Gripe: Recon-Med Armor Costs

[rosebrallier]

Costs of these armor rigs is out of proportion with both the current starting wealth of the classes that use them and the amount of protection they offer.

[bleehu]

I'll take this one. Could we also add future gripes to the gripe project board? It helps me keep track of what our burndown looks like.

[bleehu]

@Turtlelord26, I could use some help with the advanced statistics. I believe if you're shot at 20 times and make 20 armor rolls, the probability of hitting armor is a normal distribution with mean=50, and standard deviation dependent on the number of shots, but I can't remember what relation gives the st.dev.

Regardless, the jump in protection from 40% to 50% after more than 6 incoming shots is staggering. I think the solution would be to narrow the spread of coverages.

Or will this problem disappear entirely with static armor, and I shouldn't bother?

[Turtlelord26]

First of all, if there's a meaningful statistical analysis we can do here, we should do it for both old & new armor if only to make sure the new system doesn't break anything with unintended consequences.

For the record, the Variance of a normal distribution = E[X^2] - E[X]^2 (I had to look it up, for shameTM). In this case, for X is a uniform distribution on integers from 1-100 inclusive, the variance is 883.5 and the st.dev. is about 29.7237.

Long story short, I suspect you're overthinking it. Since every incoming shot is an independent event, I believe it really is as simple as saying a 40% coverage armor expects to block 40% of shots, and a 75% coverage armor expects to block 75% of shots. Also, your problem statement isn't super parsable to me at the moment; I may need to know more about what you mean by the probability of hitting armor after setting up a scenario of many shots fired.

Mathness starts here. For one shot, we have a fairly ordinary Uniform distribution on Integers from 1 to 100 inclusive. Lets take an armor with coverage C ϵ N <= 100. Straightforwardly, one shot against this armor has probability C of being blocked. So let's look at a unit of n ϵ N shots. We're interested in how many of these shots hit our armor (I've assumed), so let's define our variable B to be the number of blocked shots. B is somehow distributed on integer values between 0 and n inclusive. B looks an awful lot like a Binomial variable to me. At this stage, we could use the Central Limit Theorem to say that a distribution of many averages of samples of B will be approximately a normal distribution. I don't see a need to though, since it won't change our average number of shots being blocked. That average is nC, as stated above. The internet has some binomial distribution calculators that show us the variance of B. As long as n is usefully large, we see the variance is largest at C=.5 and gets somewhat smaller as C gets farther away from that middle, and the distribution always skews inward. Example for n=50: C=.9 -- var 4.5 C=.8 -- var 8 C=.75 -- var 9.375 C=.6 -- var 12 C=.5 -- var 12.5 Note that massive variance dropoff doesn't happen until after 80% coverage, a range that isn't relevant to current armors (we know what happens with heavy armor at 100% coverage). The takeaway here, I think, is that armor performance on average scales linearly with coverage, but that heavier armors are slightly more consistent in their performance.

For production statistics, this took me an hour and a half, two thirds of which was also overthinking, and caused 8 tabs of chrome to be opened.

[Turtlelord26]

Static armor will also lend itself to an analysis with Binomial distributions, but the starting conditions will be more complex.

[bleehu]

Ah crap. I think you're right. I was overthinking it. And I'm not sure we can expect the players to take enough bullets for the binomial distribution to approximate the normal distribution in the first place. I'll continue with my price analysis on armors using a linear value scale for coverage. I don't think your hour and a half of analysis was a waste; I think we'll come back to this math several times in future. Going forward, I'll make sure not to ask for this level of stats unless the benefit outweighs the work for sure.

[Turtlelord26]

Stats are fun! In fairness I think I gave you a bit more than you really asked for.