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Pooling equilibriumDefinition and propositionIf both types of opponent announce the same platform r, (i.e., a pooling strategy), the expected utility of candidate i of type t when the opponent wins is _{p}^u[_{X}f (7)  */)  (l  />")иЫ (=,)  xj ). where i,j = L.R, /t j, ana t = M,E. Tne utility oi candidate i of type t after he/she wins is (zj j  jv'jj  cr  (r' )j. If the utility when candidate / of type t wins is strictly lower than the expected utility when his/her opponent wins [p^{M}^xf (7)  */)  (1  p^{m})"(x? (7)  */) > "^{M}( x! (/) */)  ~Xi (/))j» candidate i prefers the opponent winning to him/ herself winning. On the other hand, in the inverse case (i.e., V'« (xf (=j) 4  ('/>«)  (rf(v) v/) <«  4  c c/ ~x'i candidate / prefers to win. In a pooling equilibrium, a moderate type chooses a platform r/^{w}* such that she is indifferent between winning and losing. That is, the above two expected utilities are the same for a moderate type: where zi* and zr* are symmetric = л,„ An extreme type mimics a moderate type by choosing the same platform.^{3} Definition 3.2 In a pooling equilibrium, zf^{1}* is chosen by a candidate regardless of hist her type. Then, a pooling equilibrium exists if the prior belief that a candidate is of a moderate type, p^{M}, is sufficiently high. The parameter, p, will be defined later (Equation (3.6)). Proposition 3.3 Suppose that the offpatli beliefs of voters are p, (M z,) = 0. If p^{M} > p, the pooling equilibrium defined in Definition 3.2 exists. Proof: See Appendix З.А.1. An extreme type’s choiceIntuitively, a moderate type is indifferent between winning and losing, with c/^{W} so she does not deviate. On the other hand, this subsection shows that an extreme type prefers his winning to the opponent winning, with r/^{w} . Let / denote the cutoff platform, where the utility when candidate i wins and the expected utility when opponent j wins are the same for type t candidate /, given the opponent’s pooling strategy, z,. That is, a candidate is indifferent between winning and losing:
For example, if the opponent chooses z¥*, /^{w}* is the cutoff platform of moderate candidate / [z^ {zf*J = r/ j. If a candidate approaches the median policy, the disutility after winning and the cost of betrayal increase. Thus, if type t candidate i announces a platform that is further from his/her ideal policy (i.e., more moderate) than zj fry), then his/her utility after winning is low'er than the expected utility when the opponent wans, and vice versa. In this chapter, when I use the term “more moderate platform,” this means that “this platform is further from the candidate’s ideal policy.” In Figure 3.1, =r(=l) is further from .vjf and xr than zr (r/J. Therefore, rjf (r/,) is “more moderate” than zr (r^). Note also that “approaching the median policy” means that a candidate announces a platform such that an implemented policy given this platform approaches the median policy. Note that, first, a more moderate platform does not mean a more moderate implemented policy, given this platform. As shown in Figure 3.1, since an extreme type will betray his platform to a greater extent than will a moderate type, the extreme type’s implemented policy, Xr (cf (яг,)) is more extreme than that of the moderate type, Xr [^{:}r (£.))• N°^{te} also that a more moderate platform may not mean that this platform is closer to the median policy, because there is a possibility that a platform encroaches on the opponent’s side of Figure 3.1 Lemma 3.4. While an extreme type's implemented policy given the cutoff platform Xr(^{:}r (^{r}i)) is more extreme than a moderate type’s one Xr (^{:}r (г))> ^{an} extreme type's cutoff platform гд (zi_) is more moderate than a moderate type’s one (:l). the policy space (i.e., z'_{R} < x_{m} < z'_{L}) like as the model in Chapter 2. The justifications discussed in Subsection 2.3.3 can be also applied here. If an extreme type’s zf (ry) is always more moderate than a moderate type’s r,^{M}(r_{y}), given any _{=j} (=l (=r)<=l(=r) and =%(=_{L}) Lemma 3.4 Suppose that an opponent announces the same platform zj regardless of type. Given any p^{M}, (i) an extreme type's cutoff platform is more moderate than that of a moderate type (=l {=r) < =1 (/?) ^{an(}l R (/.)< =r(=l) but (ii) an extreme type's implemented policy, given the cutoff platform, is more extreme than that of a moderate type (xl(=l{=r)) < Xl (=L (=R)) and Xr(=r (=_{L})) > Xr (=r {=l )))• Proof: See Appendix 3.A.2. Lemma 3.4 shows the following two facts.
If a candidate approaches the median policy and wins against the opponent with certainty, this candidate will pay a cost of betrayal with certainty. This marginal cost of approaching the median policy depends on the cost of betrayal, ci^z  x [= j On the other hand, this candidate can avoid the opponent’s victory and decrease his/her expected disutility from the implemented policy. This marginal benefit depends on the difference in the (expected) disutilities when the candidate wins and when the opponent wins, Following an election, an extreme type will betray the platform more severely and pay a higher cost of betrayal. However, at the same time, the ideal policy for an extreme type is further from the median policy than that of a moderate type, which means that his ideal policy is also further from the opponent’s implemented policy. Thus, an extreme type has a higher disutility from the opponent’s victory. Therefore, he finds it especially costly if the opponent wins, more so than a moderate type does. As a result, an extreme type has higher marginal benefit and cost than a moderate type. Since an extreme type has a higher marginal cost of betrayal, he does not have an incentive to choose a more moderate implemented policy than a moderate type (Lemma 3.4(ii)). However, an extreme type has a higher marginal benefit, so he has an incentive to announce a more moderate platform than a moderate type (Lemma 3.4(i)). Therefore, when both candidates announce a pooling strategy, sf* , an extreme type prefers his winning to the opponent winning. Thus, an extreme type does not have an incentive to lose. However, there is a possibility that an extreme type has an incentive to further approach the median policy and win with certainty, since he prefers his winning to the opponent winning at zf^{4} . To check this incentive of an extreme type, the offpath belief of voters must be specified. 
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