This blog has discussed conflict statistics before, as well as some of the widely acknowledged problems with adapting "physics models" to the social sciences. To provide some context to that debate, I thought I would share an example that I recently came across. The example I present here is interesting for its historical relevance, and is not put forth as a prototype for the kind of work that political scientists ought to be doing.

The model is F.W. Lanchester's Square Law for Modern Combat, and it comes to us by way of Martin Braun's Differential Equations and Their Applications. Lanchester's model describes the rate at which casualties will occur in a two-sided battle with modern weapons, and takes its name from the idea that the power of each side is proportional to the square of its size. Rather than modeling when international conflicts will occur, as many modern scholars do, the model is intended to predict which side will win a battle.

Wikipedia offers this example, which I have modified slightly to match the later discussion:

Suppose that two armies, Red and Blue, are engaging each other in combat. Red is firing a continuous stream of bullets at Blue. Meanwhile, Blue is firing a continuous stream of bullets at Red.

Let symbol y represent the number of soldiers in the Red force at the beginning of the battle. Each one has offensive firepower α, which is the number of enemy soldiers it can knock out of battle (e.g., kill or incapacitate) per unit time. Likewise, Blue has x soldiers, each with offensive firepower β.

Lanchester’s square law calculates the number of soldiers lost on each side using the following pair of equations [3]. Here, dy/dt represents the rate at which the number of Red soldiers is changing at a particular instant in time. A negative value indicates the loss of soldiers. Similarly, dx/dt represents the rate of change in the number of Blue soldiers.

dy/dt = -βx

dx/dt = -αy

A less abstract example, discussed by Hughes-Hallett et al (p. 606), is the battle between US and Japanese forces at Iwo Jima. The authors conjecture that α=0.05 and β=0.01. They further assume that the US had 54,000 troops and 19,000 reinforcements (whom we will ignore for now), while the Japanese had 21,500 troops with zero reinforcements. These numbers roughly match the historical record.

The first picture below shows the predicted change in forces over the course of the battle without reinforcements:

[caption id="attachment_924" align="aligncenter" width="640" caption="US vs. Japanese Forces at Iwo Jima"][/caption]

The battle starts at the initial values listed above and lasts for sixty time periods, ending in the complete annihilation of the Japanese troops (also close to reality). Note that the axes are scaled in units of 10,000 troops. The plots were created in Python with matplotlib, and the source code can be found here.

What happens when we add the US reinforcements? I created a second scenario in which the 19,000 reserve troops are committed to the battle when the Japanese force dwindles to 9,000 troops (at about t=30). The addition of reinforcements is indicated by the red arrow in the plot below.

[caption id="attachment_925" align="aligncenter" width="640" caption="US-Japanese Battle at Iwo Jima, with US Reinforcements"][/caption]

As you can see, the battle ends more quickly (at t=50 instead of 60<t<65), with fewer US casualties overall (losses of 32,000 in the first scenario versus 27,000 in the second). In actuality, Wikipedia reports, "Of the 22,060 Japanese soldiers entrenched on the island, 21,844 died either from fighting or by ritual suicide. Only 216 were captured during the battle. According to the official Navy Department Library website, 'The 36-day (Iwo Jima) assault resulted in more than 26,000 American casualties, including 6,800 dead.'" By changing the time period at which reinforcements are added, this result could be closely approximated by Lanchester's model.

This is an admirably simple model, which seems to approximately describe actual events when tested. So what is the problem? The biggest issue, which Martin Braun mentions in his discussion of Lanchester's work, is that it is almost impossible to determine the values of α and β before the battle actually occurs. There has been work on estimating those parameters as Markov transition probabilities, but for the most part contemporary scholars of conflict do not analyze individual battles. One important exception is Stephen Biddle's work, linked below.

Further reading:

Stephen Biddle. 2001. "Rebuilding the Foundations of Offense-Defense Theory." (ungated PDF)

Modeling the Iwo Jima Battle, by one of the co-authors of the Hughes-Hallett text

Kicking Butt by the Numbers, by Ernest Adams