Table 1: Summary of p( ) and t( ) for Different Asymmetric Information Environments
p( )
t( )
Bound
r
decreasing if t( ) decreasing
increasing if p( ) increasing
p (r) ≤ p(r)ρt (r)
w
∆
increasing if t( ) increasing
decreasing if p( ) decreasing
p (w
∆
) ≥ p(w
∆
)ρt (w
∆
)
ρ ≤
1
t(k)
increasing if t( ) decreasing
increasing if p( ) decreasing
k
p (k) ≥ -
p(k)t (k)[1−ρt(k)]
t(k)
ρ >
1
t(k)
increasing if t( ) increasing
decreasing if p( ) decreasing
the rising state is impatient, Lemma 4 asserts a similar relationship between a decreing t(k) and
However, when the rising state is sufficiently patient, if the expected time of
conflict is decreasing at any value for k, the overall probability of conflict at that value must be
increasing.
Now consider p( ) to be decreasing in the R’s private information. When the private information
is the initial strength of R, increasing w
∆
makes war more likely sooner rather than later. However,
if it is the speed of the transition that unknown, for sufficiently patient rising states higher k’s that
pull down the probability of conflict push the expected time of conflict into the future. Thus which
parameter is privately known to the rising state can make a difference between what relationships
can and cannot occur in equilibrium.
It is also interesting that there is no result for these cases of the form “if X is increasing, Y
is decreasing.” This is due to the nature of the compound inequalities that follow from incentive
compatibility. These inequalities simply do not allow for statements of this form. That is not to
say that such situations cannot arise. Situations of these sorts are going to be products of specific
bargaining situations. As such we should be wary of their robustness across
Now consider the bounds on p( ) given for p, t differentiable Bayesian equilibrium. As men-
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It also claims the same relationship between an increasing p() and increasing t() holds in these two situations.
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Convention