Morlic wrote: ↑Tue Sep 10, 2019 5:10 pm

Without doing proper math, my intuition is as follows: One relevant number seems to be the expectation value of "wasted shots", i.e. number of shots which hit an already destroyed target.

This seems like a good approach.

Assuming we start with equal (hitpoints*damage), after first round, whoever wasted more shots is now behind in the (hitpoints*damage) metric and will subsequently lose. By this naive reasoning, we would only have to calculate the first round to estimate the winner if (and only if) the discrepency between wasted shots is high.

In your example, fleet C wastes 26.9% and fleet D wasted 36.7% of shots. So fleet C is expected to win.

I'm still a bit concerned here that there are at least two additional factors at work that may need to be considered:

- At the start of the first bout fleet D is composed entirely of ships with 2 min_hits_to_die, whereas in subsequent combat rounds it has some ships with varying structure values. Because the composition has changed, the expected fraction of wasted shots will also change. (In round 2 C lands 41/73 shots, wasting 43.8% and D lands 56/80, wasting 30%). I think what you are saying is that the first round imbalance is going to tip the scales enough that future rounds don't matter, but that brings me to the second consideration:

- When you talk about wasted shots, another way to think about it is that any shot that doesn't actually kill a ship is wasted. Your metric of wasted shots is that any shot that doesn't reduce structure is wasted. But the other metric is that any shot that doesn't reduce the number of weapons is wasted.

In this example, in round 1, Fleet C's size was reduced to a factor of 36.5% (73/200), while wasting 26.9% of it's shots (by your shots wasted metric). Fleet D's size was reduced to a factor of 40% (40/100), while wasting 36.7% of its shots. The reduction in fleet size corresponds to decrease in offensive capacity (# of weapons), while the number of wasted shots corresponds to a decrease in defensive capacity (structure/min_hits_to_die) of the opposing fleet (i.e. if Fleet X wastes fewer shots, Fleet Y will end up with less structure).

In this example, the discrepancy between ship losses is quite small compared to the discrepancy in wasted shots.

I think i'm not really structuring well what i'm trying to say (i've tried rewriting this three times now, so i may just settle for this for the time being), but basically consider Fleet E [100 ships 1 min_hit and 1 weapon] vs a single ship with 1 weapon and 10,000 min_hits. Here no shots are wasted, but obviously the single big ship prevails easily (in a really long fight). I think this kind of thing will come into play in calculations as well.

This may be what you were getting at here:

In current balancing, structure comes cheaper than damage so asymmetric designs are often favored. Now, when total damage >> total structure, the number of wasted shots in the first rounds are negligible. Instead, we find that the fleet with higher ship count takes earlier losses which reduces the firepower while the tankier ships keep full fire power.

I didn't quite understand that part.

With the number of ships and thus shots (i.e. the number of samples) growing, the probability distribution of the hits taken for a single ship in the first round converges to the poisson distribution, i.e. it is eventually defined only by the expectation value (= number of shots / number of targets). The expected number of "wasted shots" is given according to this very probability distribution (and the parameter of ship hitpoints). Therefore, when comparing N ships with N shots and 1 "Hitpoint" vs. N/2 ships with N shots and 2 Hitpoints, it shouldn't really matter if you are looking at N=200 or N=1000 or N=1e6 ships: The ratio of "wasted" shots remains equal and thus after first turn, there will be a significant enough imbalance to decide the battle. "Unfortunately", with relatively small fleet size, the choice of N does actually influence the outcome significantly as your example for N=2 shows (as is expected - the poisson approximation is not valid in that regime).

Agreed. That first analysis i did was to test the difference between very small fleets and larger fleets like that, but i hope to do other analysis varying different parameters to see if there are other statistical phenomena that emerge past my limited current horizon of understanding. I think it still may be pertinent to look at what value of N we need to look at for this smoothing to become "enough". Clearly it's somwhere between 2 and 100.

My apologies if this post lost a bit of steam at some point. I had some things i wanted to say, but it didn't seem to come out quite right. Still some decent material here, hopefully.