Having established sequentially rational strategies, beliefs must be updated consistently by
Bayes’ Rule on the equilibrium path. Each time the challenger observes a rejection, it truncates
the support of its beliefs by dropping those low-resolve types that would have accepted the offer in
equilibrium. While the upper bound d on the challenger’s beliefs is unchanged, the lower bound
rises with each observed rejection. If the challenger’s prior beliefs θ
D
1
support a lower bound of d at
the beginning of Stage 1, then an observed rejection revises the posterior lower bound to d
1
. Thus,
at the beginning of Stage 2, the challenger believes that d > d
1
with probability P r(d > d
1
) = 1.
Following Bayes’ Rule and the defender’s equilibrium strategy,
P r(d > d
1
|R
1
) =
P r(R
1
|d > d
1
)P r(d > d
1
)
P r(R
1
)
=
1(
d−d
1
d−d
)
(
d−d
1
d−d
)
= 1
(22)
Updating follows an identical pattern after observed behavior in each stage.
I now test for the existence of a pure pooling equilibrium in which all types of defender reject
all offers and the challenger makes no demand in equilibrium, or x
t
= 0. (Any equilibrium in which
defenders that prefer disagreement to x
t
choose to accept the offer violates incentive-compatibility
constraints, as do equilibria in which defenders pool on rejection and the challenger always makes
high demands. I rule out all such equilibria here.) For an equilibrium in which all defenders reject
and the challenger makes no demand, begin with the choice over {A
3
, R
3
} in Stage 3. Pooling on
rejection is not rational for any type of D
1
or D
2
for which x
3
> d. In the second stage, however,
pooling on rejection is sustainable only if the minimum support of the challenger’s beliefs d
1
receives
more from disagreement than the per-period value of the good, or that d
1
> 1.
U
D
1
(R
2
|d = d
1
) = d
1
+ e(1) > U
D
1
(A
2
|d = d
1
) = 1 + e(1)
(23)
Solving for d
1
yields d
1
> 1. In this case, the challenger is effectively deterred from making any
demands because, even when uncertainty over the defender’s type, its beliefs do not support the
38
Convention