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Bargaining Over Power: When Do Rapid Shifts in Power Lead to War?
Unformatted Document Text:  of u i (·), u B (1 − x ∗s ) = u B (1 − p s ) ≥ 1 − p s , and u A (x ∗s ) ≥ p s . Then, V A s (σ ∗ |c s ) ≥ δ s u A (x ∗s ) + V A n (σ|c n ) ≥ δ s p s + T t=n δ t (p n (E s [c n |c ∗s ]) − δ n l A > T t=s δ t p s − δ s l A since δ s > δ n = V A s (σ|c t ) By lemma 5.2, and V B s (σ ∗ |c s ) ≥ δ s u B (1 − x ∗s ) + V B n (σ|c n ) ≥ δ s [1 − p s ] + T t=n δ t (1 − p n (E s [c n |c ∗s ])]) − δ n l B > T t=s δ t (1 − p s ) − δ s l B = V B s (σ|c s ) Hence, there exists at least one strategy profile such that both players have an incentive to deviate from σ. But this contradicts the assumption that the use of the outside option was an equilibrium strategy. Proof of Lemma 5.5 I prove here the case where T = ∞. Clearly, a similar argument can be made by backward induction for T < ∞. Note first that we can define p n = β t p t without loss of generality. Then the whole 31

Authors: Chadefaux, Thomas.
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background image
of u
i
(·), u
B
(1 − x

s
) = u
B
(1 − p
s
) ≥ 1 − p
s
, and u
A
(x

s
) ≥ p
s
. Then,
V
A
s
|c
s
) ≥ δ
s
u
A
(x

s
) + V
A
n
(σ|c
n
)
≥ δ
s
p
s
+
T
t=n
δ
t
(p
n
(E
s
[c
n
|c

s
]) − δ
n
l
A
>
T
t=s
δ
t
p
s
− δ
s
l
A
since δ
s
> δ
n
= V
A
s
(σ|c
t
)
By lemma 5.2,
and
V
B
s
|c
s
) ≥ δ
s
u
B
(1 − x

s
) + V
B
n
(σ|c
n
)
≥ δ
s
[1 − p
s
] +
T
t=n
δ
t
(1 − p
n
(E
s
[c
n
|c

s
])]) − δ
n
l
B
>
T
t=s
δ
t
(1 − p
s
) − δ
s
l
B
=
V
B
s
(σ|c
s
)
Hence, there exists at least one strategy profile such that both players have an
incentive to deviate from σ. But this contradicts the assumption that the use of the
outside option was an equilibrium strategy.
Proof of Lemma 5.5
I prove here the case where T = ∞. Clearly, a similar argument can be made by
backward induction for T < ∞.
Note first that we can define p
n
= β
t
p
t
without loss of generality. Then the whole
31


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