# Asymmetric costs of betrayal

Assume that A is not the same for both candidates, and Candidate / has A,. However, their ideal policies and disutility functions are symmetric.

**Corollary 2.11**

*Suppose Assumption 1, and that two candidates run. Suppose also that candidate i has a higher relative importance of betrayal (i. e.*, A, > Ay A *but that the candidates are symmetric in all other respects. Then, in equilibrium, candidate i wins with certainty. The expected utility from winning is higher than, or the same as, the expected utility from losing.*

**Proof: **See Appendix 2.A. 10.

When a candidate has a lower A,, he/she will betray the platform more severely, and hence, the realized cost of betrayal is higher, as shown in Proposition 2.8. Therefore, such a candidate has a lower degree of incentive to win because he/she wishes to avoid paying the high cost of betrayal. As a result, the candidate with the higher A, wins.

# Asymmetric political motivations

Suppose that the level of political motivation, */5,* differs from 1, and Candidate *i* has Д. Furthermore, assume that Д is not the same for both candidates. However, their ideal policies and cost functions are symmetric. That is, the utility following a win is

)- *i|) - Ac(|z,- - *Xi*(-/)|) + *b* and the utility when the opponent wins is -Дм(|я/(-у)^{_ A}'/|)- Thus, the degree of incentive to win is

%(--/, * ^{z}j)~ ~Pi^{u}* (| А/ (-/) - */1) - Ac

*(=i-Xi(zi))+Piu(Xj(=j)-Xi) ■*

**Corollary 2.12**

*Suppose Assumption 1, and that two candidates run. Suppose also that candidate i is less policy motivated (i.e.,* Д < /3, *), but that the candidates are symmetric in all other respects. Then, in equilibrium, candidate i wins with certainty. The expected utility from winning is higher than the expected utility from losing.*

**Proof: **See Appendix 2.A.11.

A less policy-motivated candidate is less concerned about policy and does not betray the platform so severely, and hence has a lower cost of betrayal and a higher degree of incentive to win. By contrast, a more policy-motivated candidate will betray the platform more severely, which induces a higher cost of betrayal. As a result, a less policy-motivated candidate wins the election.

# Functional example

This subsection shows a functional example as an overview of the implications described so far. Suppose a linear disutility function, Дм(|^-лу|) = *pjx-Xj,* and a quadratic cost function,

V(|=/ - *Xi* (-< )|) = A- (-/ - *Xi i=i* ))^{2} •

From (2.1), the policies to be implemented are

assuming -l~ Pl / (2 *X*_{R}) > *xj_* and *: _{R} + p_{R} I(2X_{R})< x_{R}* (Corollary 2.2). The cost of betrayal is

*fir*/ (4Я,), which decreases with Я,- (Proposition 2.8).

Then, the degrees of incentive to win are

First, when *b* is sufficiently high that *b > pp* / (4Я,) for both *i = L* and *R, *both candidates have an incentive to win, even if Xr{^{:}r) ^{=} Xl (-/.) = * ^{x}m- *Thus, both candidates commit to implementing the median policy, and they tie (Lemma 2.4).

Now, suppose that at least one candidate has *b < pf t (*4A,), and the ideal policies of *L* and *R* are symmetric. For a symmetric pair of Xl(^{:}l) ^{an}d /«(-/?)’ *'* i^{s} indifferent between winning and losing if i*j/j*(r*zj) **+ b =* 0, that is,

which is *2dj* according to (2.3) when Д/(4A, )-/> / Д > 0. If *max{0,p _{L} I(4X_{L})~bI p_{L}] < P_{R} I(4X_{R})~bI p_{R} [d_{L} *

or

Note that as *b < p%* /(*4X _{R}), p_{R} l(4X_{R})-bI p_{R}* > 0. In equilibrium,

*L*commits to implementing Xl (-/.), which satisfies the above condition. Then,

*R*does not have an incentive to commit to Xr{

^{:}r)

^{suc}h that Xr{=r)~

*Xl{*

^{x}m =^{x}m~^{=}l)’

^{anc}* thus,

*L*wins with certainty. More precisely, in equilibrium,

*L*commits to implementing Xl{=l which satisfies

and *R* commits to implementing *х _{ш}* + (а

_{ш}- Xl{

^{=}l))

*+*(Proposition 2.9). In this case,

^{e}*L*announces ~

_{L}], such that

The value of Д *l(4Xj)-bl* Д decreases with A, and increases with *p,.* Thus, if *Xi* is higher than *X _{R}* (with

*p*wins (Corollary 2.11). If

_{R}= p_{R}), L*p*is lower than

_{L}*p*(with

_{R}*Xi = X*

_{R}),

*L*also wins with certainty (Corollary 2.12).

Suppose that *x _{R} - x_{m} > x_{m} -* л

*i*that is,

*R*is a more extreme candidate than L, and

*X*and

_{L}= X_{R}*p*In this case, both candidates tie in equilibrium if

_{L}= p_{R}.*R*still has an incentive to win (Vr{=r, zi)

*+ b>*0) when

*L*chooses

*xi =*=l

*=*Xl{=l)- this equilibrium, candidates commit to implementing

which satisfy (2.2). By contrast, if *R* does not have an incentive to win against *L* when *L* chooses = =l = Xl{^{=}l)' that is,

then *L* chooses xr = :l *= *Xl (=/.), and Л commits to implementing a more extreme policy than *xl-* In this case, *L* wins with certainty in equilibrium (Corollary 2.10(ii)).