Exponentials and Unpredictability
Compounding interest is one of the most unintuitive features of human existence. Many a fortune has been lost by gambling on big returns, instead of patiently accumulating the steady returns of interest. Similar dynamics make warfare unpredictable. Small advantages compound into larger ones, sometimes creating runaway victories over closely-matched opponents.
Lanchestrian Combat
A simple example of this is provided by Lanchester’s equations, which estimate combat losses as a function of both sides’ numerical strength and lethality. Although not intended to predict the outcomes of battles, which are determined by a number of factors, they can model the outcome of attritional fights: all else being equal, the advantage of having even slightly greater numbers compounds over time. The relative strength of the larger side increases as both suffer losses, leaving it with more guns concentrated on fewer targets; eventually, the stronger side stops taking casualties as the weaker collapses.
This effect is even more striking when the advantage is not in quantity but in quality. The belligerent whose troops are more lethal (capable of individually inflicting a higher kill-to-loss ratio) can defeat a much stronger foe, so long as it can maintain its coherence long enough to inflict sufficient damage. In reality, of course, an army will begin to collapse long before it has sustained 100% casualties.
Real-world conditions rarely match this highly-simplified model, but the 1973 battle for the Golan Heights came close. It was remarkable for being decided mostly by tank fire, which caused 70% of Israeli vehicle losses and 90% for the Syrians. At the outbreak of war, Israeli armor was outnumbered nearly eight to one: the Syrians had roughly 1370 tanks in two echelons against 177 Israeli. The latter had a crucial advantage in gunnery, however, capable of accurate fire at up to 4000 meters—1000 meters farther than their adversaries. This made the kill ratio even more lopsided than raw numbers would suggest, as Israeli units could often destroy an equal number of enemy tanks before a single one could close to within firing range: the Syrians had to have a considerable local advantage to inflict any casualties. Although other tactical factors played a role, gunnery proved decisive, allowing the Israelis to hold out until their own reserves arrived to shore up the situation.
Attrition
Victory in battle does not automatically translate to victory in the larger conflict, as military practice has evolved over the ages to stem the runaway effects of defeat. As far back as antiquity, formations were drilled to prevent them from routing, and fine-grained command-and-control allowed for orderly withdrawals in the event of defeat. As field armies grew in size, generals learned to prepare fallback positions. So long as they could keep their forces intact past the attacker’s culminating point, they could avert a quick decision and turn the war into a contest of attrition—a Lanchestrian battle at a much larger scale.
Wars of attrition should be easier to predict, a straightforward calculation of relative strength and loss ratios. Yet the fog of war makes things much more obscure. The Germans at Verdun thought that they were inflicting a better than 2:1 ratio and “bleeding France white”, when in reality the ratio was barely above even. This represented a greater proportional loss for Germany than it did for the Allies as a whole, especially after America entered the war the following year. Yet when viewed from the frontlines, the cumulative effect of these losses did not become apparent until the very end: as late as summer 1918 the Germans maintained a manpower advantage on the Western Front, but they lacked the reserves to withstand the weight of the Hundred Days offensive.
Attrition extends beyond manpower to financing, manufacturing, and materiel. Not only are these individually subject to compounding, but their combined effects reinforce one another. For example, the quality of both German and Japanese aviation declined precipitately in the last year of World War II as oil shortages made it impossible to train replacement pilots to an adequate standard, and materiel shortages meant they had fewer aircraft of worse quality. By the Battle of the Philippine Sea in June 1944, the Americans were able to destroy around 600 Japanese aircraft for a loss of 123 of their own, wiping out a large chunk of remaining Japanese aircraft and opening the skies to American bombers. By comparison, the Guadalcanal campaign less than two years earlier saw a loss of 615 American aircraft against 683 Japanese.
A Race Against Oneself
Crude first-order estimates of national strength are the best predictors of victory in attritional wars, but they say nothing about when a war becomes attritional. In the opening phase of a war, the attacker is racing to achieve a decision before hitting his culminating point, where the logistical burden of sustaining the offensive becomes too great to bear. Logistical costs are highly compounding, as each additional requirement brings requirements of its own, creating demands that can quickly spiral away from even the best-resourced armies.
Donald Engels, in his classic study on the logistics of Alexander the Great’s army, illustrated this with a simple model to calculate the baggage train required to feed the Macedonian army as it marched across the desolate expanses of the Persian Empire. Pack animal had to carry everything they ate, which on longer marches left them with less spare capacity to carry an increasing overall burden. Engels calculated that on a 12-day march, they would eat through almost their entire loads, leaving just a little room for anything else—pushing the total requirement of animals to infinity. In practice, this capped marches at about a week, as it was not practical to manage so large a supply train.1
These effects, already non-linear, are compounded further as the army size increases. There is no better example of this than the French invasion of Russia. Contrary to the popular myth that Napoleon neglected logistics, this was his most carefully-organized campaign. He began preparing over a year and a half in advance, buying up vast numbers of draft animals and wagons while establishing a line of magazines along the Russian border. When the French army crossed the Niemen in June 1812, it advanced in multiple columns to spread the burden over a wider area, and established supply depots along the way.
Yet none of these measures could compensate for the runaway effects of mishaps. The sheer size of the army caused horrific traffic jams, aggravated by bad roads and autumn rains. High concentrations of men and beasts caused disease to tear through the army, causing horses to drop dead at an alarming rate, which further degraded the logistical apparatus. After initial attempts to encircle the Russians failed, the entire Grande Armée was funneled onto a single main axis of pursuit, further clogging up the supply lines. To cap it all off, Russian scorched-earth strategy and a general lack of forage left very little margin for error when supply broke down, meaning everything had to be moved over a thousand kilometers from the rear depots to Moscow. These factors all compounded one another, causing horrific shortages along the route and inflicting heavy losses before the French even made contact with the enemy. And by the time Napoleon realized he was past his culminating point, he was far from any position where he could retrench to go on the defensive.
Does this mean that the invasion was doomed from the start? Could more aggressive action around Vitebsk or Smolensk have encircled the Russian army? Could the commitment of the Old Guard at Borodino have destroyed it? It is impossible to rule out some battlefield miracle or characteristic stroke of genius turning the situation around. But as things stood, the French were left scrambling up a steepening slope, something which even Napoleon’s formidable intuition was unprepared for.
Seeing through the Haze
These types of models can illustrate the problem of compounding effects, but they cannot predict the future. Even when exponential dynamics can be foreseen, it is difficult to predict just where they will take effect. Germany invaded France in 1870 with a million-man army and reached the outskirts of Paris six weeks after crossing the border; by the same point in World War I, the Germans had already withdrawn from the vicinity of the capital after being defeated at the Marne. The army of 1914 was just 30% larger and the marches were of comparable length, and in some respects the situation was more favorable: it was dispersed across a wider front and supported by a much more extensive rail network. But the exponential growth in shell weight and consumption overtaxed their logistics network, increasing from less than 1% of all supplies by weight in 1870 to a majority in 1914. Although the Germans’ supply system held up all the way through the Marne, it also forced them into a rigid scheme that precluded the expedients which had won the Franco-Prussian War.2 Thus, although Schlieffen and the other General Staff planners had an intellectual appreciation of the problem, like Napoleon, they failed to grasp how it would overwhelm all other factors.
“For want of a nail” captures the inherent unpredictability of war, wherein a minor setback rapidly snowballs into a catastrophe. But there is rarely such a legible sequence of events originating from a single happenstance. The real dangers and opportunities lie in the mass of barely-perceptible factors which reinforce and build upon one another. These compounding effects are not only unintuitive, but often invisible.
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Donald Engels, Alexander the Great and the Logistics of the Macedonian Army, p. 14-22. Engels estimated that each animal could carry about 250 lbs. If X is the total weight of the daily requirements for the army, Y is the daily requirements for each animal (about 10 lbs. of grain when forage and water were available), and n the number of days marching, then the total number of animals required was n*X/(250-n*Y) (this is simplified somewhat, not including what the men themselves would carry).
Martin van Creveld, Supplying War, p. 139-40. On relative ammunition weight, see p. 233.
Great post. The way an attritional struggle betweeen even seemingly closely matched enemies can suddenly hit a slope for the loser, and the loser’s fighting capacity seems to slide right off the table, is an observation which is historically well founded, but it nonetheless seems counterintuitive, the way die continuous events often do. The disintegration of the Germans in 1918 was a shock to everyone involved, but particularly for people back on the homefront. The people actually close to the fighting could probably feel that the breaking point was approaching. On this model that breaking point was bound to happen. Of course lots of mistakes were made, but the basic logic that it was a slugging match and the Allies simply had more men and materiel meant the question was not if but when that collapse what happen.
https://arxiv.org/abs/1401.3281