Analyzing shields vs Armor in FTIII

As part of a continuing series, here is an analysis
of shields vs armor in FTIII.

First, we have to calculate how much damage is expected
from one die of beam damage versus no shields, one
shield, and two shields.

With the reroll of 6's, calculating average damage
requires calculating the limit of a series.

How much damage, on average, is done by the reroll?
Call this amount 'x'.
x = 4/6 + 1/6*x, or
5/6 x = 4/6
x = .8 points.

So, 1 dice of beam damage can be expected to do
4/6 + .8/6, or 4.8/6, or 4/5 versus no shields.

Versus 1 shield, 1 dice of beam damage averages to
3/6 + .8/6 or 19/30 or .63333

Versus 2 shields, 1 dice of beam damage averages to
2/6 + .8/6 or 7/15 or  .46666

24/30, 19/30, 14/30 is the progression.

A ship with 2 shields and 14 Structure/Armor
would have same beam resistance as a ship with 1
shield and 19 Structure/Armor as a ship with 0
shields and 24 structure/armor.

However, a final calculation of the efficiency of
shields depends on the size of the ship they are
installed on and the ratio of beam weapons to weapons
which ignore shields their expected opposition has.

One thing that quickly becomes obvious is that shields
work a _lot_ better when a high percentage of the
ship's mass is devoted to Armor and Structure.

Consider:  A Mass 60 ship with a certain amount
of Mass devoted to Armor, Structure and Shields
compared for the different shield levels.
Beam resistence for:
30 Mass ASS, no shields (30/.8)  37.5 Beam Dice
30 Mass ASS, 1 shields (27/.633) 42.6 Beam Dice
30 Mass ASS, 2 shields (24/.466) 51.4 Beam Dice
20 Mass ASS, no shields (20/.8)    25 Beam Dice
20 Mass ASS, 1 shield  (17/.633) 26.8 Beam Dice
20 Mass ASS, 2 shields (14/.466)   30 Beam Dice
15 Mass ASS, no shields (15/.8)  18.7 Beam Dice
15 Mass ASS, 1 shield (12/.633)    19 Beam Dice
15 Mass ASS, 2 shields (9/.466)  19.3 Beam Dice

If one is going to go for shields at all, it usually
pays to have two of them.

It is a bit complicated figuring out a formula for
calculating when shields are effective in the face
of an enemy with shield penetrating weaponry.  If
an enemy fleet has over half its weapons' mass in
shield penetrating weaponry, it probably isn't worth
buying shields, but the precise balance is affected
by the balance of interceptors, the level of skill of
one's opponent in placing salvo missile bursts, the
average thrust of your fleet, etc...

Michael Sandy