Scott responds:
> When the simulation is run with the "no break" rule in effect, the
> addition of a peasant always helps the stack [...]
> Therefore I conclude that the fears that there exists a counterintuitive
> situation where adding units to a stack makes it weaker is groundless.
Arrrgh... the breaking rule. Forgot about that. That does make it difficult
to construct counterexamples.
Without breaking, an additional melee troop seems likely to always help
your chance of winning with the standard model. With 4-to-1, an extra
peasant will probably still improve your chances of winning -- though it
may change the nature (quantity and quality) of your expected casualties.
Missle troops are a different story. Here's a gedanken-experiment for
your consideration. [I hope that Scott can construct some plausible
combats out of it.]
4-in-1 rule, no breaking
========================
Side A: L lions (100 att/defense)
Side B: 1 super-dragon (1000 attack/defense)
M blowpipers (2 attack & defense, 25 missle), behind 9
You want M/L to be large enough so that the 4-in-a-row rule is fairly common.
M/L ~= 4-6 should do it.
It will take (on average) 11 attempts for the lions to kill the dragon. The
blowpipers will kill about 4*11*.25 = 11 lions. Choose values of L & M such
that M blowpipers are a near-even match to L-11 lions.
Then consider what happens if you add rock-throwers (1 att/def, 5 missle)
to Side B. The "good" 25-point missle attacks are replaced with 5-point
missle attacks. Fewer lion casualties are done, and then the lions chew
up the combined weak troops. I *think* there is a point where the decreased
number of missle-killed lions overbalances the rock-throwers value as extra
fodder, but perhaps not. [Simulations anyone?]
4-in-1 rule, with breaking
==========================
Same setup as before. The lions need 11 "meta-rounds" to kill the dragon
to break side B. Side A must kill L/2 lions to win.
M blowpipers will kill ~11 lions before Side A can "expect" to win.
But M blowpipers & M rock-throwers will only kill about
4*11*(1/2*.25 + 1/2*.047) ~= 6.5 lions.
So L=15 and M=100 should show significantly different results for this combat.
The standard combat model doesn't have this *particular* problem, but it
does have other peculiarities associated with the breaking rules.
IMO, critical hits (e.g. smashing) should avoid some of these problems (you
can't get away from strangeness associated with the breaking rules), and come
closer to the results that you want.
Ed Bailey
P.S. Yes, I know that I originally argued in favor of the 4-in-1 rule,
but I've changed my mind now that I've seen the numbers. I do like
the sqrt suggestion. Has anyone run sample combat suite using that
system?
P.S. Actually, I'm a mathematician, not a physicist (of any flavor).
--
Ed Bailey | Voice: (512) 471-4198 Fax: (512) 471-6715
Inst. for Fusion Studies | Internet: bailey@{hagar,ziggy}.ph.utexas.edu,
Univ. of Texas at Austin | u70262@c.nersc.gov, or pnab643@chpc.utexas.edu
Austin TX 78712 | "No pithy quotes. Just email addresses."