After the

deviation, it is slightly lower than *-u(Xi* (z,) - л,|) - Acflz,- - *Xi* (-, )|) + *b- *This candidate can increase his/her utility by slightly less than 0 / ^{2})[«(|*Xj {=j*)~*/|) - *»(|Xi* (=,■)“ */|) - *he* (|-■ - *Xi (ч*)|)+* from this

deviation. Since-ф,.(-.)- *.| j- Ac(|-■ - * _{Xi}(z,*)|) +

*b*

*> -u(xj(=j*)-*/|) and the policy space is continuous, there exists a platform that can

increase the candidate’s utility, and hence, either candidate has an incentive to deviate.

Finally, suppose that a candidate chooses a mixed strategy. Denote *Zj* as the platform under which the utilities, should either candidate win, are the same. That is, *-uy^Xi* (z,)- x,|j - Ac(|z,- - *Xi* *(=i* )|) + *b = -u^Xj(=j)-XjU.* If this mixed strategy is discrete, a candidate whose mixed strategy includes a more extreme platform than z,- has an incentive to deviate slightly to approach the median policy, because the probability of winning increases discretely with only a slight increase in the cost of betrayal and the disutility. If all strategies in a discrete mixed strategy are more moderate than *: _{h}* a candidate deviates to lose. If a mixed strategy is distributed on a continuous policy space, the probability of winning is zero when a candidate announces the most extreme platform in his/her mixed strategy, given that the two candidates’ positions are symmetric. Then, a candidate never chooses such a platform. As a result, equation (2.2) is the necessary condition.

**Existence and uniqueness**

As I have shown, for both candidates, the policies to be implemented must be symmetric in equilibrium. Here, 1 show that such a unique, symmetric equilibrium exists. To prove this, 1 consider the simultaneous and symmetric move of both candidates’ policies. From condition (2.2),

When *Zj = Xj* for both candidates, r, = л, = *Xi-* Therefore, the left-hand side of (2.9) is u(xr -*xi) + b.* When *Xi ^{= x}m* for both candidates, the left-hand side is

*b.*The value of the left-hand side continuously and strictly decreases to

*b*as

*Xi(*

^{:}i)^{an}d

*Xj[*approach

^{=}j)*x,„.*When

*Zj = Xj = Xi,*the cost of betrayal is zero. The right-hand side is positive, continuous, and increasing as

*Xi{*approaches

^{:}i)*x*

_{m}, from Corollary 2.3. There exists a point at which the value of the left-hand side is the same as the cost of betrayal, because I assume

*b <*Яс(|г, (л„,)-л„,|). The left-hand side strictly decreases, and the cost of betrayal increases as

*Xi*(r, ) approaches

*x*Hence, this point is unique.

_{m}.**2.A.6 Corollary 2.7**

When a candidate chooses = *x _{t},* the expected utility is

*-u(xR-xi)/2 + b/2,*because и(х,-лу) = 0. When the candidate commits to a policy that is slightly closer to the median policy than his/her ideal policy, and wins, the expected utility is slightly lower than -Ac(|z,- (a,-) - x,-|) +

*b.*As a result, if (1 /2)(и(лд

*-xi)-b)*< Acf |r,•(*,■)-the candidate has no incentive to deviate to be certain of winning when he/she chooses

*z, =*x, =

*Xi{*Otherwise, candidates have an incentive to commit to implementing a policy that is more moderate than лу. Therefore, they choose

^{:}i)-*{zi,*гд}, which satisfies (2.2).

50 *Electoral promises as a commitment device ***2.A.7 Proposition 2.8**

Fix and denote it as *Xi-* Denote *:,(Xi)* as the platform that commits to *Xh* that is, *Xi* = *Xi{=t(Xi))-* Differentiate Ac(|z,(*,)-£/|) by A, yields

Differentiate equation (2.1) by A, yields Thus,

Moreover, A = m^{,}(|^,-x,|)/c^{,}(|z,(^,)-^_{/}|) in equilibrium, from Lemma 2.1. Substitute these into (2.10). Then, (2.10) becomes

which is negative, from Assumption 1.

From condition (2.1), Ac(|z,-^(z,)|)=[c(|z,-x(w)|)

м'(|^(.-,)-А_{7}|)/с^{/}(|г/-^ (r,**■)!)]• **If Я goes to infinity, |z, - *x,i=i)* converges to 0 from Lemma 2.1, and thus, c(|z,--Я/(-/)|)/-Я/(-/)|) decreases to zero from Assumption 1. From Lemma 2.5, |я;(-,)-А/| does not exceed |x_{m}-A_{(}| in equilibrium, and as such, |я,(-/)-a,-| goes to a certain positive value (|я, (с,)-a,| g(0,°°)) when A goes to infinity. Therefore, a,|) goes to a certain positive value

when *u"^Xi{-i*)~*/|)> 0, which is a constant positive value when * ^{u}"lXi(^{=}i)~^{x}ij* = 0- As a result, the cost of betrayal, Ac(.), approaches zero as A goes to infinity, and then,

*p*> Ac(|z,(a„,)-a„,|) = 0 since

*b>* 0. Hence, both candidates choose *Xi{ ^{=}i) ^{=} Xj(^{z}j) ^{= x}m* from Lemma 2.4.

**2.A.8 Proposition 2.9 Sufficient condition and existence**

If the pair of platforms satisfies condition (2.4),

*Xi* (-/)^{= x}*m ~[,Xr*(-Л)^{—x}*m*) + ^{6} **if ***i — L,* and *x _{m}* + (л/

_{И}— Xl{

^{=}l))~

*if*

^{6}

**i***= R,*then this pair is in equilibrium. Candidate^' has

*yfj(zj, z A + b*< 0, which means that

*j*does not have an incentive to deviate by approaching the median policy and winning against (or tying with)

*i.*From any other possible deviation,

*j*has the same expected utility, because he/she still loses. Thus, there is no profitable deviation for

*j.*Candidate / has

*zj^ + b>*0, which means that

*i*does not have

an incentive to deviate and lose. Moreover, / cannot find a policy that is more extreme than *Xi( ^{:}i)* but that still wins against

*j,*because no such policy exists when

*Xi{*

^{:}i)^{= x}m ~{Xr(^{:}r)~

*if*

^{x}m)^{+ 6}*i*

*= L,*and

*x*+(л,„ -Xl(

_{m}^{=}l))~

*if*

^{6}*'*

^{=}*R*Thus, there is no profitable deviation for /'. As a result, this is a state of equilibrium. Such an equilibrium exists since

*d*, <

*dj*and

*dj >*0.

**Necessary condition**

*To show the necessary condition, 1 use a contradiction. First, there is no equilibrium in which the winner commits to implementing a policy that is closer to x_{m} than the opponent’s policy by more than e. That is, Xl{=l)^ x_{m} ~{Xr{=r)~x_{m}) + 2*)) -

*2e■*The winner has an incentive to deviate by choosing

Xl{=l) = X„i -[XR(* ^{:}R)-^{x}m) ^{+ e}* or

*Хя(*

^{:}я)^{= х}т +[^{x}m ~ Xl{=l))~^{6}, that is, committing to a policy that is closer to his/her ideal policy and still wins. In what follows, I exclude such cases.

Second, because *d, < dj,* there is no possibility of satisfying both *y/Jzj, zj)+b =* 0 and *цгj(zj, z^+b =* 0 at the same time with (;,•) — л'_{ш}| = *Xj(=j)~ ^{x}m*|- Therefore, no symmetric equilibrium exists in which both candidates have the same probability of winning, according to Proposition 2.6.

Third, suppose that *Xi(=i) = ^{x}m ~{xr{=r)~^{x}m) +?* if

*i = L,*and

*x*(л„, - Xl (=l ))

_{m}+^{_ e}if

*i = R.*Because

*dj < dj,*and

*c*is very small, it is not possible to satisfy both yr, (r,-,

*=j) + b <*0 and y//(ry,

*z,^+b>*0 at the same time, or to satisfy both i/r,■(:,■,

*=j)+b*< 0 and (///(- ■,

*:^ + b*> 0 at the same time. If i/r, (r,-,

*=j) + b*< 0 and t-,) + (>< 0, this is not a state of equilibrium because / wants to deviate to lose. Furthermore, if !//, (-,•,

*=j) + b*> 0 and ) + > 0, this is not in equilibrium

because *j* wants to win with certainty (or with a 50% chance) by approaching the median policy.

Asaresult,condition(2.4),alongwith^, (г,) = л_{/)(} - (с_{л}) - a-,„) + e if *i = L,* and *x„, +(x„, - Xl* (=/.))-f if / = *R* is the necessary condition.

**2.А.9 Corollary 2.10**

Suppose that the two candidates are originally symmetric (i.e., they have symmetric cost and disutility functions, and their ideal policies are equidistant from the median policy), and they announce symmetric platforms. Thus, they will implement Xl ^{anc}* Xr-> which are also symmetric. Moreover, both candidates initially have V,- *=j)+b =* 0.

Then, consider that *R* becomes more extreme than *L.* (i.e., *x _{R}* increases). If

*y/*decreases, then from Proposition 2.9, a more moderate

_{R}(=_{R}, z_{L})*L*wins against a more extreme

*R*, with certainty.

Denote sr(Xr) ^{=} Xr* ^{1}*(Xr)i which is the platform committing a candidate to Xr- Fix Xl

^{a}°d Xr>

^{a}nd assume that Xl

^{ar}>d Xr

^{are }symmetric. Differentiate

*u{x*

_{R}~Хь)-фя ~Xr)-^{xr ~=r{Xr))by *x _{R},* which gives

*u'(x*

_{R}-Xl)-W(x_{r}~Xr) + ^'{Хя ~=r{Xr))

*(d=*

_{r}{xr)IЭл:r). Now, differentiate equation (2.1) by

*x*Then,

_{R}.

Moreover, A = *u'(x _{R}* - Xr)/cxr ~

^{:}r(£«)) >

^{n}equilibrium, from Lemma 2.1. After substituting these into the above equation, we have

If (2.11) is negative, V_{r}(=r, becomes lower than r/^when

*R* becomes more extreme, which means that *R* will lose. Equation (2.11) is negative if

*Electoral promises as a commitment device* 53 Since this was originally *w (-- - )+Ь = 0* and Я = *u'(x _{R}* - Xr)

Ic'{xr-=r(Xr)

From Assumption 1, c'[xr-=r{Xr))Ic"{Xr-=r{Xr))>c{Xr-

that is, the right-hand side of (2.12) is higher than the right-hand side of (2.13). If

(2.12) holds. This equation can be changed to

If ^{X}R ~ Xl ^{=x}r~XR' both sides are the same. *If x*_{r}~Xl increases, the left-hand side decreases or does not change. The reason is as follows. Differentiate the left-hand side with respect to *x _{R}* - Xl, which gives

*u"{x*This value is non-positive because

_{R}~Xl)I^{u}"{^{x}R ~ Xr)~^{u}'{^{x}R ~Xl)I^{u}'{^{x}R ~ Xr)-*u'(x*- Xr ) / u"{

_{R}^{x}r ~ Xr)-

^{u}'{

^{x}r ~ Xl ) / u"(

^{x}r - Xl ) when

*x*~Xl>

_{R}^{x}r ~ Xr from Assumption 2. As a result, the left-hand side of (2.12) is lower than or the same as the left- hand side of (2.13). Therefore, (2.12) holds, and the candidate with a lower |Ay

*-x*in this case) wins.

_{m}(LConsider | vy -a„,| < |лу -a,„|. Then, candidate / wins with certainty and, for candidate /, *Wi ^{z}i> ^{=}j) ^{+} b* is not negative. That

is, *-u(xi* (=,-)- *,-|) - V(|-- - *Xi i=i*)|) + *b* > *-u(xj **~* where

*Xj* satisfies * ^{x}m~ Xi(=i) = ^{x}m~Xj-* Since к - *

_{m}|

*Xi(=i)ty + b >*-м(|А/(-/)-

*Xj*|j. The left-hand side is the (expected) utility for candidate

*i*(utility from winning) and the right-hand side is the (expected) utility for candidate

*j*(utility from losing).

When the candidates have a linear utility function, *u'( ^{x}R~ Xl) = u'(*

^{x}r~ Xr)

^{a}nd д=д(х

_{х})/дхл = 0, the change in both sides of the first-order condition is zero as

*x*changes. Thus, regardless of the position of the candidates, they still tie if they have an incentive to approach

_{R}*x*more than лу. If both candidates (or either) do not have such an incentive and choose r,- =

_{m}*x, = Xi(*

^{=}i^{a}more moderate candidate is certain to win.

54 *Electoral promises as a commitment device *2.A.10 Corollary 2.11

Suppose that the two candidates are originally symmetric (i.e., they have symmetric cost and disutility functions, and their ideal policies are equidistant from the median policy), and they announce symmetric platforms. Thus, they will implement Xl and *X&* which are also symmetric. Moreover, originally, both candidates have i//_{(} *( _{=h} -j ) + h =* 0- Then, consider that

*R*'s relative importance of betrayal, Я, becomes lower than that of L’s (i.e.,

*X*decreases). If

_{R}*y/*decreases, from Proposition 2.9, L, which has a higher A/_, always wins against

_{R}(:_{R}, :_{R})*R,*which has a lower

*X*

_{R}.Now, fix Xl and *xr*, and assume that Xl and Xr are symmetric. Differentiate *u(x _{R}* - Xl)~

^{u}(

^{x}r ~Xr)~ Яrc(yr ~=r{Xr ))with respect to

*X*which yields -c(xr ~ =r{Xr)) +^rc'[Xr ~ =r{Xr))[^=r{Xr) ^^-r)- Differentiate equation (2.1) with respect to

_{R},*X*Then,

_{R}.

Moreover, *X _{R} = u'(x_{R} - *Xr)

^{/c}'{Xr

*~*=r{Xr))

^{in}equilibrium, from Lemma 2.1. Substituting these values in (he above equation, it becomes

*-c[x*- =

_{R}_{R}{Xr)) + c'[Xr

*~*=r(Xr))' /c"{Xr ~=r{Xr)), which is positive, from Assumption 1. As a result, the candidate with a higher А,- (

*L*, in this case) always wins.

For candidate *i,* the utility when he/she wins is higher than or equal to the utility when the opponent wins; that is, *~ ^{u}(Xt* (-г)

^{-}*«j)-

*^t*where

^{c}(^{=}i - Xi{=i1 + b > - u(xj-x,y*%j*satisfies |

^{Л}Ш

^{—}

*Xi*(

*~i*)|

^{=}|

^{Л}Ш

^{—}

*Xj-*Since |лг,- — A'

_{w}| =

*x,*—

*x*, —

_{m}^{м}(|я/

^{— л}/|)

^{=}

- -w
*(|Xt (=i*)“Therefore,^{x}j)-*-u(xi{=,*)-*,|j - V(|=/ -*X,*(-/ )|) +*b >*-м(|^, (с,)-л_{;}|). The left-hand side is the (expected) utility of/' (from winning), and the right-hand side is the (expected) utility of*j*(from losing). - 2.A.11
**Corollary 2.12**

Suppose that two candidates are originally symmetric (i.e., they have symmetric cost and disutility functions, and their ideal policies are equidistant from the median policy), and they announce symmetric platforms. Thus, they will implement Xl and Xr, which are also symmetric. Moreover, both candidates initially have *:j)+b =* 0. Then, consider that *R* becomes more policy motivated than *L* (i.e., *p _{R}*

increases). If *4**/ _{R}(:_{R}, :_{L})* decreases, then from Proposition 2.9, the less policy-motivated

*L*always wins against a more policy-motivated

*R.*

Now, fix Xl ^{an}d Xr and assume that Xl and Xr are symmetric. Differentiate *P _{R}u{x_{R}-Xl)~ Pr“(xr~ Xr)-*c(xr-:r{Xr))* with respect to

*p*which yields

_{R},*u(x*«(

_{R}- Xl)~^{а}л ~ Xr)~ ^'{Xr ~=r{Xr))• {P=r{Xr) ^Pr) Differentiate equation (2.1), A =

*p*- Xr)/ ^{XR-^RiXRl), with respect to

_{R}n'(x_{R}*fi*We get

_{R}.

Moreover, *X = p _{R}u’(x_{R}- Xr)I c'(xr-=r{Xr))*

^{in}equilibrium, from Lemma 2.1. On substituting these into the above equation, we have

Since this was originally *y/j(z _{h} z*

*j)+b*

*=*0 and Я = Pru'(x

_{r}- Xr) /

^{c}’[Xr~-r{Xr) (213) is satisfied in the proof of Corollary 2.10. From Assumption 1, c'[xr ~ =r{Xr))/c"{xr ~=r{Xr))

> - =r {Xr))/ c’{Xr ~ =r {Xr))> and thus (2.14) is negative. Therefore, the candidate with a lower Д (L, in this case) always wins.

Consider д. < *p*.. Then, / always wins, which means that in equilibrium, I/O(r,-, *=j) + b =* 0 is not negative for /'. That is,

- -Piuixi{=i)-x,§-Xi{:i))
*+ b>-pj*и(|яу-лу|), where*Xj*satisfies |лг_{т}*- Xi (=i) =*Note that if Vo(=,,^{x}m ~ Xj-*zj) + b<*0,*i*does not have an incentive to win in equilibrium. Since Д < Д,, - -Pi
*“(Xj-Xi)>~Pj u(Xi{=i)~Xj),*and hence, -Ди(|я,(=,)-*,]]-*-Xc^zf- Xi{=i)) + b>~Pj*м(|я/(-|)^{_А}7|)- The left-hand side is the - (expected) utility of
*i*(from winning), and the right-hand side is the (expected) utility of*j*(from losing).

- [1] In cases (1) and (2), L wins with certainty or has a 50% probability of winning. Both candidates have an incentive to deviate tochoose Zj = Xj = Xi{:i) and so definitely win with the maximizedexpected utility from winning (i.e., b). • In cases (3) and (4), R wins with certainty. Here, R has an incentiveto deviate to choose zr = xr = Xr (-/?) but still definitely wins withthe maximized expected utility from winning.
- [2] In cases (1) and (2), L wins with certainty or has a 50% probability of winning. Both candidates have an incentive to deviate tochoose Zj = Xj = Xi{:i) and so definitely win with the maximizedexpected utility from winning (i.e., b). • In cases (3) and (4), R wins with certainty. Here, R has an incentiveto deviate to choose zr = xr = Xr (-/?) but still definitely wins withthe maximized expected utility from winning.
- [3] In cases (1) and (2), L wins with certainty or has a 50% probability of winning. Both candidates have an incentive to deviate tochoose Zj = Xj = Xi{:i) and so definitely win with the maximizedexpected utility from winning (i.e., b). • In cases (3) and (4), R wins with certainty. Here, R has an incentiveto deviate to choose zr = xr = Xr (-/?) but still definitely wins withthe maximized expected utility from winning.
- [4] In cases (1) and (2), L wins with certainty or has a 50% probability of winning. Both candidates have an incentive to deviate tochoose Zj = Xj = Xi{:i) and so definitely win with the maximizedexpected utility from winning (i.e., b). • In cases (3) and (4), R wins with certainty. Here, R has an incentiveto deviate to choose zr = xr = Xr (-/?) but still definitely wins withthe maximized expected utility from winning.
- [5] In cases (1) and (2), L wins with certainty or has a 50% probability of winning. Both candidates have an incentive to deviate tochoose Zj = Xj = Xi{:i) and so definitely win with the maximizedexpected utility from winning (i.e., b). • In cases (3) and (4), R wins with certainty. Here, R has an incentiveto deviate to choose zr = xr = Xr (-/?) but still definitely wins withthe maximized expected utility from winning.
- [6] In cases (1) and (2), L wins with certainty or has a 50% probability of winning. Both candidates have an incentive to deviate tochoose Zj = Xj = Xi{:i) and so definitely win with the maximizedexpected utility from winning (i.e., b). • In cases (3) and (4), R wins with certainty. Here, R has an incentiveto deviate to choose zr = xr = Xr (-/?) but still definitely wins withthe maximized expected utility from winning.
- [7] Suppose (C): • In cases (1) and (2), L wins with certainty. Here, L has an incentive to deviate to choose Xl{=l) such that Xl{='l) is closer toxi than Xl{=l) but is still closer to xm than Xr{=r)-
- [8] Suppose (C): • In cases (1) and (2), L wins with certainty. Here, L has an incentive to deviate to choose Xl{=l) such that Xl{='l) is closer toxi than Xl{=l) but is still closer to xm than Xr{=r)-
- [9] Suppose (C): • In cases (1) and (2), L wins with certainty. Here, L has an incentive to deviate to choose Xl{=l) such that Xl{='l) is closer toxi than Xl{=l) but is still closer to xm than Xr{=r)-