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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