The Mathematics of War

I am still under the weather, so I am bringing this one out from the reserves.  It looks like it needs some work; but then so do I! LOL

The Economist has a review of a paper that Niel Johnson of the University of Miami in Coral Gables, Florida submitted to Science.  What the paper shows is that not only the overall scale (size) of wars follows a mathematical pattern (a power law), but that timing and severity of attacks does as well.  In this case it follows a curve known as a progress curve.  Progress curves occur when people adapt to circumstances and learn how to do things better.

Cry Havoc! And let slip the maths of war
Warfare seems to obey mathematical rules.  Whether soldiers can make use of that fact remains to be seen.

Dr Johnson’s proposal rests on a pattern he and his team found in data on insurgent attacks against American forces in Afghanistan and Iraq. After the initial attacks in any given province, subsequent fatal incidents become more and more frequent. The intriguing point is that it is possible, using a formula Dr Johnson has derived, to predict the details of this pattern from the interval between the first two attacks.

The formula in question (T_n = T₁n^-b) is one of a familiar type, known as a progress curve, that describes how productivity improves in a range of human activities from manufacturing to cancer surgery. T_n is the number of days between the nth attack and its successor. (T₁ is therefore the number of days between the first and second attacks.) The other element of the equation, b, turns out to be directly related to T₁. It is calculated from the relationship between the logarithms of the attack number, n, and the attack interval, T_n. The upshot is that knowing T₁ should be enough to predict the future course of a local insurgency. Conversely, changing b would change both T₁ and T_n, and thus change that future course.

In the case Dr Johnson is examining the co-evolution is between the insurgents and the occupiers, each constantly adjusting to each other’s tactics. The data come from 23 different provinces, each of which is, in effect, a separate theatre of war. In each case, the gap between fatal attacks shrinks, more or less according to Dr Johnson’s model. Eventually, equilibrium is reached, and the intervals become fairly regular.

The obvious conclusion to draw-if you have an advantage- use it.  This would be somewhat similar to what the Germans did through much of the early parts of World War 2, and occasionally in World War 1 as well.  In World War 2, the Germans had a couple of non-shooting practice invasions (Austria and Czechoslovakia) that were a mess:  units going  wrong way, running out of gas, etc.  This helped them to prepare for the first shooting war with Poland.  They outnumbered the Poles greatly, and certainly used some of their new radio-controlled combined arms techniques well.  But again, they struggled.  They took a lot of causalities given the size and armament disparities.  However, by the time the French campaign came around they were ready.  Although actually fairly evenly matched in size and equipment, the German’s at an operational level were able to take the initiative any time they choose.  The French tanks may have been better than the Germans, and they may have had more of them, but at the point of contact they were often isolated and disorganized.

This advantage was repeated in North Africa, the Balkans, and the Soviet Union.  However, because the allies did not collapse, they eventually closed the disparity gap.  When factoring in allied fighter bomber support, the western allies were actually slightly superior to the Germans in heads up fighting - at least according to that great number cruncher Dupey.  By the time of the (air)Battle of Germany, the Germans would lose as many fighter planes in a day as they did in the entire Battle of Britain.

If we go back to World War 1 you see the same effect at the Battle of Caporetto and the invasion of Romania.  Romania was so bad it was an example of the addition of an ally weakening ones side.

However, also in World War 1, the allies through fire power and mechanization were barely able to hold off the Germans, and by the end of the conflict probably had a very slight edge in offensive capabilities.

So where does this lead us:

1. If you are outgunned and outfought, but can hold on, your team will eventually get to be in a better position.  You personally may not survive, but I will not presume that there are no causes worth dying for.
2. If on the other hand, you have the upper hand, you better go for the knockout blow.  Letting a war that you have won drag on at a lower level (aka. When the U.S. turned to Iraq before completing the rebuilding of Afghanistan), will cost you dearly in the long run.  Even the most overwhelmed and foolish enemy will eventually learn from their mistakes and come back to cost you dearly: the will kill more often and in greater numbers.