Cry havoc! And let slip the math of war

Warfare seems to obey mathematical rules. Whether soldiers can make use of that fact remains to be seen.

In 1948 Lewis Fry Richardson, a British scientist, published what was probably the first rigorous analysis of the statistics of war. Richardson ahd spent seven years gathering data on the wars waged in the century or so prior to his study. There were almost 300 of them. The list runs from conflicts that claimed a thousand or so lives to the devastation of the two world wars. But when he plotted his results, he found that these diverse events fell into a regular pattern. It wasas if the chaos of war seemed to comply with some hitherto unknown law of nature.

At first glance the pattern seems obsious. Richardson found that wars with low death tolls far outnumber high-fatality conflicts. But that obvious observation conceals a precise mathematical description of the link betweenthe severity and the frequency of conflicts to follow a smooth curve known as a power law. One consequence is that extreme events such as the world wars do not appear to be anomalies. They are simply what should be expected to occur occasionnally, given the frequencywith which conflicts take place.

The results have fascinated mathematicians and military strategists ever since. They have also been replicated many times. But they have not much impact on the conduct of actual wars. As a result, there is a certain "so what" quality to Richardson's results. It is one thing to show that a pattern exists, another to do something useful with it.

In a paper currently under review at Science, however, Neil Johnson of the university of Miami in Coral Gables, Florida, and his colleagues hint at what that something useful might be. Dr Johnson's team is one of several groups who, in previous papers, have shown that Richardson's power law also applies to attacks by terrorists and insurgents. They and others have broadened Richardson's scope of inquiry to include the timing of attack, as well as the severity. This prepared the ground for the new paper, which outlines a method for forecasting the evolution of conflicts.

Progress, of a sort

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 a given province, subsequent fatal incidents become more and more frequent. The intriguing points 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 aprogress 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. (}T_{1 is therefore the number of days between the first and the second attacks.) The other element of the equation, b, turns out to be directly related to} T_{1. 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} T_{1 should be enough to predict the future course of a local insurgency. Conversely, changing b would change both} T_{1 and} T_{n, and thus change that future course.}

_{Though the fit between the data and the prediction is not perfect (an example is illustrated at the beginnng of this post), the match is close enough that Dr Johnson thinks he is onto something. Progress curves are a consequence of people adapting to circumstances and learning to do things better. And warfare is just as capable of productivity improvements as any other activity.}

_{The twist in warfare is that two antagonistic groups of people are doing the adapting. Borrowing a term used by evolutionary biologists (who, in turn, stole it from Lewis Carroll's book, "Through the Looking-Glass"), Dr Johnson likens what is going on to the mad dash made by Alice and the Red Queen, after which they find themselves exactly where they started.}

_{Personal note: I shall explain how to defeat the Red Queen effect in another post.}

_{In biology, the Red Queen hypothesis is that predators and prey (or, more often, parasites and hosts) are in a constant competition that leads to stasis,a s each adaptation by one is countered by an adaptation by the other. In the case of Dr. Johnson is examining the co-evoluation 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, an equilibrium is reached, and the intervals become fairly regular.}

_{The mathematics do not reveal anything about what the adaptations made by each side actually are, behond the obvious observation that practice makes perfect. Nor do they illuminate what the value of b variesso much from place to place. Dr. Johnson has already ruled out geography, density of displaced people, the identity of local warlords and even poppy production. If he does find the crucial link, though, military strategists will be all over him. But then such knowledge might perhaps be countered by the otherside, in yet another lap of the Red Queen race.}

_{Source: The Economist}