# Discussions

## Other equilibria

The previous sections only discuss a pooling equilibrium in which both types choose r/w and a semi-separating equilibrium in which a moderate type chooses z*. This section discusses other equilibria that may exist.

### Separating equilibrium in which a moderate type wins

If a separating equilibrium exists in which a moderate type wins against an extreme type, both types must choose different platforms, and the moderate type must implement a more moderate policy than does the extreme type. However, such a separating equilibrium does not exist.

Proposition 3.8

There is no separating equilibrium in which a moderate type wins against an extreme type, regardless of off-path beliefs.

Proof: See Appendix 3.A.5.

This result is true because an extreme type always has an incentive to pretend to be moderate.

If a separating equilibrium exists in which a moderate type wins, the moderate type must prefer her winning to the opponent winning in this equilibrium. However, this means an extreme type also prefers to win over having the opponent win at the moderate type’s platform, for the same reasoning given in Lemma 3.4. Moreover, if an extreme type deviates by pretending to be a moderate type, he will have a higher probability of winning. Thus, an extreme type has an incentive to deviate by choosing the moderate type’s platform within this separating strategy.

This result contradicts that of Banks (1990) and Callander and Wilkie (2007) who show that there exists a separating equilibrium in which a moderate type wins.

### Separating equilibrium in which an extreme type wins

A separating equilibrium in which an extreme type wins against a moderate type exists if off-path beliefs are pj{M:j) = 0. For example, suppose that an extreme type announces a platform, zf, such that he is indifferent between winning and losing. To be precise, ~j satisfies —м(|*'(£')-л-'|)-с(|_-' -X't (=/)|) =-«(|*; (='j )"*/|) where

|л„, -r/| = |л„, -ry|. Suppose also that a moderate type announces zf1, such that the moderate type will implement a more extreme policy than the extreme type. That is, xm - xf (3,£)| < |*„, - X™ (:!^')[ As a result of the above off-path beliefs, although a moderate type approaches the median policy, voters believe that this candidate is an extreme type. To increase the probability of winning, a moderate type needs to approach the median policy in a significant way. This may decrease her expected utility. An extreme type does not deviate either, because he is indifferent between winning and losing. As a result, a separating equilibrium in which an extreme type wins exists.

However, this separating equilibrium is less important, since it exists in a very restricted set of off-path beliefs. For example, suppose that the off-path beliefs are pj[Mzi) = pM if the platform is more extreme than rf4. Then a moderate type has an incentive to choose zf4, where she is indifferent between winning and losing. If a moderate type announces zf4, an extreme type does not have an incentive to win against a moderate type by committing to implement

which is more moderate than £/v/(^W), from Lemma 3.4. Therefore, this separating equilibrium does not exist with such off-path beliefs.

### Other pooling and semi-separating equilibria

In the previous sections, I assume off-path beliefs as p, (M|= 0. Under this assumption, there exist multiple pooling and semi-separating equilibria. First, there could be a pooling equilibrium in which both types announce a platform, say zf4 , that is more extreme than r/w (-l * < zf4* and :r < zr* ). Since the off-path belief is p, (Mzj) = 0, a moderate type needs to approach the median policy significantly to be sure of winning because voters believe that the candidate is extreme when he/she deviates from r/w , regardless of the real type. Thus, a moderate type may not want to deviate. If an extreme type also has no incentive to deviate, a pooling equilibrium with zf4 exists. Second, there could be a semi-separating equilibrium in which a moderate type (and a pooling extreme type) announces a platform, say r, , that is more extreme than z,. A moderate type needs to approach the median policy significantly to win against an opponent who announces zj , because this deviation leads voters to believe that the candidate is extreme. Therefore, a moderate type may have no incentive to deviate from z, .

These equilibria (including a separating equilibrium in which an extreme type wins) have several problems. First, a moderate type does not want to approach the median policy because voters will likely misunderstand the candidate to be an extreme type. Second, if the off- path beliefs, /у exceed zero for some off-path platforms, many

of the above equilibria will be eliminated.8 Thus, these equilibria exist for restricted values of off-path beliefs.

On the other hand, the equilibria analyzed in the previous sections exist in broader values of off-path beliefs than do other equilibria. As I discussed, a pooling equilibrium with zf4 exists when /у (M|c,) = pM

if the platform is more extreme than r/w*. A semi-separating equilibrium with z* exists when /y(M|r,) = pM l[pM +o;V/ (l — />л/)] if the platform is more extreme than z*.

## Differences from past papers

### Assumptions

As indicated in Chapter 1 (Section 1.3.2), there are two important differences between my model and those of Banks (1990) and Callander and Wilkie (2007): (1) candidates choose a policy to implement strategically and (2) they care about policy when they lose.

In a semi-separating equilibrium, an extreme type reveals his type by approaching the median policy with some probability since he can obtain a higher probability of winning. However, if candidates implement their own ideal policies automatically, voters only believe that an extreme type will implement his ideal policy, so a separating extreme type cannot increase his probability of winning by revealing his type. That is, a strategic choice of an implemented policy provides a way to win for an extreme type.

Suppose that a candidate does not care about policy when he loses. From (3.1), the expected utility of candidates is

Certainly, candidates will announce their ideal policy as platforms (and his/her expected utility is zero) because if not, they will incur disutility from the policy and a cost of betrayal, and the expected utility becomes negative. Thus, benefits from holding office should be introduced to induce candidates to approach the median policy. However, even with benefits from holding office, a semi-separating equilibrium does not exist when candidates do not care about policy after losing. An extreme type has a stronger incentive to prevent the opponent from winning in my model but to a lesser extent when they do not care about an opponent’s policy. Thus, caring about policy after losing provides an extreme type with an incentive to win.

### Universal divinity

Banks (1990), Callander and Wilkie (2007), and Huang (2010) employ universal divinity, as introduced by Banks and Sobel (1987).

If universal divinity is applied to my model, in short, as Lemma 3.4 shows, an extreme type always has a greater incentive to announce a more moderate platform than does a moderate type. This means that a moderate type always has a stronger incentive to announce a more extreme platform than does an extreme type. Suppose a pooling equilibrium at :1й*. With universal divinity, if a platform is more moderate than Zj1*, Pi (M|r,-) = 0. If not, p, (M|r,) = 1. With these off-path beliefs, both have an incentive to deviate to a more extreme platform than zf1 , be thought a moderate type by voters, and win. For the same reasons, a semi-separating equilibrium does not exist, and only a separating equilibrium exists in which an extreme type wins against a moderate type, which is discussed in Section 3.5.1. However, such a separating equilibrium seems peculiar, as I discussed. Moreover, in this separating equilibrium, an extreme type always wins against a moderate type, so my main result does not change.9

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