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...