Let's continue the conflict between the first group of normal damage ships;
First attack: PlayerA (200) does 20 ships damage (10%) to PlayerB (100), who does 10 ship damage (10%) to PlayerA.
Second attack: PlayerA (190) does 19 ships damage to PlayerB (80) which does 8 damage to PlayerA.
Third attack: PlayerA (182) does 18 ships damage to PlayerB (61) which does 6 damage to PlayerA.
Fourth attack : Player A (175.9) Player B (42.8 )
Fifth attack : Player A (171.62) Player B (25.21)
(Use of non-integer numbers is a minor approximation in terms of damage output as damage obviously does not scale with a unit's health).
Notice the similarity between the remaining number of ships between normal battle after 4 iterations and the double damage battle after 2 iterations? The difference exists only because you are using a discrete method to model the number of ships.
Given that there are normally a large number of ships that all come into range at different times, and fire at difference rates, it is more appropriate to use a continuous model as follows;
Let t be the time after the start of the conflict, A be the number of player A's ships at time t, B be the number of player B's ships at time t, and k be a constant determining the amount of damage the ships in question inflict.
The situation can be modeled by the pair of simultaneous differential equations;
dA/dt = -kB (1)
dB/dt = -kA (2)
Multiply both sides of equation (1) by A, and both sides of equation (2) by B and subtract the two resulting equations from each other, this yields;
A(dA/dt)-B(dB/dt)=0
By the Chain Rule;
(1/2)(d(A^2)/dt)-(1/2)(d(B^2)/dt)=0
Integrating with respect to time;
A^2-B^2=c
Assuming at t=0, A=A_0 and B=B_0;
A^2-B^2=(A_0)^2-(B_0)^2
Note how the constant k does not feature in this equation and hence the total remaining ships after the conflict will not depend on whether the game is in Normal or F&D mode. This can be seen as follows;
When B=0 (They are all dead);
A=((A_0)^2-(B_0)^2)^(1/2)
In the case of your example A = 173.2.
All doubling k (playing on F&D mode) ultimately achieves is to end the battle in half the time, its effects on ship losses are negligible. A similar result holds even when the damage constants for the two different groups are different, as I will show if you wish. This model becomes increasingly accurate as fleet size increases.
There are actually a number of interesting strategic pieces of information that can be gleaned from the above equation, such as the fact that a force's fighting strength is proportional to the square of its size. This rule of thumb is taken into account by military strategists - After the Battle of Trafalgar, Lanchester used this model to demonstrate why Nelson's tactic of splitting the enemy's fleet into 2 was so effective.
Your argument regarding ships going through wormholes being more dangerous on F&D mode is similarly inaccurate.
Your model
can be accurate, assuming two conditions are met (see 1 and 2 below), however, if you continue the conflict to the end you will still find relatively minor differences for changes in the damage rate;
1. All ships start within range of each other and start firing immediately.
2. All ships are the same type and fire at the same rate.
If I had time I could put together a spread sheet to work this all out, (no doubt X4k could tweak his balance analysis program to handle the case to see how different it all works), but there are a lot of situations in the game where this sort of thing happens. I'm sure there are situations where you loose less ships then normal (Ion Cannons are the prime example), but I'll almost certainly expect to loose a lot more on F&D just because of this.
If you do decide to create your spreadsheets, try a smaller and smaller time gap between each iteration - you should find that the value your spreadsheet outputs at the end of the conflict will approach a limit, which is the value for A I gave above.
I apologize if it's difficult to follow the derivation, it's tough to made it particularly legible using only ascii characters - x4000, can we have LaTeX support on this forum?
EDIT: I've just found a page with some more information on this subject if you're interested. The proof I gave above is a simplified version of
Lanchester's Square Law.
