# Beliefs

Suppose that when a candidate deviates from zf1*, voters believe with a probability of one that the candidate is an extreme type. That is, I consider simple off-path beliefs as p, = 0. This simple off-path belief is partially based on the idea of the intuitive criterion in Cho and Kreps (1987). In a pooling equilibrium, a moderate type is indifferent to winning and losing at r/w*. Thus, a moderate type never chooses a more moderate platform than zj11 since her expected payoff decreases from the equilibrium expected-payoff even if voters believe this candidate to be a moderate type as a result of this deviation.4 As a result, if a platform is announced that is more moderate than , this candidate must not be a moderate type. On the other hand, an extreme type has an incentive to announce a platform more extreme than his cut-off platform, given zf* (i.e., zf(zj**jj. Lemma 3.4 shows zf^zjf*J is more moderate than zf* , so the off-path belief for a platform that lies between zf1 and =? y=j* ) should be /y(M|r,) = 0. The intuitive criterion cannot apply to any other off-path beliefs, so I simply assume that pi (Mzj) = 0 for all off-path strategies.

# The existence of the pooling equilibrium

Voters do not know' candidates’ types in a pooling equilibrium, so their expected utility is the weighted average of the utility between a moderate and an extreme type. On the other hand, if an extreme type deviates by approaching the median policy, voters believe that this candidate’s type is extreme because of the off-path belief. If an extreme type deviates to announce a sufficiently moderate platform, this extreme type can win over an uncertain opponent who chooses zf1 . Here, I denote z such that it satisfies

The left-hand side is the utility of the median voter when extreme candidate i, who deviates to z- wins. The right-hand side is the expected utility of the median voter when candidate j, who announces the pooling platform wins. That is, the median voter is indifferent between candidates who announce z- and . If an extreme candidate announces a platform that is slightly more moderate than z,-, this candidate wins over an uncertain opponent. Figure 3.2(a) shows г', in the case of R when voters have linear utility. Note that because voters are uncertain about the type of the candidate who announces zf1 , an extreme type who deviates would need to implement a more moderate policy than an extreme type who chooses a pooling platform

Figure 3.2 Pooling Equilibrium.

Let E(x) = pM Xr (-jqf')+(l-/’,W);t« [ZR*) denote the expected policy implemented

by a candidate announcing :'¥*■ Suppose that voters have a linear utility. If an extreme type’s platform is more moderate than :'r, such an extreme type wins over the opponent L announcing rrjK*.

[xf (=j**)) blit does not need to implement a more moderate policy than a moderate type (ry**))’ as swn in Figure 3.2(a).

An extreme type does not deviate by announcing a platform that is more moderate than z if

The right-hand side is the extreme candidate fs expected utility when he stays in a pooling equilibrium. His expected utility from this deviation is slightly lower than the left-hand side. If (3.5) holds, this extreme type does not deviate, so a pooling equilibrium where all types announce -V exists. Now, denote

Then, condition (3.5) can be represented by pM > p, as shown in Proposition 3.3. This p is always positive and less than one.

Corollary 3.5

pe (0,1).

Proof: See Appendix 3.A.3.

If pAi >p, a pooling equilibrium exists. The next section shows that if pM semi-separating equilibrium exists. Thus, the previous corollary implies that for all parameter values and functional forms, if pM is large enough, a pooling equilibrium always exists; otherwise, a semi-separating equilibrium exists.

The intuitive reasoning is as follows. Suppose that L chooses z‘i as a pooling equilibrium, and R is an extreme type who originally announces :r*. If pM is high, extreme type R needs to announce a very moderate platform to win with certainty, because the expected utility to the median voter of choosing L is quite high given that there is a strong possibility that L is moderate and will implement a moderate policy. Thus, as in Figure 3.2(b), there is a significant distance between :'r and R’s ideal policy, so this deviation decreases his expected utility.

As a result, a pooling equilibrium exists. However, if pxl is sufficiently low, the expected utility of the median voter who chooses L is quite low. Thus, if R slightly approaches the median policy, the policy he implements improves for the median voter. That is, z'r is closer to zr , as in Figure 3.2(c), so the extreme type will deviate from a pooling strategy.

Note that this pooling equilibrium exists in the broader value of the off-path beliefs. For example, suppose p, (M|=,-)= pM if the platform is more extreme than zf1 . Then a candidate still has no incentive to deviate to a more extreme platform than , since he/she will then be certain to lose and the expected utility decreases or is unchanged by this deviation. A pooling equilibrium can exist when the off-path belief, pi (M=i), is lower than pM for r,-, which is more extreme than zf1 .

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