A couple of months ago I ran across an article by the mathematician John Allen Paulos on the ABC News web site, questioning Defense Secretary Rumsfeld's approach of using small but heavily armed forces in Iraq instead of the large forces traditionally required for a military invasion. The article was written in March 2003, while the conflict in Iraq was very much under way and the outcome undecided. Paulos' analysis introduced me to something called Lanchester's Square Law, a mathematical formula for calculating the strength of military forces. I've been interested in this issue for some time, because I'm curious about the problems of balancing strategy games, so I decided to find out a little more about it.
The Theory of Predictable Victory (or Defeat)
There are now quite a number of books on the market about videogame design. To my surprise, I haven't discovered any that address Lanchester's laws; to my embarrassment, that includes my own book  an error that I intend to rectify if I can persuade my publisher to bring out a second edition. In the meantime, here's a very brief introduction to a huge topic called Operations Research. Theorists of military strategy can stop reading here; you're not going to learn anything you don't already know.
Suppose you have a certain amount of artillery, and your enemy has a different amount of identical artillery, and both are steadily shooting at each other and reducing each other's numbers. If everything else is equal, who's stronger? Obviously, whoever has more guns. How much stronger? Obviously, the proportion between the size of the first army and the size of the second, right? Blue has three times as many field guns as red, so blue is three times as strong.
"Wrong!" said Frederick William Lanchester, a British engineer and mathematician who lived from the middle 19th century through the end of World War II. Lanchester was one of those energetic Victorians who made the British the masters of the world  he built the first British gasolinepowered automobile, invented power steering and disc brakes, studied problems of aerodynamics, and more or less created the field of Operations Research (OR). OR was originally devised to solve problems associated with logistics, such as moving large quantities of troops and materiel around, although now it's used as much for business and industrial simulations as it is for warfare.
Back in 1916, during the height of World War I, Lanchester devised a series of differential equations to demonstrate the power relationships between opposing forces. Among these are what is known as Lanchester's Linear Law (for ancient combat) and Lanchester's Square Law (for modern combat). In 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, as you might expect (assuming identical weapons). This is the "obvious" conclusion I arrived at above.
In modern combat, however, with artillery pieces firing at each other from a distance, the guns can attack multiple targets and can receive fire from multiple directions. Lanchester determined that the power of such an army is proportional not to the number of units it has, but to the square of the number of units. This is known as Lanchester's Square Law.
Here's why. Suppose the blue army is three times the size of the red army. This means that it is concentrating three times as much firepower on red as red is firing. Just as importantly, red's firepower is diluted over three times as many blue units. The combined effect of these two conditions is that blue is nine times as strong as red although it only has three times as many units. Similarly, the number of units remaining at the end of the battle is the square root of the difference between the squares of the numbers of units on each side. In other words, if blue has five units and red has three, the number of units left at the end will be four. (Five squared minus three squared is 16; its square root is four.)
Squarely Beaten
This leads to what Lanchester called his Principle of Concentration: in modern warfare, bigger armies are better  in fact, a lot better than you might think at first. This may seem trivially obvious, but it has important strategic consequences. During the Napoleonic Wars, British naval strategy depended on keeping the French and Spanish fleets from uniting so that they could not mass against Britain. Admiral Nelson's victory at the Battle of Trafalgar depended on splitting up his enemies' superior force, and destroying the smaller groups before they could reunite.
Let's look at another consequence of Lanchester's Square Law. We in America are very proud of the quality of our military technology  what Lanchester would have called "weapon efficiency," or, more bluntly, "killing rate." What are the consequences of one side having better technology than the other?
If red and blue have equal numbers of units, but red's weapon efficiency is twice that of blue, then red is obviously going to win the battle: they can kill two blue units for every one red unit they lose. But if blue has three times as many units, the Square Law still holds in spite of red's technological advantage. Blue's mathematical advantage is reduced from 9 times as strong to 4.5 times as strong by red's technology, but blue still defeats red. In order to compensate for the other side's numerical advantage, the killing rate of the technology must advance in proportion to the square of the number of opponents it is meant to kill. As Dr. Paulos put it, it takes an Nsquaredfold increase in quality to make up for an Nfold increase in quantity. That's a tall order.
To Whom Does the Glory Belong?
Now, Lanchester's Laws are far from perfect. For one thing, they only apply to battles of attrition, in which the object is to wipe out the other side. If "winning" a battle is defined in some other way  and modern Western militaries don't normally consider slaughtering every last opponent to be a legitimate objective  then Lanchester's Law has nothing to say about who wins. Napoleon "won" every battle he fought in his march into Russia, but he still lost 98 percent of his men and was forced to retreat without achieving his objective. Similarly, the British and French dispute who won a naval engagement called (by the British) the "Glorious First of June." On June 1, 1794, a British squadron of 26 menofwar attacked a French squadron of similar size, sinking or taking seven ships and driving the rest away in confusion. The British had a little over 1,000 casualties; the French, closer to 7,000, so it seems like a British victory. But the French squadron was protecting a convoy of 117 cargo ships laden with American grain bound for starving, revolutionary France. All the cargo ships escaped and made it to port. So far as the French are concerned, the heroic sacrifice of their squadron produced a strategic victory for them.
Lanchester's Laws also don't take into account such considerations as terrain, morale, weapon range, movement and maneuver, surprise, weather, and many other issues that have decided battles over the centuries. In fact, various people have argued that in actual battle situations Lanchester's laws are well nigh useless. (See the works of Col. Trevor N. Dupuy for further criticism; Dupuy favored an empirical model of combat based on analysis of real, historical battles  not numerical analysis.) In other words, mathematical models and computer simulations are all well and good, but because they don't model actual battle conditions, it's difficult to know what value they really have to military planners.
A Fair Fight
However, to us as game designers, the Lanchester Laws are still extremely useful. We know we're not modeling actual battlefield conditions  certainly not if we're designing war games about elves and trolls, or robots and aliens. All we have are mathematical models. All we produce are computer simulations. It doesn't really matter if our results don't reflect historical reality, because historical reality is irrelevant to our purpose.



Questionable though Lanchester's Laws may be in reallife military scenarios, it should serve as a good mathematical model for strategy game designers in balancing opposing forces (image from Rise of Nations, Microsoft). 
Lanchester's Laws should be particularly useful in problems like setting the cost and production rate of different unit types. Suppose the aliens can produce twice as many Slime Leeches as the humans can produce Robot Tanks in a given amount of time, and both sides pay the same amount for each unit. Suppose further that the weapons efficiency of a Slime Leech is identical to that of a Robot Tank. Lanchester's Square Law tells us that at the end of five minutes of production by both sides, the aliens will have a force twice as large, but four times as strong, and costing only twice as much, as the humans' force. Clearly, the aliens have a big advantage. To compensate for their doubled production rate, each Slime Leech should cost twice as muchin theory, at least. Only playtesting can determine whether it really works in practice, given all the other factors in the game.
There's
a great deal of debate about the merits of combat
analysis, and many different theories have emerged
over the years. Logistical, business, and industrial
practices are more suitable for operations research
methods than actual battles, because they're more
rational activities, less influenced by the emotions
of the participants or the charisma of the leaders.
But in war games, we don't usually want to model soldiers'
emotions or leaders' charisma too much anyway  they
introduce too great a random factor. For our purposes,
operations research, starting with Lanchester's Laws,
seems seems like a valuable tool.
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