Analysis of sequential exchanges between vulnerable forces Page: 4 of 11
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ANALYSIS OF SEQUENTIAL EXCHANGES BETWEEN VULNERABLE FORCES
Gregory H. Canavan
A multi-stage and -step analysis of sequences of crises or exchanges
shows that aggressiveness on one side can induce rapid counter-value strikes by
the other as well and knowledge that opponents will later become less aggressive
does not mitigate the tendency to strike early in crises.
The single-strike, first strike formalism developed earlier is extended to treat multi-stage
and -step sequences of crises or exchanges.' For unsymmetric objectives, aggressiveness on one
side can induce rapid counter-value strikes by the other as well. For symmetric forces and
objectives, non-aggressive opponents do not engage, while aggressive opponents strike to the
maximum against value. The knowledge that opponents will later become less aggressive does
not mitigate the tendency to strike early in crises, which is surprising in view of the fact that
inter-stage optimization provides a rudimentary knowledge of the future and hence an
appreciation of the marginal cost reductions obtained.
Exchange Model. The illustration uses an simplified version of a aggregated,
probabilistic treatment of the interaction between two vulnerable missile forces denoted by prime
and unprime, in which outcomes are evaluated in terms of the first and second strikes each side
could deliver. Each side initially has M = M' = 100 vulnerable missiles with m = 3 weapons each
and 1/k - 100 value targets at risk; thus, the weapons could saturate either the missile launchers
or value targets. At each step the unprime side launches dM missiles, a fraction f of which is
directed towards the other side's remaining vulnerable missiles. which gives a counter force
r = fmdM/M, (1)
The remaining weapons constitute a first strike on value targets of magnitude
F = (1 - f)mdM, (2)
The average survival probability of a prime vulnerable missile is Q' = qr, where p = 1 - q is the
missile single shot probability of kill. Prime's r' and Q are found by conjugating Eqns. (1) and
(2). Thus, unprime and prime's missiles decrease as
dM/dt = -dM - m'dM'(1 - Q'), (3)
dM'/dt = -dM' - mdM(1 - Q), (4)
where the variable t can be interpreted as either time or as the step in a multi-stage sequence of
conflicts and exchanges. At each step, each attacker minimizes the cost for that step and the
whole sequence, which might be terminated at any step, by minimizing the cost of executing the
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Canavan, G. H. Analysis of sequential exchanges between vulnerable forces, report, September 4, 1998; New Mexico. (digital.library.unt.edu/ark:/67531/metadc680312/m1/4/: accessed September 23, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.