As much as I believe that 3:1 ratio is too vague to be of any pratical value, as much I'd be surprised to learrn that the attacking party routinely has fewer assets than the defending. But that would be a happy surprise.
The 3:1 rule is at best roughly applicable at the “tactical level” (describing the number of troops and weapon systems at the point of impact, but even at this level many qualifications apply, such as posture, preparedness, terrain, quality of forces and so on.
At the higher levels (operational / strategic) it does not apply at all. While
some superiority is usually required, because in its absence superiorities at the tactical level cannot be sustained, historically a measure of superiority much less than 3:1 was often sufficient, sometimes near parity, sometimes even inferiority (but these normally involving a large qualitative difference).
But I guess we agree on this, so please excuse my lecturing.
If you could point to the relevant book(s) by Mearsheimer and Biddle, it'd be even better.
(the only Mearsheimer thing I read is the essay on Liddel Hart).
Stephen Biddle, “Military Power” includes sketches on force relations for several WW1 and WW2 battles (be warned though, he belongs to the "incompetent Brits" school of thought).
Upon re-reading, Mearsheimer (in his article: “Assessing the Conventional Balance: The 3:1 Rule and Its Critics”, available online) appears more of a measured supporter of the 3:1 rule, because he stresses the need for “special circumstances” in cases where it does not apply (I was speaking from memory).
That is incorrect though. Fresh German divisions were being committed to the fighting in Normandy at a rate of about one every three days during July. It was the German mistake, and the Allied intention, that most of these were sent to 5th Panzer Army, leaving only the line of German 7th Army 'thin'.
This argument is missing the critical context that these “fresh” divisions, apart from their often unfinished state, or that they were sometimes only battlegroups, also served as a substitute for casualty replacements. The large bulk of casualties was not matched by replacements, because at the time of the Normandy battle, the Germans were facing their yet greatest disasters and crisis on the Eastern front. So, seven or eight divisions were needed to nominally replace the 100,000 or so casualties not replaced by normal replacement personnel (disregarding for simplicity that these “fresh” divisions included a lot of non-combat personnel, while combat losses drained almost exclusively combat personnel).
A similar argument could be made for newly arriving armoured divisions, because armour losses in the already committed divisions were not replaced, either.
In addition, there was not much alternative to sending the majority of newly arriving forces to the British sector. A breakthrough there would have cut off all German forces fighting in Normandy, while a breakthrough in the Western sector allowed at least a more orderly retreat to the East (disregarding here the Mortain stupidity, which is another problem).
Also, the terrain around Caen was much more suited for armoured warfare then the Bocage country to the West, so it was quite natural to send the bulk of the armour there, as long as there was no real breakthrough attempt by the Americans.
And despite all arriving German divisions, the British still retained a comfortable numerical superiority, with their 3,000 tanks, and in other key weapon systems (artillery, air) as well.