Depending upon the specific parameter values, one of three policy equilibria may exist: Convergent, moderately divergent, and extremely divergent. I begin by discussing the incentives of the challenger. I consider only non- negative values of a, as the analysis for the negative values is symmetric.
π2
π1
EB(1) =c
EB(2) =c
c 4−E(v)
c (1−γ)|vL| c
3−E(v)
c 1−E(v)
I II
III IV
Figure 2.1: In region I, the voter does not observe the state for any position of the challenger. In region II, he observes the state only fora= 2, while in region III, he observes the state only ifa= 1. In region IV, the voter observes the state for either position.
The office motivated candidate acts to maximize her probability of winning the voter’s support given her conjecture about whether or not the voter will pay attention given her choice. The function P(a, y) provides these probabilities:
P(a, y) =
1−γ ifa= 0;
π1+π2 ifa= 1, y(1) = 1;
0 ifa= 1, y(1) = 0;
π1(1−γ) +π2 ifa= 2, y(2) = 1;
0 ifa= 2, y(2) = 0.
To reduce the number of cases to be considered, note thatP(0, y)> P(1,0)andP(0, y)> P(2,0). That is, the challenger does not take a divergent position unless the voter pays attention. Further, P(1,1)> P(2,1), so the challenger chooses a= 2 only if y(1) = 0 and y(2) = 1. In the event that the challenger is indifferent between a= 0 and any other position, I assume the challenger chooses a6= 0.
The incentives of the challenger given π1 and π2 are depicted in Figure 2.2. In region I, the challenger strictly prefersa= 0 to all other positions, regardless of the voter’s choice of attention because the probability of an extreme state is low relative to the probability of a negative valence shock. In region II, she prefers the position a= 1 to a = 0 conditional on the voter’s attention because there is a sufficient chance that the the state observed by the voter will be favorable to her. In region III, she also prefers to the position a = 2 to a= 0. This preference requires a larger value of π1, π2, or both, reflecting the increased risk associated with a= 2 relative toa= 1.
2.4.1 Equilibria
Equilibria in whicha= 1ora= 2are considered to bemoderately divergent and extremely divergentequilibria respectively. An equilibrium in which a= 0 is considered to be aconvergent equilibrium.
In order for an equilibrium witha6= 0to exist, it must be the case that (i) voters are willing to observe the state when a= 1 or a= 2 and (ii) the probability of election conditional on attention is greater at a= 1 or a= 2 than the probability of a negative valence shock. If either of these conditions fail, then the equilibrium must be convergent. Intuitively, convergence occurs either because there is not enough benefit to the voter of observing the state, so that y(1) = y(2) = 0, or because there is not enough of a benefit to the challenger to
π2
π1
1−γ 1−γ
1−γ
P(2,1) = 1−γ
P(1,1) = 1−γ
I
II
III
Figure 2.2: The challenger’s preferences over positions given the values ofπ1 andπ2.
a divergent position. Both benefits depend negatively upon the values of π1 and π2. As π1 or π2 decrease, the expected benefit of observing the state decreases whenever a 6= 0. Further, the probability of election conditional ona= 1 and a= 2 also falls because it is less likely that the state favors the challenger’s position.
Consequently, a convergent equilibrium is likely when voters have little uncertainty about the correct policy.
Proposition 2.1. The unique Bayes-Nash equilibrium is a= 0 if and only if any of the following are true:
1. π1+π2 <(1−γ).
2. π1(1−E(v)) +π2(3−E(v))< c and π1(1−γ)|vL|+π2(4−E(v))< c.
3. π1(1−E(v)) +π2(3−E(v))< c ,π1(1−γ)|vL|+π2(4−E(v))≥c and π1(1−γ) +π2 <(1−γ).
In the first case of Proposition 2.1, the combined probabilities of states 1 and 2 are small relative to the chance of a negative valence shock. The challenger would rather take the positiona= 0and hope for a negative valence shock than gamble by diverging to either a= 1 or a= 2, even if the voter observes the state. In the second case, the voter drives convergence because she is unwilling to observe the state whena= 1ora= 2. In the third case, the voter and the challenger are mismatched; while the voter only wants to observe the state if a= 2, the challenger is unwilling to gamble on such an extreme position.
Now consider when a = 1 may be an equilibrium. In that case, it must be that π1+π2 ≥ (1−γ) and y(1) = 1, which occurs if π1(1−E(v)) +π2(3−E(v))≥c. Recall that P(1,1)> P(2,1)> P(2,0), so the value ofy(2)is irrelevant.
Proposition 2.2. The unique Bayes-Nash equilibrium is a = 1 if and only if π1+π2 ≥ (1−γ) and π1(1− E(v)) +π2(3−E(v))≥c.
Lastly, consider when a= 2 may be an equilibrium. In order for the challenger to prefer this toa= 0, it must be that π1(1−γ) +π2 ≥(1−γ) andy(2) = 1, which occurs if π1(1−γ)|vL|+π2(4−E(v))≥c. Further, asP(1,1)> P(2,1), it must be the case thaty(1) = 0or, equivalently, that π1(1−E(v)) +π2(3−E(v))< c. Proposition 2.3. The unique Bayes-Nash equilibrium is a= 2 if and only ifπ1(1−E(v)) +π2(3−E(v))< c, π1(1−γ)|vL|+π2(4−E(v))≥c, and π1(1−γ) +π2 ≥(1−γ).
Intuitively, either divergent equilibrium relies on a probability of re-election conditional on the voter paying attention that is greater than the probability of a negative valence shock on the part of the incumbent. Because a= 1 offers a policy advantage for the challenger over the incumbent when either ω= 1 orω= 2, whilea= 2
π2
π1
EB(1) =c
EB(2) =c
c 4−E(v)
c (1−γ)|vL| c
3−E(v)
c 1−E(v)
I II
III IV
Figure 2.3: The equilibrium positions of the challenger given the values ofπ1 and π2.
provides a policy advantage only whenω = 2, the challenger prefersa= 1whenever possible. However, if the voter is not willing to pay attention toa= 1, then the challenger is willing to take a larger risk by deviating to a= 2.
Voters are willing to pay attention to divergent candidates only when divergent states are sufficiently likely;
that is, whenπ1 and π2 are sufficiently large. Whether they are more willing to pay attention to positions at a= 1and a= 2depends critically on the value ofπ2 relative toπ1. Positiona= 1provides a policy advantage when eitherω = 1orω = 2. Position a= 2 provides a larger advantage whenω = 2, but a smaller advantage when ω= 1. Consequently, EB(2) may be greater thanEB(1)ifπ2 is high relative toπ1, and forcsuch that EB(1)< c≤EB(2), the voter may only pay attention to positiona= 2.
The incentives of the voter to pay attention to divergent positions and the incentives of the challenger to gamble by taking a divergent position both depend in a similar fashion on π1 and π2. Consequently, when determining the type of equilibrium that may exist, the incentive constraint of the challenger may be redundant in light of the attention constraints of the voter. In particular, as the probability of a negative valence shock falls, the challenger is more willing to take a risk by diverging from a= 0. When the probability of a negative valence shock falls below a threshold, both of the incentive constraints of the challenger are redundant in determining the equilibrium.
Corollary 2.1. If (1−γ) ≤ 3−E(v)c , then EB(1) ≥ c =⇒ P(1,1) ≥ 1 −γ. If (1−γ) ≤ 4−E(v)c , then EB(2)≥c =⇒ P(2,1)≥1−γ.
If the first condition of Corollary 2.1 is met, then whenever the voter is willing to pay attention toa= 1, the challenger prefers the position a= 1 toa= 0. If the second condition of Corollary 2.1 is met, then whenever the voter is willing to pay attention toa= 2, the challenger prefers the positiona= 2 toa= 0.
Figure 2.3 illustrates the dependence of the equilibria on π1 and π2, in the special case in which (1−γ)≤
c
4−E(v). This may be interpreted as reflecting an election in which the incumbent is highly advantaged. In this case, the challenger prefers to take the positiona= 0only if the voter will not pay attention to either a= 1 or a= 2, which is true if EB(1) and EB(2)are less than c. This complete inattention occurs when both π1 and π2 are sufficiently small, as in region I. If the voter is willing to pay attention to onlya= 2, then the challenger is willing to gamble by offering the position a = 2. This attention only to extreme states occurs when π2 is relatively high andπ1 is relatively low, as in region II. However, whenever the voter is willing to pay attention toa= 1, then the candidate prefers this to a= 2 regardless of whether the voter pays attention toa= 2, as a= 1has a strictly higher election probability. This can occur either because π1 is large or because π2 is large, as in regions III and IV; the voter’s benefit to attention at is increasing in both.