$Find H $\langle H \rangle=P+\frac{1}{AD} \sum_{i=0}^{D} ( \sum_{j=0}^{A} ((j-i)Step(j-i)))$$

Question

$Find H $\langle H \rangle=P+\frac{1}{AD} \sum_{i=0}^{D} ( \sum_{j=0}^{A} ((j-i)Step(j-i)))$$

Hi all, I was reading a post about how certain stats work in a game. Most expressions where simple, but then there was this one and I am completely stumped on how to do it. After some research I now know how to do the entire expression, Except for the " ( (j-i)STEP(j-i) ))" part! From what i've gathered I believe they are using either the Step (staircase) function or the Heaviside Step function. However even after knowing this I am still unable to get a number close to theirs.
I'm very much so a math "noob." So I understand that possibly this is very simple, this is not even a correct expression or somthing else.
If anyone can help me on this that would be much appreciated! ! !

How it was written in the post:

<H> = P + (1/AD) * SUM(i=0 to D)(SUM(j=0 to A)( (j-i)STEP(j-i) ))

In one example they define A=30 P=5 D=80 with an answer of "~10"
In another they define A=20 P=6 D=75 with an answer of 8.4

My main goal is to get the same outputs he got with the same variable definitions, so I hope thats possible.

Link to post: https://www.reddit.com/r/totalwar/comments/aolarg/applying_math_to_combat_mechanics/

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Ian Todd

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M F H

0

well, parts of this is reproducable but that after giving that formula the OP writes "However, the results are much messy, highly dependent on if A > D or not...not able to solve...". (Side note, the name of the variables [and functions P and H=avg.dam.!] is extremely awkward, and the same letters stand for different things in that formula compared to the other one (for P)...) So I fear that if I accept to answer the question you won't be satisfied.
Erdos

0

Could you please clarify the definition of the step function: Step (x)?
Erdos

0

Do you mean the Heaviside step function?
M F H

0

I think I have found the explanation which I think might be correct, and I'm now able to reproduce the figures -- not exactly but up to a difference of less than 1 unit. Unfortunately, if you consider my answer only partially correct, then it might even cost me money to answer, therefore I'm a bit reluctant to accept to answer the question ...

Answer

The answer is accepted.

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M F H

0

well, parts of this is reproducable but that after giving that formula the OP writes "However, the results are much messy, highly dependent on if A > D or not...not able to solve...". (Side note, the name of the variables [and functions P and H=avg.dam.!] is extremely awkward, and the same letters stand for different things in that formula compared to the other one (for P)...) So I fear that if I accept to answer the question you won't be satisfied.
Erdos

0

Could you please clarify the definition of the step function: Step (x)?
M F H

0

I think I have found the explanation which I think might be correct, and I'm now able to reproduce the figures -- not exactly but up to a difference of less than 1 unit. Unfortunately, if you consider my answer only partially correct, then it might even cost me money to answer, therefore I'm a bit reluctant to accept to answer the question ...

Answer 1

Answers can be viewed only if

The questioner was satisfied and accepted the answer, or
The answer was disputed, but the judge evaluated it as 100% correct.

View the answer

OK, I found an explanation which I believe to be correct, and which reproduces the figures up to a small discrepancy of less than 1 unit, often much less.

First off, let me note (even if it may be trivial...) that for the attack and damage values you have to take those of the attacker, but for defense and armor you have to take those of the defender. (That's quite clearly stated for P(hit) on the reddit page, but less clear for the more obscure <H>.)

So if you define the function:
(I'll use the PARI language which you can copy-paste into the online interpreter at http://pari.math.u-bordeaux.fr/gp.html, but I'll also give the numerical results after comment characters "\\", below)

P_hit( A, D ) = if ( A < D, (A + 1) / D, (2*A - D + 1) / A) / 2.

then we get all the mentioned percentages: (copy-paste 1 by 1 into the PARI interpreter, not all in 1)

P_hit( att=47, def=57 ) \\ = 0.421... = 42% for Principe vs. Hoplite
P_hit( att=34, def=53 ) \\ = 0.330... = 33% for Hoplite vs. Principe
P_hit( att=47, def=51.) \\ = 0.470… = 47% for Principe vs Axeman
P_hit( att=32, def=53.) \\ = 0.311... = 31% for Axeman vs. Principe

Now concerning the main problem, <H>, I think that the O.P. made an error in his formula.
With A = normal damage, P = armor piercing dam., D = armor (although to me it looks weird to use A for Damage and D for Armor, but I guess the O.P. chose this because Damage ~ Attack and Armor ~ Defense...),
I think that the correct formula should be:

H( A, P, D ) = P + 1/A/D * sum( i=0, D, sum( j=0, A, j*(j>=i)))

i.e., my correction to the formula is to use not (j-i)*STEP(j-i), but rather j*STEP(j-i), where

STEP(j-i) = Heaviside(j-i) = [ j >= i ] (Iverson_bracket) = { 1 if j >= i ; 0 if j < i }.

My reasoning/explanation, apart from the fact that this reproduces very well the given values, is that the inflicted damage should be equal to the damage value j (rather than difference : damage j - armor i) once the armor is pierced, i.e.,when damage j > armor i).

With this formula I get: (note, I append a "." to some values to get decimal numbers rather than exact fractions)

avg_dam ( dam=30, apd=5, arm=80.) \\ = 9.1333... for Principe (dam.30+5) vs. Hoplite (arm=80), compared to "~ 10 (just under 5 + 5)" given on reddit

avg_dam( dam=20, apd=6, arm=75.) \\ = 8.0533... for Hoplite (dam.20+6) vs. Principe (arm.=75), compared to "8.4 damage" given on reddit

avg_dam( 30, 5, 75.) \\ = 9.4088.. for Principe vs. Axeman ("delivers an average of 10.3 per hit")

avg_dam( 16, 10, 75.) \\ = 11.360... for Axeman vs Principe ("...11.5 expected damage. ")

So, although this does not reproduce exactly the given values, it is very close (less than +- 1, less than 0.15 in the last case) and the explanation seems quite satisfying to me - and, I hope, to you too !
(I hope I don't have to regret having spent quite some time to find this correction, explanation & values... :-?)

M F H

263

Ian Todd

0

Your an absolute genius, thank you so much!
M F H

0

Thanks a lot!
M F H

0

Oops! actually I had changed the name of the function H() to avg_dam(), but then changed back to H() in the definition, but not in the subsequent lines. Instead of " avg_dam(...) " it should be " H(...) " (or you can define " avg_dam = H " in the PARI interpreter).
Ian Todd

0

That had me hung up for a bit, while inputting it on the PARI interpreter site, but I figured it out.

$Find H $\langle H \rangle=P+\frac{1}{AD} \sum_{i=0}^{D} ( \sum_{j=0}^{A} ((j-i)Step(j-i)))$$

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