# Semi-separating equilibrium

If pM < p, a pooling equilibrium does not exist since an extreme type will deviate from r/w* to z. In this case, a moderate type still announces one platform with certainty (a pure strategy). On the other hand, an extreme type chooses a mixed strategy: With some probability, an extreme type announces the same platform as a moderate type. With the remaining probability, an extreme type approaches the median policy. That is, if pM < p, a semi-separating equilibrium exists. I call an extreme type a “pooling extreme type” when he imitates the moderate type, but a “separating extreme type” when he approaches the median policy.

## Definition and proposition

In a semi-separating equilibrium, a moderate type announces and an extreme type announces r,- with probability oM e (0,1), and 3} with the remaining probability, 1-сгл/. I call this type of a semi-separating equilibrium a two-policy semi-separating equilibrium. This equilibrium is shown in Figure 3.3(a) and is defined as follows.

Definition 3.6

In a two-policy semi-separating equilibrium, a moderate type chooses Zj, and an extreme type chooses zt with probability aM g (0,1), and Zj with probability 1 -<7 V/.

Figure 3.3 Semi-Separating Equilibrium.

Now, denote (and zj) such that it satisfies

where z* and :* are symmetric ||л„,-г,*| = |лшThe left-hand side

is the expected utility of moderate candidate i when the opponent promising r, wins. The right-hand side is the utility of moderate candidate i when she wins. That is, a moderate type is indifferent between winning and losing and announces r ,.5

Next, denote г,- such that it satisfies

That is, the median voter prefers r) to :) when r, is announced by an extreme type. The right-hand side is the expected utility of the median voter when candidate j wins and announces r,. The left-hand side is the expected utility of the median voter when extreme candidate i wins and announces Moreover, r, denotes the most extreme platform that satisfies (3.8). More precisely, because the policy space is continuous, there is no maximal (minimal) value of (=i) that satisfies (3.8). Instead, it is possible to define r, such that a platform satisfies (3.8) with equality, and to assume that if an extreme type announces :h he defeats an opponent who announces - ,-.6

Then, the following proposition is derived.

Proposition 3.7

Suppose that the off-path beliefs of voters are p, (M|r, ) = 0. If pM r,. Proof: See Appendix 3.A.4.

## Each player's choice

The details of the semi-separating equilibrium are given in Appendix

3.A.4; hence, I provide the intuitive reasoning in this section. Suppose again that when a candidate deviates from the equilibrium platform, voters believe with a probability of one that the candidate is an extreme type. That is, p, (M|r,) = 0. This off-path belief is also partially based on the idea of the intuitive criterion. Since a moderate type is indifferent between winning losing (and announcing =*Л, a moderate type never chooses a platform more moderate than =*. On the other hand, from Lemma 3.4, an extreme type prefers his winning to the winning of an opponent who announces when he announces ? Thus, an extreme type has an incentive to choose a more moderate platform (until his cut-off platform, given ry). The intuitive criterion cannot apply to any other off-path beliefs, so I simply assume that Pi (M|=,-) = 0 for all off-path strategies.

The following summarizes each player’s rational choice.

### Voters

From the definition of z, (Inequality (3.8)), an extreme type who announces Zj can win against an opponent who announces z -. (and tie with an opponent who announces zj). Suppose that candidate R (an extreme type) announces ~zR, while candidate L announces zL. Voters can know that the type of R is extreme, but remain uncertain about the type of L who announces zL, because an extreme type L will still pretend to be moderate and announce zL. with probability GM. Therefore, to defeat L (i.e., to satisfy (3.8)), R does not need to implement a more moderate policy than a moderate type L. That is, Xm- xI[=l)> Xr {=R)-Xm>Xm-XL [=l ln other words, for the median voter, a moderate type L will implement the best policy Xl (-1)]- However, if L wins, there is the possibility that L is an extreme type who implements the worst policy for the median voter (/f£ (-!))• Thus, the median voter forgoes the chance of electing a moderate type L to avoid electing an extreme type L, and chooses the second-best candidate, /?, who is a separating extreme type.

### A moderate type

As a result of this off-path belief, a moderate type has no incentive to deviate from r( . If a moderate type deviates to a more extreme platform than z* or Zj e (c,,r/j, she will be certain to lose and her expected utility will remain unchanged as she is indifferent between her winning and the opponent winning at r,. A moderate type has no incentive to approach the median policy by more than z, since her utility from winning will then become lower than her utility when the opponent wins from Lemma 3.4.

### An extreme type

An extreme type prefers his winning to the opponent winning and announcing zj when he announces zt. Thus, he does not have an incentive to deviate to a more extreme platform than z* or z, e (c;,r*) since he will be certain to lose and his expected utility will decrease. When an extreme type announces zt, his disutility following a win and the cost of betrayal are higher, but the probability of winning is greater than when he announces r,. Thus, an extreme type can be indifferent between Zj and Zj when pM < p.

## The existence of the semi-separating equilibrium

A two-policy semi-separating equilibrium exists if pM < p and

When (3.9) holds, an extreme type has no incentive to defeat with certainty an extreme opponent who announces r, by approaching the median policy. This is because (3.9) means that, for extreme candidate the utility when extreme opponent j who announces r j wins is higher than the utility when / wins. However, an extreme type with z, does not want to deviate to a more extreme platform because that would mean he would also lose to an opponent with zj and his expected utility would decrease.

If (3.9) does not hold, an extreme type still has an incentive to converge by more than z, to beat an extreme opponent who announces zj. Therefore, a two-policy semi-separating equilibrium does not exist, but a continuous semi-separating equilibrium does exist. In a continuous semi-separating equilibrium, an extreme type’s mixed strategy includes Zj and a connected support, [zl,=i.] for L and [су?,гл] for R, as shown in Figure 3.3(b). An extreme type chooses any platform in this support with probability 1 - о л/ and has a continuous distribution function, F{.), within the support. More specifically, the distribution is (1-сгл/ )F(.). Platform Zj is defined in the same way as z, in a two-policy semi-separating equilibrium, so the basic results are the same as those of a two-policy semi-separating equilibrium. That is, a separating extreme type defeats an uncertain type (a moderate type and a pooling extreme type).

A semi-separating equilibrium exists in the broader value of the off- path beliefs.Forexample,supposep, (M|r,) = pM l[pM +aM (l- PM)

if the platform is more extreme than z*. Then a candidate still has no incentive to deviate to a more extreme platform than r,-, since he/she will be certain to lose and the expected utility decreases or is unchanged by this deviation. Thus, a semi-separating equilibrium can exist when

the off-path belief p,■ (M|r,) is lower than pM l[pM +oM (l — />л/)] for Zj, which is more extreme than r,.

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