03 December 2010

Blue versus Gray, or maybe Federation versus Rebels?

"Mr. Business" left me a good question on my last post about Gratuitous Space Battles, and it deserves a good answer.
N.B. writes: As an alternative, what about special anti-fighter weapons? Perhaps a frigate type ship that can act as picket against fighter waves. Specialized weapons can overcome numbers in many cases.
Like American Civil War. 100 Grays with rifles vs 3 Blues with a Gatling gun. On the open fields of a typical A.C.W. battlefield, your 3 Blues are going to win.
To the first part about specialized weapons, GSB lets you do just that. There are anti-fighter missiles and tractor beams to help shoot down those pesky little space-piranhas. Of course, big ships can carry heavy armor that is nearly impenetrable to fighter weapons, so it helps to have a few big ships with big guns to soften up those hard targets. Real battles are complicated, with many factors influencing the outcome. Lanchester's Laws are a simplification down to the bare basics of the number of men and how effective the are at killing the other guys. This demonstrates a principle of warfare; the advantage of superior numbers and why it exists, but it doesn't begin to cover all the possibilities that real situations allow.

The second part though - the Civil War example - is something that can be described nicely by Lanchester's Law (squared law). I'll change the number just a bit for my convenience though.

Suppose we start off with equal sides: 100 Blue and 100 Gray soldiers, each side equivalently armed with Rifle Muskets, and each able to shoot freely at any soldier on the other side. These sides should be approximately equal, with 100 soldiers on one side going to receive about as many casualties as they deal out to the other.
A game-like example will be useful: If we set this up as a game played in rounds, and give each soldier a 2% change of killing one of the other soldiers each round (completely arbitrary, just bear with me), then each side will lose about 2 soldiers each round (at the start) and the winner is probably just a matter of chance.

Now lets change it up - lets give the Blues Gattling guns (or maybe Uzi's?), and lets say that these BIG guns inflict casualties on the other sider 25 times faster that the old muskets of the Gray's. Now the Blue team is 25 time more effective, and therefore 100 Blues should now be equal to 25 times 100 equals 2500 Gray soldiers. Right?

Nope, 'fraid not, at least not according to Lanchester. The Squared law tells us that the advantage of greater number is proportional to the square of the ratio of the number of soldiers on each side. The Gray's can balance the Blue advantage with just 500 soldiers, not 2500, because a 500-to-100 is a ratio of 5, and 5 squared is 25. The reason behind this is that ever time Blue suffers a single casualty, they lose a greater portion of their total firepower.

Back to my game-like example: If each Blue soldier now has a 50% chance of killing a Gray soldier each round, and the Grays are still at 2% but there are now 500 of them, then there should be about 50 Gray and 10 Blue casualties in the first round. Now the ratio is 450 to 90, still 5-to-1, and the sides are still balanced. Under the differential equations of Lanchester's Laws, these to forces will grind each other down always at this 5-to-1 ratio until both sides are dead.

Now if this game has any bit of randomness, or as in reality things are not always equal, that ratio will not stay at 5-to-1 for very long. Before many rounds of fire are exchanged one side will gain a slight advantage, and that advantage will expand in to a victory without killing all the soldiers on both sides. If Gray bring 600 soldiers instead of 500 they should win this game every time, and this still isn't half way to 2500.

We could play this game differently too. Instead of giving Gray more soliers we can take away some of the Blues. With their BIG guns, just 4 Blues should be equal to 100 Grays. Now the difference in the casualty ratio really shows, because after just one round Blue will have 2 soldiers left, and Gray will still have 98. 20 Blue to 100 Gray should be about even (but don't hold me to that, because I make mistake when I'm tired and do math in my head).

That said, I haven't really done justice to N.B.'s Civil war example. There probably were Civil War examples of 100 Confederates charging a Union Gattling gun and getting torn to peices. As I said before, it's complicated! I'm no student of Civil war history, but I know it was a time of changing technology and tactics. A few decades earllier and 200 musket-wielding Grays charging straight at 100 similarly armed Blues was probably a winning tactic, but 200 Grays charging 100 Blues - who by-the-way have a few of Mr. Gattling's latest inventions - I think that might not work out so well for the Grays.
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 Ashley said...

All of this fits in rather well with army SOPs about force ratios for the attack and the expected casualties ratios, which are linked together. These SOPs give rules of thumb about defense factors etc, and IIRC at 4:1 one expects about 10% casualties when over running the opposition.

J. Merton Walkley said...

It may seem like fun and games to you, but wait until they become sentient and come after your family and job.

EastwoodDC said...

@ J. Merton: LOL! Who says they haven't already?
Interesting blog you have there too: http://jmerton.blogspot.com/

@ Ashley: This is in-part how they come up with those force ratios, along with a fair amount of wargaming and other methods. At a 4:1 ratio the Squared laws gives 1/16 or 6.25% casualties, but in reality most battle have some "linear law" component and you don't do quite as well. You can approximate this by taking the ratio to a power a little less than 2. For instance 4^1.7 = 10.55 or about 9.5%, which is pretty close to your recollection. The approximation value "1.7" is close to the value found in some analysis of historical data, but it's not a "law". I practice, a 4:1 ratio is probably enough to nearly guarantee no more than 10% causalities.

EastwoodDC said...

PS: Ashley has an interesting blog too!

Anonymous said...

Interesting series of posts - linked here from Starship Combat News or I'd never have seen them. If you can find a copy, you might want to look at Steve Jackson's ancient Space Gamer article "The Fuzzy-Wuzzy Fallacy" or better yet, his whole book on Game Design: Theory & Practice. Has some reasonably cogent discussion of the principles you're talking about, as well some insight on how to assign combat statistics to game elements, both real and fictional.

nunya business said...

As noted this gets complicated. Be it Battletech or Space Fighters, or anything beyond single shot rifle volleys of the 19th and earlier. When Target accusation, Rate of Fire, and lethality are introduced the basics need to be expounded on. What ratio is needed in order to succeed in assault can be modified by the defenders capbilites. If a defender can engage you at long range with high rates of lethal fire you will need a higher ratio and/or other weapon systems to help dampen the effects of that defensive fire. To take the Civil War example a bit further, A Blue force holding ground with Gattling Guns. You would either need to be willing to send a lot of guys to their death to overcome the large rate of lethal fire or, find a way to silence those guns before you rush the Blue positions. In comes Artillery. Delivering equal or better lethality at a safe distance and negate the Blue force multiplier. Even though artillery rate of fire is lower its range and lethality are both high and make it superior.

As technology progresses, how much you can kill is sometimes as important if not more important then how much you have. Another example is Rorkes Drift a famous battle of the Zulu Wars. The movie Zulu is excellent and based loosely off that battle. For my purposes the Movie’s account is not historically accurate as to what happened but the battle scenes are an accurate enough depiction of that kind of warfare so Ill use it. 4500 Zulus attacked 140 A ratio of ~32:1 in the Attackers favor. Yet the British held after several days of fighting. Why? Because the British weapons were Rifles and Zulus were almost all spears. The Zulus had to rush open ground in full view of swift volleys of long range rifle fire. Range, lethality and British will beat and vastly numerically superior force whom by appeared quite willful on there own right.

EastwoodDC said...

@Anon: Alas for my long gone collection of the Space Gamer (complete from issue #15 too, *sob*)! I think my copy of Steve Jackson's Game Design book must have been lost with it - BUT - I see it's available for just $8 as a PDF. SOLD!

The Fuzzy Wuzzy Fallacy I remember, but had all but forgotten. Fortunately it was republished in The Ogre Book, which I still have - and read again last night.

@Nunya: The situations you are describing go beyond the simplifying assumptions of Lanchester. Recall this assumes every unit can attack any other, ignores movement, morale, ammunition, and a host of other factors that definitely matter. If one side is completely unable to respond, then the advantage of the other side in infinite. The "Fuzzy Wuzzy Fallacy" that Anon mentioned is similar situation, where fast-moving units use hit-and-run tactics to avoid effective retaliation. The conditions that allow such one-sided battles can be very situation-dependent, and may not lend themselves to this sort of simplified mathematical modelling - At least not at this scale. Historically, such entirely one-sided battles generally do not occur, or are not remembered.

Despite the limitations of the Lanchester model, these methods appear repeatedly in the Miltary Science literature over the last 50 years, and continue to be used extensively in military wargaming and simulation models.

nunya business said...

Yes, I do take a more involved look that perhaps was not intended in your posts. The mentions of force scoring led me to make the further comments. Lanchester’s ideas have been a foundation expounded on to try to account for the more detailed issues. They are also the basis for many more detailed models that use different procedures. I have been toying around with this sort of thing for some time. I find in making my games that the combat variables and assigning combat value to individual weapons used in the game are very necessary as they can greatly influence force ratio.