War Chests and Information
For incumbents in the upper region (I > ˆ
I
), raising more money just adds to their overall
cost without affecting challenger entry. Raising less money may encourage the challenger to
enter, and if she does not, no longer maximizes utility. Incumbents in this region also do
not wish to imitate each other as there are increasing differences in spending and incumbent
strength.
22
Thus, no incumbent has incentive to deviate.
Step 4:
σ
∗
is a best response to
(ρ
∗
, α
∗
, β
∗
).
The reasoning is similar to ρ
∗
. Incum-
bents in the lower region are maximizing their utility, given the high quality challenger will
enter the second election. Raising and spending enough money to deter the challenger is too
costly; raising and spending less reduces utility. Incumbents in the upper region raise and
spend enough money to make the challenger indifferent. Raising more money or spending
less money reduces utility, raising less or spending more induces challenger entry. As above,
incumbents in this region also do not wish to imitate each other as there are increasing
differences in spending and incumbent strength. Thus, they do not deviate.
¤
A graphical illustration of the two-election perfect-information equilibrium is in Figure
6.This equilibrium is constructed with incumbents distributed on [0, 1] = [I, ¯
I]
, and with
ˆ
I ≈ 0.7. In the both elections, incumbents weaker than ˆ
I
will face the high quality challenger;
incumbents stronger than (or equal to) ˆ
I
will not face the high quality challenger. Looking at
fund-raising behavior in the first election (the solid line), stronger incumbents raise enough
funds to deter the high quality challenger from entering and that no other incumbents wish
22
More technically, the utility function is supermodular in incumbent spending and incumbent strength. A
common feature of signalling games is that it is incentive compatible to separate when the utility function is
supermodular. See Milgrom and Shannon (1994) for a discussion of supermodularity and references to other
applications.
28
Convention