The Lanchester Model for Combat explained with an application to unit activation mechanisms

First of all, this is a long post, so I apologize for those who are used to short snippets.
If you don't want to read the whole thing, skip to the conclusions in section 4. Or if you don't even want to do that, here's the conclusion verbatim:

If you want to write a ruleset that rewards manoeuvring, be sure the rules allow to inflict as many casualties proportional to the number of units. If you want a ruleset that puts less emphasis on manoeuvre, and more on clever use of terrain, use a fixed number of units that can activate every turn.

Anyway, the long story:

When I was reading through some old issues of Wargames Illustrated, I came upon the model of battle proposed by Frederick Lanchester during WW1 ('Aircraft in Warfare: The Dawn of the 4th Arm'). The fundamental question he was faced with was how to judge the relative strengths of 2 opposing forces, and more specifically two groups of aircraft facing each other.

This led to the well-known 'law of squares': when 2 forces are meeting each other, and when trying to predict the outcome of battle, one should take the square of their strengths, subtract them from each other, and take the square root of the result. The outcome is the number of soldiers/units/... left to the numerically biggest side.
E.g. suppose Red with 10 soldiers (or units, tanks, aircraft ...) is facing Blue with 6 soldiers. Let's also assume the quality of soldiers on both sides is equal. Red, due to its numerical superiority, will be victorious. But how many soldiers will Red have left after the fight? Lanchester's law provides the answer: 10.10 - 6.6 = 100 - 36 = 64; sqrt(64) = 8; hence Red will have 8 soldiers left after all Blue's soldiers have been eliminated. This is a simple example, but Lanchester's model also allows for the difference in troop quality to be factored in.

The classic application of Lancester's model is to show how a large force can be defeated by a smaller force, by splitting the large force in 2 smaller forces and defeat them one by one. Real-world examples cited in this context usually are Trafalgar or a number of Napoleonic battles. Suppose Red has 8 soldiers/unit/ ... and Blue has 10. Through clever manoeuvring Red has managed to split Blue's force in 2 halves, one of 4 strong and another of 6. Red takes on the first half first, emerging victorious with sqrt(8.8-4.4) = sqrt(48) = 7 soldiers. Red then continuous to battle the remaining Blue force of 6 soldiers, again victorious with sqrt(7.7-6.6) = 3.6 units left.

This model is simple enough, and relatively well-known, but as I was reading through the article, I was wondering what the underlying mathematical model was; exactly what assumptions were being made; and how it could be applied to some issues of games design. Often, the assumptions stated are along the lines of: 'What's important is that all units involved have an equal opportunity to kill or to be killed'. This seemed a bit vague to me, so I thought I could quickly reconstruct the mathematics myself by applying some discrete probablistic Monte Carlo model, but never arrived at the square law. So, I was even more intrigued about the underlying mathematics.

Theory

A quick search on the net provided me with the following article (2006), which explains the mathematical model underlying Lanchester's model (download pdf from http://arxiv.org/abs/math.HO/0606300 - Warning: this article uses differential equations, so if you're not familiar with those, stay away :-)). The fundamental assumption is as follows: During a unit of time, the number of casualties lost on side A, is proportional to the number of troops on side B. Or, in other words, if your opponent would have twice the number of troops as he has now, he would inflict twice as many casualties on you. The number of casualties inflicted scales linearly with your troop number. Is this is a valid model? The idea is that ALL of your troops add to the casualties inflicted: every tank/soldier/plane shoots at the enemy all the time during the battle. No troops are held in reserve, no troops are kept away from the fight.

Now, over time, both sides will lose troops, linearly proportional to the number of troops on the other side. By integrating the underlying equations, the square law as stated above is achieved. More specifically, the square law states: R(t).R(t) - B(t).B(t) = constant, with R(t) and B(t) being the number of troops at any point in time t. Since the constant value will never change, the sign of the difference never changes, and the strongest side always wins. Since the equation holds at time 0 (when the battle starts), we can say that for any point in time:

R(t).R(t) - B(t).B(t) = R(0).R(0) - B(0).B(0)

When the battle ends at time tfinal (and without loss of generality, we assume R is the biggest force, and hence Red wins), we have:

R(tfinal).R(tfinal) = R(0)*R(0) - B(0).B(0)

or

R(tfinal) = sqrt(R(0).R(0) - B(0).B(0))

The interesting thing is, that is you changes the assumptions, a different model comes out. Suppose that the number of casualties inflicted over a unit of time is not proportional to your strength, but is a fixed number. E.g. you have 100 soldiers, but only 5 are ever engaged, and when these are eliminated, others step in. Your actual number of 100 soldiers is not relevant for the casualties inflicted, but only the 5 you throw in the fight during a given time unit.
Using similar principles, the following equation is arrived at:

R(t) - B(t) = constant

or when the battle ends:

R(tfinal) = R(t) - B(t).

In other words, Red is still victorious, but only with the difference of actual numbers in troops left. Hence the linear comparison of strengths, not the quadratic comparison.

So, in order to keep as many troops alive at the end of the battle, you should throw in ALL of your force (square law), and not enter the battle piecewise (linear law).

A quick example:
Suppose Red attacks a force of Blue of 6 with 10 soldiers.

• According to the quadratic law (ALL units are engaged in battle all the time and inflict casualties), Red wins with sqrt(100-36) = 8 soldiers left. Tactically, this means all 10 soldiers of Red are thrown at all 6 soldiers of Blue in one big attack.
• According to the linear law (fixed number of casualties per time unit, e.g. units pick each other off 1 by 1), Red still wins, but only with 4 soldiers left. Tactically, this means that e.g. both sides send in 3 soldiers first (they eliminate each other). Then, both sides send in the next 3 (kill each other as well). Red remains with 4 soldiers. Note that the fixed number (3 in this example), is rather irrelevant (the example works with any number small enough).

Tactical Consequences

As stated above, the main advantage of the square law is that you can defeat a bigger force by trying to split the force in half. Would this also work in case of the linear law? Working out a few examples show that it doesn't. There is no advantage in cleverly outmanoeuvring your opponent - at the end of the battle the linear difference is always the result. But it does pay off to do that when the square law is in effect.

Now, let's go to the wargaming table. Our unit of time is the turn. So, during a single turn, can you inflict a number of casualties equal to your strength, or only a fixed number of casualties irrespective of your strength? And will it affect your tactics?

• Suppose the number is fixed (linear law): clever manouvring doesn't help (see above). The only thing you can do to gain an advantage on top of the linear difference is to try to prevent your opponent from killing his quotum in any given turn. How can you do that? By making it more difficult for your opponent to hit you (e.g make use of terrain and go into cover).
• Suppose the number is proportional to your strength (quadratic law). Now, if you are the weaker force, you have every interest to try to outmanoeuvre your opponent and hitting with everything you can on small chunks of his force. So taking the initiative is much more important than when the linear law would be in effect.

Whether we are working in one model or the other is governed by the rules system. So, the rules used might affect your tactics on the wargaming table.

Application to unit activation

Let's consider 3 broad categories of unit activation in wargaming rules:

1. All units are activated during your turn all the time (classig IGO UGO).
2. Only a FIXED number of units are activated during your turn. Examples are card activation such as Memoir44/Battlecry: typically on average, you can expect to activate 2 units, irrespective of the total number of units in your force. Or e.g. in rules like Black Powder or Blitzkrieg Commander (roll against command value to activate a unit), you can activate an (expected) fixed number of units per turn per commander. It doesn't matter whether the actual number activated is by itself a stochastic variable - what does it matter it that it does not depend on the total number of units in your army.
3. A number of units PROPORTIONAL to your total force is activated. E.g. in a card-driven system, you can vary the # cards played, or change the content the single card you can play during your turn. When using rules that use general activation as the central mechanism, command quality of generals or the number of generals should scale with the number of units. However, it should remain proportional over the course of the battle. E.g. if you lose units, the number of units you can activate in one turn should scale down proportionally (less cards can be palyed, generals should also be lost, etc...). Otherwise, if this last thing doesn't happen, you're really in case b.

In case 1, the quadratic law applies: all units can attack in every turn, and you can expect to inflict casualties equal to the number of troops. So, manoeuvring and splitting the opponent's force is a valid tactic.

Case 2 results in the linear law. It doesn't matter that much how many units you have, the stronger force usually will win (everything else being equal, of course). In a game such as Memoir44 this is very obvious: since you only can activate a fixed number of units each turn, your excess units are actually a reserve - attacking in force and using your numerical advantage is often not possible in the sense of the quadratic law.

Case 3 is again the quadratic law, providing the amount of units that can attack each turn scales down linearly as you lose units. Keeping the same #cards (or quality), or keeping the same #generals (or quality), actually moves you to case b.

The analysis above makes of course abstraction from other factors: initial setup, the amount of units that can be thrown at the enemy given their dispositions at the start of the turn, etc. But it's always possible to make the analysis on subparts of the battlefields.

Conclusion

If you want to write a ruleset that rewards manoeuvring, be sure the rules allow to inflict as many casualties proportional to the number of units. If you want a ruleset that puts less emphasis on manoeuvre, and more on clever use of terrain, use a fixed number of units to activate every turn.

I know this is a long piece of text, but I hope that some game designers did find this useful. I was mostly intuitively familiar with most of the concepts outlined here, but having written them up concisely actually clarified my thinking a bit.