Lanchester's Laws and Attrition Modeling, Part I

Consider of problem of two armies facing each other of a field of battle. One army is larger, but the other army is better armed. Once battle is joined, the two sides wear each other down until one is completely destroyed, or more likely, until one has suffered so many casualties it can no longer hold the field, and so retreats in defeat. There is a mathematical way to describe this sort of battle, a battle of attrition, and it is a topic I've been wanting to introduce here since I first started. I hesitated though, because it's a difficult topic with few really satisfying answers. Rather that reinvent the wheel, I'm going to link to a source articles and add comments of my own. This should easily make a series, maybe a long series if I don't get tired of it first.

So let's get things started - this first link is good introduction to the topic. After you get back, I'll repeat the definition in my own words, and explain where the Wiki article, and most articles on this topic, get it wrong.

Wikipedia - Lanchester's Laws
[edit April 2010: Of course Wikipedia changes sometimes! The article linked above is still good, but I have quoted the relevant proportion of the classical interpretations here.]

Lanchester's Linear LawIn ancient combat, between phalanxes of men with spears, say, one man could only ever fight exactly one other man at a time. If each man kills, and is killed by, exactly one other, then the number of men remaining at the end of the battle is simply the difference between the larger army and the smaller, assuming identical weapons.[...]Lanchester's Square LawWith firearms engaging each other directly with aimed fire from a distance, they can attack multiple targets and can receive fire from multiple directions. The rate of attrition now depends only on the number of weapons firing. Lanchester determined that the power of such a force is proportional not to the number of units it has, but to the square of the number of units. (reference 6 below).

My turn. This is a very simplified model of combat; Each side has identical soldiers; Each soldier has an identical probability of killing a soldier on the other side, if they can (the probability does not have to be the same for both sides). Range, terrain, movement, and all other factors that might influence the fight are either abstracted to the probability of a kill or ignored entirely. In the derivation from differential equations casualties are inflicted continuously over time, but it also works to think of casualties inflicted in rounds or turns, which will be a more familiar setting to most gamers.

Note on some abbreviations I may use:
Pk = "Probability Rate of a kill", with subscript k1 or k2 if it matters. This should be a small value relative to the size of the force and time interval. [Correction: There is no probability involved here. This should be the proportion or rate of casualties that results from combat over a short length of time.]
N1, N2 = The initial force sizes (before combat) of each side.
C1, C2 = The total casualties suffered by each side at the end of combat.


The Linear Law applies when one soldier can only fight one other soldier at a time. If one side has more soldiers, some of them won't be fighting all the time as the wait for an opportunity to attack. In this setting, the casualties suffered by both sides are proportional to the number actually fighting (and the relative probability of a kill). If the Pk is the same for both sides, then both sides will suffer casualties equaly to the size of the smaller force. This was originally called Lancherster's Law of Ancient Warfare, because it tries to model what happens if neither side has ranged weapons, and so are fighting with swords or spears (but it works equally well with ba'tleth or light-sabers).

The Squared Law, sometimes known as Lanchester's Law of Modern Warfare, is intended to apply to ranged combat, and it quantifies the value of the relative advantage of having a larger army. With the Linear Law, this advantage is proportional to the size of the forces, but when the entire force of both sides can engage the other simultaneously, the relative advantage is a function of the square of the force size. Again assuming equal Pk, the casualties of the larger army will be proportional to the ratio of the squared forces sizes.

So for example, if N1 = 3000 and N2 = 2000, then this ratio C2/C1 is equal to (dropping zeros, 3^2 / 2^2 = ) 9/4 or 2.25. By the end of the battle side-2 will have suffered 2.25 casualties to every 1.0 on side-1. Conversely, side-1 will lose 4/9 or 44.4% of side-2's loses, for total casualties of (4/9)*2000 = 889 soldiers.

Now for the bug. There is nothing wrong with the mathematical derivation, but there is considerable confusion about the interpretation. That the Squared Law describes the advantage of superior numbers in ranged combat is the practical interpretation, but range is not even considered in the derivation. The Squared Law really has nothing to do with range - what really matters is the rate of acquiring new targets. Having ranged weapons generally let's your soldiers acquire new targets as fast as they can shoot, whereas with a spear or sword (Linear Law) you have to locate a target and then move to engage them. In real life this may be a trivial distinction, because the "advantage of range" interpretation makes sense in most situations. However, games offer some alternatives where the Squared Law applies, but it clearly has nothing to do with range. Some examples of the Linear and Squared Laws in action, both on the gaming table and in real life:

1. For a platoon of Battle-axe wielding Dwarven warriors the Linear Law would generally apply, but make that a platoon of motorcycle mounted Battle-axe wielding Dwarven warriors that can move to engage any target on the gaming table, and suddenly the rate of target acquisition is as high as that of unit with ranged capability, and so the Squared Law applies.
2. (For the Battletech players) Consider a Battlemech like the Dasher H that carries powerful but short range weapons. This would usually imply the Linear Law. The limitation of short range weapons is irrelevant here because it moves so fast it can effectively engage most targets immediately.
3. In a game where some units may be effective invisible, either through stealth technology, "cloaking", or magical invisibility, then the advantage of range for acquiring targets may be effectively nullified, and the Linear Law will prevail in the battle.
4. The US military is increasingly making use of battlefield information systems to give field commanders knowledge of where the enemy is - to allow them to acquire targets first, and in the most advantageous way possible. This gives the advantage of the Squared Law to the US military, where the opposition with limited information is effectively fighting under the Linear Law.
5. Guerrilla warfare, in setting such as Iraq and Afghanistan, it is much easier for insurgent to find US targets than is it for the US to find insurgent targets, and the Square Law applies.

And here is the take-home lesson: Lanchester's Laws are NOT about range. Range doesn't matter, "ancient" or "modern" doesn't matter - It's all about the rate of target acquisition. It's OK to think about range being the key concept in most settings, because that is the mechanism which allows new targets to be attacked immediately. However, if you want to apply Lanchester's Laws when designing a game, or in understanding how game balance works, this distinction may be important.

There is more, much more, which is why I'm spliting this up into a series of posts. I have some references below, some of which didn't even get mentioned here, so if you don't want to wait for me you could peek ahead at some upcoming topics. Finally, here are some closing notes that didn't make it into the text above:

1. Though Lanchester generally gets the credit, a Russian mathematician named Osipov also wrote on the same topic at about the same time.
2. In application of Lanchester's Laws to historical data, it is generally found that some mix of the Linear and Squared Laws is the rule, not one or the other exclusively. I don't have any references on this below yet, but a Google search on Lanchester and Helmbold ought to turn up something relevant.
3. This has implications on point systems for balancing game scenarios, such as Battle Value in Battletech. Such point systems tend to have serious flaws, and Lancherster's Laws illustrate why: no single point system can be correct in every setting.
4. I implied, but did not state, that it is possible for one side/army to be operating under the Linear Law and the other under the Squared Law. This may be a topic in coming posts.
5. The assumptions for Lanchester's Laws are rarely true in a game setting, much less in reality. However, they do demonstrate the superiority of numbers principle in combat, which is a very important lesson, even if somewhat obvious.

References and Reading
(Please pardon the hodgepodge of styles. Organizing my math & gaming references is an ongoing project.)

1. Ernest Adams, "Kicking Butt By the Numbers: Lanchester's Laws", a Designer's Notebook, Gamasutra webzine, August 4, 2004.
2. Bruce Fowler, De Physica Belli: An Introduction to Lanchestrial Attrition Mechanics Part One, DEFENSE MODELING SIMULATION AND TACTICAL TECHNOLOGY INFORMATION ANALYSIS CENTE R HUNTSVILLE AL, 1995. [Early versions of this series can be found online at DTIC: 1,2,3.]
3. Michael J. Artelli and Richard F. Deckro, Modeling the Lanchester Laws with System Dynamics, The Journal of Defense Modeling and Simulation: Applications, Methodology, Technology 2008 5: 1-20
4. Niall MacKay, Lanchester combat models, arXiv:math/0606300v1 [math.HO] (2006)
5. Wikipedia contributors. "Lanchester's laws." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 3 May. 2010. Web. 23 Jun. 2010.
6. Wikipedia contributors. Lanchester's laws. (2011, March 30). In Wikipedia, The Free Encyclopedia. Retrieved 00:19, April 3, 2011.

Update: Part II is up.

Epilogue: I'm getting a lot of hits via Facebook, maybe thru Networked Blogs, or possibly posted on a group somewhere. If you came to this page via Facebook, please leave me a comment about how you got here. --- Thanks --- Dan

1/24/2024: I just found this article on Lanchesters Laws and applying them to plan encounters for Dungeons & Dragons. Go read it!