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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

%(--/, zj)~ ~Piu (| 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 XR) > xj_ and :R + pR I(2XR)< xR (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 (-/.) = xm- 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) and /«(-/?)’ ' is indifferent between winning and losing if ij/j(rzj) + b = 0, that is,

which is 2dj according to (2.3) when Д/(4A, )-/> / Д > 0. If max{0,pL I(4XL)~bI pL] < PR I(4XR)~bI pR [dL R), there exists a symmetric pair of Xl (-z. ) and Xr (-/?) such that

or

Note that as b < p% /(4XR), pR l(4XR)-bI pR > 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) such that Xr{=r)~ xm = xm~ Xl{=l)’ anc* thus, L wins with certainty. More precisely, in equilibrium, L commits to implementing Xl{=l which satisfies

and R commits to implementing хш + (аш - Xl{=l)) + e (Proposition 2.9). In this case, 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 XR (with pR = pR), L wins (Corollary 2.11). If pL is lower than pR (with Xi = XR), L also wins with certainty (Corollary 2.12).

Suppose that xR - xm > xm - лi that is, R is a more extreme candidate than L, and XL = XR and pL = pR. In this case, both candidates tie in equilibrium if 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)).

 
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