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Subject: Are there any mathematical formalisms in war fighting?
jrn44    12/2/2006 6:34:58 AM
As a recent graduate in engineering, I have discovered that no one can truly understand something unless he can describe it in mathematical terms. I have always wondered if this philosophy has ever been applied to war fighting. I realize that war fighting is a highly nonlinear and uncertain process, but the mathematical tools still exist to at least attempt a mathematical theory of warfare. In particular, the tools of stochastic calculus (probability theory with derivatives, integrals, and vectors in it) can be used to generate crude models that predict a well-defined probability for victory in some arbitrary engagement. Does anyone know if this has ever been attempted? Any references? It probably would not be immediately useful from a practical perspective, but at the very least it could help to quantify certain relationships in mathematical terms.
 
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swhitebull       12/2/2006 1:03:08 PM
Check out the name Dupuy Brothers Military on the internet-  these brothers have tried to come up with matrices along these lines;  also, James Dunnigan's (editor of this website)  How to Make War, has a lot of numbers crunching on the subject.
 
swhitebull
 
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HIPAR       12/3/2006 8:44:22 PM
Frederick Lanchester asked the same question and published a paper on the mathematics of warfare during World War One. Now, his methodology is generally considered to be outdated.

Modeling  warfare today is much more complex.  From order of battle, command & control,  operational characteristics of weapons, human factors, geography, climate, weather, battlefield sensors and countless other factors are played into complex computerized simulations.  Modeling asymmetrical warfare is especially challenging.

---  CHAS


 
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jrn44       12/6/2006 8:44:13 PM
I have read Dunnigan's book, and the "numbers" in there are not quite useful.  It does not make much sense to assign some unit of combat power to an army and expect victory just because A > B.  Since war fighting is so chaotic, it makes much more sense to assign a probability of victory rather than think of it as a deterministic outcome.  Certain situations actually lend themselves to probabilistic modeling very well.  As a hobby, I have been toying around with this idea and I am wondering if no one has ever thought of this before.  It actually leads to some interesting outcomes that mathematically prove certain intuitions (example:  All things being equal, the probability of victory is greater for the unit which fires the first shot). 
 
I am curious about the computer modeling comment.  Does anyone know if there are any references on how this is done?  I have extensive experience with numerical modeling to solve problems in electromagnetics, so I am curious what the equations of war look like. 
 
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Jeff_F_F       1/2/2007 12:29:56 PM
This should work every time : Probability Blue Victory=(((Blue Force Strength*Blue Tactics) / (Red Force Strength*Red Tactics))* Terrain Coeficient) / (Murphy's Law ^ (Distance/Time))
 
The difficulty is figuring out what numbers to use. If you use the right numbers it works perfectly, otherwise you're out of luck.
 
As to Dunnigan's numbers (not related to my equation above, btw), I assumed from the number of times he emphasizes the Murphy's Law factor that he had assumed that the reader was not to take the numbers as deterministic.
 
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Jeff_F_F       1/2/2007 12:39:54 PM
Correction to my equation above : the exponential increase of Murphy's Law is by Distance/Time (that is, Distance that must be covered/Time available to do so) only for offensive battles. On the defense, it is Time/Distance (that is, Time that the enemy's advance must be delayed/Distance available to trade for time).
 
...taking a joke too far...
 
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flamingknives       1/12/2007 3:51:24 PM
I don't know, I can't understand anything in mathematical terms unless I can relate it to a graphical or physical model, and I'm an engineer too. It takes all sorts.

As for mathematical models in warfighting, there are vast numbers of probability-based systems for simulating warfare out in the public domain. They are called wargames. Some are better than others. Some are ludicrously complicated (Advanced Squad Leader), others are somewhat over-simplified (Games Workshop's Warhammer 40k). Others still have a very smooth playing style that make people focus on tactics rather than dice rolls, usually by simplifying things again, but somehow not losing the key features of combat.

Wargames also range from very technical considerations, where tanks, armour and guns come to the fore as these are easily defined, to statistically based rules that attempt to model human behaviour, either individually or en masse.

There are a few generic rules that crop up in military histories that have equations behind them somewhere, usually on the effect of odds.
 
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PowerPointRanger    Going by the numbers   1/17/2007 11:51:14 AM
The problem with treating war as a science is that strictly speaking it is also an art.  (One of only a few fields that can reasonably qualify as both an art and a science).  So what's the difference?
 
A science is something which can be objectively proven and can achieve consistent results.
An art is something which relies on creativity and for which there are no real limits or laws.  There are rules, but these can be skillfully broken to artistic effect.
 
So, for example, if we look at the German invasion of France in 1940:
         According to the science of war, all the factors were in the Allies favor.  They had more aircraft, tanks, ships and troops.  Moreover, their tanks were of superior quality.  Add to this, a heavily fortified boarder and terrain suited to the defense.  According to the science of war, it should have been a decisive Allied victory.
 
So how did the science of it fail?
       There are many important factors in war which are difficult to quantify: cohesion, surprise, leadership, morale, training, tactics, logistics, technology, etc...
       It is long-established that these things can be "force multipliers" (or dividers), yet the degree to which they multiply or divide can be rather nebulous.  In our France '40 example, the Germans had superior training, tactics, leadership, and cohesion, plus the element of surprise on their side.
 
So what use is the Science of War if it can't acculately predict outcomes?  It's still useful as a guide for the relative allocation of resources and application/economy of force.  For example, to put down a guerilla war, historically has required a 10:1 force ratio.  Can you be successful with less?  Yes, but you would have to have some of those wonderful intageables in your favor.   And even if you do have the 10:1 ratio, there's no guarantee it will be quick or easy.  It could still take years and be "untidy".
 
 
 
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SGTObvious       1/18/2007 9:49:07 AM
It's not chaotic.
 
It is, however, detailed beyond the ability of any model to, well, model.
 
This means, inevitably, any model will diverge from real world results.  When you go back an examine why, you will usually find that it diverged because of a factor the model designer did not account for, and might have been very deliberately ignored for purposes of simplicity. 
 
I could offer here a long and rambling post on information theory, simulation, and mathematics, but to sum it up, there is a respectable body of physicists who believe that there is a given maximum information content to a given unit volume of space, and the smallest volume of space capable of containing enough information to accurately model the universe is, in fact, the volume of the universe. 
 
And it has been recognized for centuries that the smallest factor can influence the outcome of a battle ("For want of a nail...") it is forever beyond our hope to perfectly model a military situation.
 
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