19
Proposition 3
: Given a time interval [t
1
, t
2
] and a level of costs b, there exists a maximum
number of re-negotiations and hence a maximum frequency of re-negotiation if war is to be
avoided.
d. Relative loss
We now want to know whether this number of re-negotiations will be sufficient to avoid war.
The answer depends on how frequently agreements need to be updated, and hence on the
growth differential in the dyad. This section shows that a large growth differential forces both
actors to re-negotiate frequently if they want to avoid war
Since re-negotiations are discrete and not continuous, D anticipates at time t
n
a relative loss
because the agreement will not be re-negotiated until t
n+1
, although D’s power will have
increased in the meantime
14
. In other words, D anticipates a relative loss whose size is a
function of the growth rate differential and of the time interval between the present
negotiation and the future one: D has a relative loss equal to the area of L
1
on figure 3,
with
dt
p
L
t
t
t
t
∫
−
=
2
1
1
)
1
(
1
β
.
15
14
Remember that both D and S anticipate a change in relative power at a rate
β
.
15
The value of L
1
is derived as follows:
(
)
(
)
[
]
(
)
( )
[
]
[
]
dt
p
dt
p
p
dt
b
c
p
b
c
p
L
t
t
t
t
t
t
t
t
t
t
t
D
t
t
D
t
t
t
∫
∫
∫
−
=
−
=
+
−
−
+
−
=
2
1
1
2
1
1
1
2
1
1
1
1
1
)
1
(
1
β
β
β
Convention