War Chests and Information
divine equilibrium (ρ
∗
, α
∗
) such that:
ρ
∗
(I) = (¯
r)
∀I ≥
ˆ
I
ρ
∗
(I) = (ˇ
r(I))
∀I <
ˆ
I
α
∗
(r) =
{not enter} if r ≥ ¯r
α
∗
(r) =
{enter}
if r < ¯
r
where ¯
r
is the amount raised such that incumbent I = ˆ
I
is indifferent between imitating
other strong incumbents (and deterring a high quality challenger) and revealing his type
(thus drawing a high quality challenger). ˇ
r(I) is the amount raised by incumbent I < ˆ
I
knowing that the high quality challenger will enter. In addition, ι is defined to solve c
H
=
E
£1−W
H
(¯
r, I)
|I ≥ ι
¤=1−R
¯
I
ι
g(I)W
H
(¯
r, I)dI.
Proof:
18
I show that (ρ
∗
, α
∗
)
are best responses. For the challenger, any incumbents
who raise less than ¯
r
reveal their strength. Given their strength and the funds raised,
the challenger wishes to enter the race (c
H
< 1
− W
H
(ˇ
r(I), I)
for I < ˆ
I
). Incumbents
who raise ¯
r
pool so that the challenger cannot distinguish between them. The challenger
updates her beliefs on the pooling incumbents; her expected payoff if she enters the race
is c
H
− E
h1−W
H
(¯
r, I)
|I ≥
ˆ
I
i=c
H
−
h1−R
¯
I
ˆ
I
g(I)W
H
(¯
r, I)dI
i≤0. Thus, she(weakly)
prefers to stay out of the race.
The cutpoint incumbent, I = ˆ
I
, is indifferent between separating (and inducing challenger
entry) and pooling (and keeping the challenger out). Incumbents below ˆ
I
, find it too costly
to pool, and thus separate, and raise funds expecting challenger entry. Incumbents above ˆ
I
strictly prefer pooling. For these strong incumbents, raising more than ¯
r
does not change
the challenger’s decision, but costs more. Raising less than ¯
r
leads the challenger to enter,
18
A more technical proof, which includes uniqueness, can be found in the appendix.
18
Convention