I guess, the first analysis i would be prone to do, is to determine the expected value of a battle between Fleet A {two ships with 1 min_hits_to_die and 1 weapons ea] vs. Fleet B [one ship with 2 min_hits_to_die and 2 weapons].

This is based on the premise that cost^2 = (weapons)*(min_hits_to_die), or in other words, that we should be building ships that maximize the value of (weapons)*(min_hits_to_die)/cost^2.

The reasoning for that formula is that if we double the cost of a single ship, we could instead double the size of the fleet (here we neglect ramp-up cost for fleet building, and some other factors). Doubling the size of the fleet, of course, doubles

*both* it's total damage (# of weapons) and total structure -- here modeled by (min_hits_to_die) * (ship count). So if we double the size of the fleet we gain a factor of four in total strength (2x for offense and 2x for defense).

If we are trying to get the same effect from a single ship (or rather by improving each ship in the fleet, instead of making the fleet bigger) we need to either double both its structure (min_hits_to_die)

*and* double its number of weapons OR multiply either structure

*or* weapons by a factor of 4. This is where the square of cost in the formula comes from. Sorry if you've already been over this, i only skimmed the thread, but it can't hurt to mention this again, in any case.

Fleet A {two ships with 1 min_hits_to_die and 1 weapons ea] vs. Fleet B [one ship with 2 min_hits_to_die and 2 weapons]

Bout one:

Fleet A lands two hits, killing Fleet B.

Fleet B lands two hits. One ship in Fleet A dies, Second shot has 50% chance to kill the second ship in Fleet A.

Expected value: Fleet A 75% losses, Fleet B 100% losses.

Will continue analysis later! Sorry if i'm going too slow or anything with my analysis. I think you've been thinking about this a while longer. I will just say what i'm thinking, but feel free to ignore it if you are already beyond this

Ultimately would like us to be able to answer your question

How much structure do you have to add if you are fighting the double amount of ships/weapons to achieve the same level of damage? So in the examples: how much more structure do the 5er fleet's ships need in order to shot as many shots as the 10er fleet.

Edit 1: Corrected 50% --> 75% in expected value above.

Edit 2: Next i'd like to take a look at what happens if we increase the size of both fleets by a factor of, say, 100.

Fleet C [200 ships with 1 min_hits_to_die and 1 weapons ea] vs Fleet D [100 ships with 2 min_hits_to_die and 2 weapons ea]

Bout One:

Fleet D lands 200 hits. (I'm starting with Fleet D's hits on Fleet C because that seems like an easier calculation). For each ship is Fleet C, its chance of survival is (chance of a given shot hitting some other ship)^(number of hits) = (199/200)^200 = .367 = 36.7%. The other 63.3% of Fleet C is destroyed in Bout One. So the expected value of ships left in fleet C after bout 1 is 200*.367 = 73.4.

Fleet C lands 200 hits. The chance of a Fleet D ship taking no hits can be calculated similarly to our last calculation: P_0 = (99/100)^200 = .134 = 13.4%

Now we will calculate the chance of a Fleet D ship taking exactly one hit:

P_1 = (number of shots)*(chance of that shot hitting)*(chance of every other shot missing) = 200 * (1/100) * [ (99/100)^199 ] = .271 = 27.1%.

The remaining 59.5% of Fleet D is destroyed in Bout One.

Bout Two (If I was being more rigorous i should really carry the whole probability distribution through to this step, but i'll just see what happens to the expected value case, at least for now):

Fleet C [73 ships with 1 min_hits_to_die and 1 weapons ea] vs Fleet D [13 ships with 2 min_hits_to_die and 2 weapons ea + 27 ships with 1 min_hit_to_die and 2 weapons ea]

Fleet C takes 80 hits. Chance to survive = (72/73)^80 = .332 = 33.2%. Total survivors = .332 * 73 ~= 24

Fleet D receives 73 hits. For the 27 ships that only have 1 min_hit_to_die left, chance of survival is (39/40)^73 = .158 = 15.8%, leaving .158 * 27 ~= 4 ships left of these 27.

For the other 13 ships, we get 13 * P_0 = (39/40)^73 * 13 ~= 2 ships with 2 min_hits_left_to_die

and P_1 = 73 * (1/40) * [ (39/40)^72 ] = .295 chance of taking exactly 1 hit, leaving another .295*13 ~= 4 ships with 1 min_hits_left_to_die from the group of 13.

Bout Three:

Fleet C [24 ships with 1 min_hits_to_die and 1 weapons ea] vs Fleet D [2 ships with 2 min_hits_to_die and 2 weapons ea + 8 ships with 1 min_hit_to_die and 2 weapons ea]

Fleet C survivors: 24 * [ (23/24)^20 ] ~= 10

Fleet D: injured ships surviving: 8 * [ (9/10)^24 ] ~= 1

uninjured ships unhit: 2 * P_0 = 2 * [ (9/10)^24 ] ~= 0

uninjured ships surviving with one min_hit_to_die: 2 * P_1 = 2 * [ 24 * (1/10) * [ (9/10)^23 ] ~=0

Final results : 10 ships from Fleet C survive (95% losses), 1 ship from Fleet D survives with 1 min_hit_to_die (99% losses)

Phew! I doubt i made it through all that without errors. I'll try to look through it again later, or maybe dig up my old college combinatorics textbook and try to remember how power series work or something, lol!