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Learning from Interested Parties
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Notice that E(ω| ¯
Ω) depends on ¯
Ω, while in turn ¯
Ω in Lemma 1 depends on E(ω| ¯
Ω).
Thus, finding the equilibrium is equivalent to finding a fixed point in this belief mapping. In
other words, the equilibrium in the signalling game between interest group and bureaucrat
requires that B chooses the optimal policy given his expectation and that B’s expectation
E(ω|∅) is compatible with G’s message strategy m
∗
. The following proposition establishes
the existence and characterizes the equilibrium.
Proposition 1. Assume w.l.o.g. x
G
< x
B
. The equilibrium in the signaling game between
interest group and bureaucrat is as follows. There exist an ω
∗
∈ [0, ¯
r] and an interval
¯
Ω = [ω
∗
, ω
∗∗
], ω
∗∗
= min ω
∗
+ 2(x
B
− x
G
), ¯
r , such that
a. G sends message m = ω iff ω ∈ ¯
Ω, m = ∅ otherwise;
b. After hearing m = ω, B chooses p
∗
(ω) = x
B
+ ω, and after hearing m = ∅, B chooses
p
∗
(∅) = x
B
+ E(ω|∅);
c. B’s posterior expectation after receiving m = ∅ is E(ω|∅) = ω
∗
.
Proof. a. follows from Lemma 1, and b. has been shown above. It needs to be shown that
there exists an ω
∗
for which beliefs are confirmed such that c. holds.
Define z(˜
ω) ≡
qF (Ω)E(˜
ω|Ω)
1−q+qF (Ω)
on [0, ¯
r], and Ω ≡ [˜
ω, min ˜
ω + 2(x
B
− x
G
), ¯
r
⊂ [0, ¯
r]. Since
F is absolutely continuous, F (Ω) =
Ω
f (ω)dω and E(˜
ω|Ω) =
1
F (Ω)
Ω
ωf (ω)dω are both
continuous in Ω. Observe further that Ω is continuously and monotonically increasing in ˜
ω,
for ˜
ω ∈ [0, ¯
r]. Hence, z(·) is continuous.
Next, notice that Ω ⊂ [0, ¯
r] implies E(˜
ω|Ω) ∈ [0, ¯
r]. Recall from (2) that z(˜
ω) is a
weighted average between 0 and E(˜
ω|Ω); hence z(˜
ω) ∈ [0, ¯
r]. We conclude that z(·) is a
continuous mapping [0, ¯
r] → [0, ¯
r]. By Kakutani’s fixed point theorem there exists an ω
∗
s.t. ω
∗
= z(ω
∗
) = E(ω|∅), where the second equality is given by (2).
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Notice that E(ω| ¯
Ω) depends on ¯
Ω, while in turn ¯
Ω in Lemma 1 depends on E(ω| ¯
Ω).
Thus, finding the equilibrium is equivalent to finding a fixed point in this belief mapping. In
other words, the equilibrium in the signalling game between interest group and bureaucrat
requires that B chooses the optimal policy given his expectation and that B’s expectation
E(ω|∅) is compatible with G’s message strategy m
∗
. The following proposition establishes
the existence and characterizes the equilibrium.
Proposition 1. Assume w.l.o.g. x
G
< x
B
. The equilibrium in the signaling game between
interest group and bureaucrat is as follows. There exist an ω
∗
∈ [0, ¯
r] and an interval
¯
Ω = [ω
∗
, ω
∗∗
], ω
∗∗
= min ω
∗
+ 2(x
B
− x
G
), ¯
r , such that
a. G sends message m = ω iff ω ∈ ¯
Ω, m = ∅ otherwise;
b. After hearing m = ω, B chooses p
∗
(ω) = x
B
+ ω, and after hearing m = ∅, B chooses
p
∗
(∅) = x
B
+ E(ω|∅);
c. B’s posterior expectation after receiving m = ∅ is E(ω|∅) = ω
∗
.
Proof. a. follows from Lemma 1, and b. has been shown above. It needs to be shown that
there exists an ω
∗
for which beliefs are confirmed such that c. holds.
Define z(˜
ω) ≡
qF (Ω)E(˜
ω|Ω)
1−q+qF (Ω)
on [0, ¯
r], and Ω ≡ [˜
ω, min ˜
ω + 2(x
B
− x
G
), ¯
r
⊂ [0, ¯
r]. Since
F is absolutely continuous, F (Ω) =
Ω
f (ω)dω and E(˜
ω|Ω) =
1
F (Ω)
Ω
ωf (ω)dω are both
continuous in Ω. Observe further that Ω is continuously and monotonically increasing in ˜
ω,
for ˜
ω ∈ [0, ¯
r]. Hence, z(·) is continuous.
Next, notice that Ω ⊂ [0, ¯
r] implies E(˜
ω|Ω) ∈ [0, ¯
r]. Recall from (2) that z(˜
ω) is a
weighted average between 0 and E(˜
ω|Ω); hence z(˜
ω) ∈ [0, ¯
r]. We conclude that z(·) is a
continuous mapping [0, ¯
r] → [0, ¯
r]. By Kakutani’s fixed point theorem there exists an ω
∗
s.t. ω
∗
= z(ω
∗
) = E(ω|∅), where the second equality is given by (2).
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