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_L. Then, it becomes

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

Differentiate Xc(z_L - Xl (=/.)) by z_L. Then,

■

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)*^xnr Xj{=j)*^xnt 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_m, the expected utility is -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[^zj) >^s further away from a,- than x_m, u[xj[~j}~^xi)

-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)~^xm = ^Xjthus, there is a tie. The expected utility of i is (112)[-u(x,(-,)-^[1],|)-^'(|-< -Xi(=i)) ^{+ h}-^u(Xj{=j)-4)- ^lf i deviates to commit to a policy, xl, which is slightly closer to x_m than Xi{=i), and which is also closer to x, than x_m, the expected utility is -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)-^xi)■ l^ Xi{=t) is further away from ^[1], than x„„ -_M(|^,(- )-x₁|)-Ac(|r,-- Xt (=i)|) < ~^U{X!-^Xi)- Ac(|--, (xi)~ X!)^If Xt(=i) is closer to л, than x_m, -n(|^/- x,-1) - Ac(|z,-(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/|) ⁺ b, and as such, i deviates in this way.

Third, suppose that Xi(-,■) ^[1]xm and Xj(=j) = ^xm- thus, i loses. The expected utility of i is -и(|х_т ~ ^л'/|). If i deviates to commit a policy x_m, the expected utility is -u(jx_m - x,|) + (1 / 2)|-Ac(|г,- (л,„) - x_m|) + bj 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 ISs policy is more extreme than his/her ideal policy, XL {=L ) < ^XL- That is, Xl{=l)< ^xL < ^xm < ^XR < ^xm + (^[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)<^xr^ (⁴) ^xr < Xr(=r)< ^xm + {^xm ~Xl(=l))l and (5) xr(=r) = ^xr• In case (5), R wins with certainty. Here, L has an incentive to deviate to choose :_L = x_L = Xl (=l ) and so has a 50% chance of winning with the maximized expected utility from winning.

For the same reasons, if x_R < ;^_л(;:_л), there is no equilibrium.

Next, suppose that Vs 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_m -{xl{^:l)~ ^Xm )<^XL < ^xm < ^XR < Xl{=L )l (B) X_m —(xl (^zL ) — ^xm ) ^{= X}L < ^xm < ^XR ⁼ Xl{⁼l )> and (C) X_L(Xl(=l)~ ^xm)^[7]#» xr- In each case, there are five possible positions of Xr(=rY (!) Xr(=r) < ^xn, ~{Xl(=l)~ ^xm}, (2) Xl(=l)< Xr(=rY (3) Xr{^:r) = ^хт~{Х1-{^:1.)-^xmp (4)

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^xm + (x_m ~Xl(=l)) -2e■ The winner has an incentive to deviate by choosing

Xl{=l) = X„i -[XR(^:R)-^xm) ^{+ e} or Хя(^:я) ^{= х}т +[^xm ~ 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)~^xm|- 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) = ^xm ~{xr{=r)~^xm) +? if i = L, and x_m + (л„, - Xl (=l ))^{_ 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/_R(=_R, z_L) decreases, then from Proposition 2.9, a more moderate L wins against a more extreme R, with certainty.

Denote sr(Xr) ⁼ Xr¹(Xr)i which is the platform committing a candidate to Xr- Fix Xl ^a°d Xr> ^and 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_R. Then,

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 ^XR ~ Xl ^=xr~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_R ~Xl)I ^u"{^xR ~ Xr)~ ^u'{^xR ~Xl)I ^u'{^xR ~ Xr)- This value is non-positive because u'(x_R - Xr ) / u"{^xr ~ Xr)- ^u'{^xr ~ Xl ) / u"(^xr - Xl ) when x_R ~Xl> ^xr ~ 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_m (L in this case) wins.

Consider | vy -a„,| < |лу -a,„|. Then, candidate / wins with certainty and, for candidate /, Wi^zi> ⁼j) ⁺ b is not negative. That

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

Xj satisfies ^xm~ Xi(=i) = ^xm~Xj- Since к - *_m| -w(|^(c,)-Ay|), and then, -г<(|*Дг,)-лу|)-А,ф,- 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'(^xR~ Xl) = u'(^xr~ Xr) ^and д=д(х_х)/дхл = 0, the change in both sides of the first-order condition is zero as x_R changes. Thus, regardless of the position of the candidates, they still tie if they have an incentive to approach x_m more than лу. If both candidates (or either) do not have such an incentive and choose r,- = 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_R decreases). If y/_R(:_R, :_R) decreases, from Proposition 2.9, L, which has a higher A/_, always wins against 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(^xr ~Xr)~ Яrc(yr ~=r{Xr ))with respect to X_R, which yields -c(xr ~ =r{Xr)) +^rc'[Xr ~ =r{Xr))[^=r{Xr) ^^-r)- Differentiate equation (2.1) with respect to X_R. Then,

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^c(⁼i - Xi{=i1 + b > - u(xj-x,y where %j satisfies |^ЛШ ^— Xi (~i)| ⁼ |^ЛШ ^— Xj- Since |лг,- — A'_w| = x, — x_m , — ^м(|я/ ^{— л}/|)⁼

• -w(|Xt (=i)“ ^xj)- Therefore, -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 ^and Xr and assume that Xl and Xr are symmetric. Differentiate P_Ru{x_R-Xl)~ Pr“(xr~ Xr)-*c(xr-:r{Xr)) with respect to p_R, which yields u(x_R - Xl)~ «(^ал ~ Xr)~ ^'{Xr ~=r{Xr))• {P=r{Xr) ^Pr) Differentiate equation (2.1), A = p_Rn'(x_R - Xr)/ ^{XR-^RiXRl), with respect to fi_R. We get

Moreover, X = p_Ru’(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 zj)+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) = ^xm ~ Xj- Note that if Vo(=,, 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/(-|)^_А7|)- 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)-