It's probably noteworthy that a different strain of thought about necessary ratios does not look at friendly / allied forces, but at friendly forces / frontage in kilometres.
This was often done in regard to breakthrough battles (guns/km, tanks/km, AT weapons/km) and there were also rules of thumb about acceptable frontages for battalions or brigades (of specific type, such as mechanized infantry) in various forms of combat (attack, defence, delay).
The latter seems to have been favoured for scenarios with unusual force densities (such as the Cold War when 26 NATO divisions were supposed to defend a 1,000 km front line - this took rather 50+ divisions in WW2).
OK, having read the free copy I see where the problem lies. The 3:1 ratio should apply at platoon and company levels and less so further up the line. By the time you get to division it would not apply as the greater battlefield intelligence picture would dictate actions. If you have no battlefield intelligence then you would not know what you are up against to apply the ratio of three against, would you?
A planning factor. Paint by numbers. Something for the observer controllers to use to abuse you with at the AAR "Well, certainly your unit was successful, but we're concerned that you did not have a 3:1 ratio over all (though your may well have been 8:1 where you through your strength against a point of weakness while the bulk of your opponent's force sat idle eslewhere...)
Robert C. Jones
Intellectus Supra Scientia
(Understanding is more important than Knowledge)
"The modern COIN mindset is when one arrogantly goes to some foreign land and attempts to make those who live there a lesser version of one's self. The FID mindset is when one humbly goes to some foreign land and seeks first to understand, and then to help in some small way for those who live there to be the best version of their own self." Colonel Robert C. Jones, US Army Special Forces (Retired)
Concepts like 3 to 1 ratios are only valid when comparing like or similar capabiity systems, weapons or units.
comparing the average WW2 infantry platoon armed with bolt-action rifles and a limited number of MGs to any of today's cutting edge infantry with magazine fed assault rifles, ICOM intra-squad comms, body armor etc will reveal that a straight comparison of numbers only is invalid. The concept of Relative Combat Power was an attempt by the US during the late 80s early 90s to address this issue. Sometimes it worked, sometimes it did not.
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