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Asymmetric costs of betrayalAssume 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 motivationsSuppose 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 (=iXi(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 policymotivated 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 policymotivated candidate will betray the platform more severely, which induces a higher cost of betrayal. As a result, a less policymotivated candidate wins the election. Functional exampleThis subsection shows a functional example as an overview of the implications described so far. Suppose a linear disutility function, Дм(^лу) = pjxXj, 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 ij/j(rzj) + 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)~ ^{x}m = ^{x}m~ 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 х_{ш} + (а_{ш}  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 X_{R} (with p_{R} = p_{R}), L wins (Corollary 2.11). If p_{L} is lower than p_{R} (with Xi = X_{R}), L also wins with certainty (Corollary 2.12). Suppose that x_{R}  x_{m} > x_{m}  лi that is, R is a more extreme candidate than L, and X_{L} = X_{R} and p_{L} = p_{R}. 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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