Will adding a peasant to a stack ever make it weaker overall?
Consider a stack of size N of unknown composition. We now add to it a
single peasant.
Defense
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[1] What's the effect of that addition on *one* attack against the stack?
* If the attack comes against the peasant, then clearly the stack N has
benefited, because it will not lose a member in that attack.
* If the attack comes against some member of stack N, then clearly N
is no worse off than it would have been without the peasant (since
without the peasant the same thing would have happened).
So defensively adding units is always a benefit against one attack.
Offense
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What's the effect of the addition of the peasant to an attack by the
stack?
* Without the peasant, the stack attacks with an average attack
value of A/N, where A is the total attack value of the stack.
* With the peasant, the stack attacks with an average attack
value of (A+1)/(N+1). If A/N > 1, then (A+1)/(N+1) < A/N.
So the addition of a peasant reduces the effectiveness of a single
attack (except in a stack whose average attack factor == 1).
Overall
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Is there a case where the attack detriment outweighs the defensive
bonus?
The defensive bonus is porportional to N/N+1. Therefore the
defensive bonus is minimized for large values of N.
The attack detriment is roughly porportional to A/N. The detriment
is maximized when the average attack value (A/N) is high.
The balance of these two factors is worst when there are many more
attacks than defenses (because the attack detriment will apply many
more times than the defensive bonus). This implies a battle where
N is much larger than the opposing side.
Therefore, we expect the worst case to be when a peasant is added to
a large stack (minimizing the defensive bonus) with a high average
attack value (maximizing the attack detriment) fighting a small
stack (maximizing the ratio of attack detriments to defense bonuses).
Accordingly, let us consider the case of 20 nazgul vs a superdragon
with (5000,5000) statistics. (The superdragon is necessary because
even a dragon gets wiped out thoroughly by 20 nazgul.) What happens
when we add a peasant to the nazgul?
With a trial of 10000 battles:
20 nazgul 90.83%
20 nazgul + 1 peasant 90.71%
So, assuming that these results are statistically reliable, the
addition of 1 peasant does slightly reduce the effectiveness of
a stack of 20 nazgul.
Note that these arguments (and the above results) are for the
*existing* combat system. The flaw of reducing a stack's
effectiveness by adding to it exists in the *existing* system.
The 4:1 rule does magnify this flaw:
20 nazgul 50.29%
20 nazgul + 1 peasant 49.77%
Conclusion
----------
Adding a peasant to a stack can reduce the stack's overall
effectiveness in the current combat system. This is magnified if the
4:1 rule is in place. However, even in extreme cases the effect is
very small.
(Someone else can do the analysis of the smash system, I'm getting
sleepy :-)
-- Scott T.