On a simple military force planning model

The main idea: graph total military personnel and per-personnel expense, and arrange your forces perpendicular to where your adversary graphs. 

0. Posts on this blog are ranked in decreasing order of likeability to myself. This entry was originally posted on 03.05.2023, and the current version may have been updated several times from its original form. 


1.1 Here’s a dirt-simple and entirely conjectural (but hopefully fecund) model of military force planning: collect, for a series of countries of interest, the total size (in personnel) of all military branches (active duty, reserve and paramilitary), as well as the total military budget (ideally on a PPP basis).

1.2 Now calculate the total spending per personnel, and you’ll have decomposed military strength into two poorly-correlated factors, quality (a proxy for which is the per-personnel expense) and quantity (a proxy for which is total personnel). Graph this on a log basis. As an example, I use Wikipedia numbers to chart the below for the US, China and Russia.


1.3 Now add a budget line ,that captures all combinations of personnel and expense per personnel at the current budget. This being a log graph, that line will be straight. 

1.4 Here’s my conjecture now: you maximise your chances of defending against a stronger aggressor one-on-one if you organise your forces to be perpendicular to theirs along your budget line (i.e. minimising your distance on the graph). 

1.5 Say, Russia and China want to maximise their separate chances of surviving the US onslaught. Simple math gives the below, reformed structures (the dots don’t look perpendicular on the graph due to the log scale, but they are). To nobody’s surprise, they’d have to get that quality metric way up.


1.6 The same probably applies in reverse as well, with the US needing to place itself likewise against its budget to maximise its chances of defeating Russia or China one-on-one in a total, ground sea and air war. Without running the calcs, it is obvious that this would require a much, much larger (and less qualitative) US force.

1.601 I am aware of the contradiction inherent in claiming that the same combination of forces will both allow the US to win against China, and allow China to win against the US. I cannot resolve this, but will update if I get any hunch.

1.61 Anyway, translating all that into a simple formula gives us, [optimal armed forces size] = adversary armed forces size * square root ([own budget] / [adversary budget]). 

1.7 Obviously a dirt-simple model, since we’re making a ton of simplifying assumptions: treating active duty and paramilitary forces as equivalent, ignoring population size, jumbling together three or more branches, ignoring the need to station troops elsewhere, geography and much, much more. Yet, I think it works.  


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