Attack optimization for unequal moderate forces

Attack allocation optimizations produce stability indices for unsymmetrical forces that indicate significant regions of both stability and instability and that have their minimum values roughly when the two sides have equal forces. This note derives combined stability indices for unsymmetrical offensive force configurations. The indices are based on optimal allocations of offensive missiles between vulnerable missiles and value based on the minimization of first strike cost, which is done analytically. Exchanges are modeled probabalistically and their results are converted into first and second strike costs through approximations to the damage to the value target sets held at risk. The stability index is the product of the ratio of first to second strike costs seen by the two sides. Optimal allocations scale directly on the opponent`s vulnerable missiles, inversely on one`s own total weapons, and only logarithmically on the attacker`s damage preference, kill probability, and relative target set. The defender`s allocation scales in a similar manner on the attacker`s parameters. First and second strike magnitudes increase roughly linearly for the side with greater forces and decrease linearly for the side with fewer. Conversely, the first and second strike magnitudes decrease for the side with greater forces and increase for the side with fewer. These trends are derived and discussed analytically. The resulting stability indices exhibit a minimum where the two sides have roughly equal forces. If one side has much larger forces than the other, his costs drop to levels low enough that he is relatively insensitive to whether he strikes first or second. These calculations are performed with the analytic attack allocation appropriate for moderate forces, so some differences could be expected for the largest of the forces considered