The maths of war

The Art of War 孙子兵法 was a treatise written by Sun Tzu who was a contemporary of Confucius and Socrates: Gautama Buddha used to be considered coeval but modern scholarship now puts him a couple of generations younger. The art of war is mostly about putting one over on your enemy. It would be no fun for boys if you could push a button and the host across the valley were vaporised. Getting to a key position firstest with the mostest could win the battle even if it meant embracing risk by stripping soldiers from another part of your front. Generals, famously Patton,  read about ancient battles to better understand the psychology of opposing generals: it didn't matter if their soldiers carried spears or carbines.

But all generals acknowledged that they needed quants to run the commissariat: no food at the front, no battle; archers need a lot of arrows; tanks need diesel.  Not only must you be confident that you have sufficient materiel at your disposal, you also have to have some idea of how much the other chap has.  The Irish word faisnéis, at the root of bithaisnéisíochta [bioinformatics] originally meant military intelligence - knowing whether there was an army the other side of that hill. And not only the existence of an army but how many regiments. In Sun Tzu's time and for the next 2500 years,
faisnéis was obtained by sending out a well-connected young chap, in a very smart uniform, on a horse to count camp-fires

In the run up to D-Day in WWII, one of the logistical problems that needed urgent solution was an estimate of the number of tanks the Germans had at their disposal.  Not only the gross number but a strong indication of what types: Tigers I & II were not as worrying as the new Panzer V.  It's a problem that could be given over to 'intelligence' but it turned out that good statistics got a far more accurate account.  The base data was serial numbers from tanks, destroyed or captured, that were accessible to the Allies in other theatres of the war: notably North Africa and Italy. The German war-machine was obsessive in its accounting and numbered each gearbox and each chassis sequentially.  The Allied quants had, therefore, long strings of numbers with a lot of gaps and were required to estimate the total size of the tank 'population' that had come off the assembly-lines in The Ruhr valley. They also needed to know how many would be available to the opposition, in 3, 6, 12 months time.  So the key parameter was how many tanks could be produced in the Panzer factories each month.  Clearly it was unlikely that the highest serial number available was the sum total of tanks up to the day that tank was destroyed. Nevertheless that was a minimum number.
Q. How much bigger was the true total?
A [simple]. N ~= m + m/k -1; where N is the total number estimated from m, the highest serial number and k the number of tanks in the sample.
A [more complicated Bayesian solution]:

where N, m and k are defined as above.  Another edge was to look carefully at the wheels of the tanks and determine how many different moulds were available to cast these out of steel. From the number of different moulds and the experience of Allied tank factories, Logistics could estimate the maximum number that could be produced per month. One of the insights from this scrutiny of the logistics, convinced the Allied high command that ball-bearings were a limiting resource for the Axis war-machine.  This resulted in numerous attacks on the ball-bearing factories of the Schweinfurter Kugellagerwerke in Schweinfurt, Bavaria. Numerous attacks resulted in the loss of numerous bombers and their crews. The Germans bought the entire ball-bearing output from neutral Sweden and also recovered thousands of usable ball-bearings from the ashes.  Similar analyses were applied to other munitions including the V-2 flying bombs. With access to German records after the war, it turned out that the statistical solution to the problem was remarkably accurate and much lower than the arm-waving worry-wart estimates from 'Intelligence'.  As with Abraham Wald, not to mention Alan Turing, a bit of insight and a lot of maths solved the Tank Problem goodo.