Calculation for train HP or SS (in General)

Wizard'sFirstRuleJune 20 2008 11:10 AM EDT

assumptions (I think they are ok, but won't mirror CB exactly):
1) no ENC
2) no other damage reduction
3) physical damage source (all AC applies)
4) constant amount of damage (it doesn't matter its 20k or 200k or higher, but it stays the same between rounds - unrealistic, but I think its ok for this academic exercise)
5) no "free XP" in the first 20 HP
6) ignore base reduction (only a few hundreds or so, very small comparing to the % reduction of millions)
7) ignore AS - SS is based on trained HP, so this might be relevant

With 100% HP, you can survive:
HP/Damage * (1 - 0.167% * AC) rounds

To train to full SS with x% penalty:
HP = SS * (1 - x%) * 5
but SS = 1 - HP (you either train HP or SS as far as for survival goes)
HP = (1 - HP) * (1 - x%) * 5
simplies to:
HP = (5 - 5 * x%)/(6 - 5 * x%)

With Max SS, you can survive:
[(5 - 5 * x%)/(6 - 5 * x%)] * HP/ [Damage * (1 - 0.21% * AC)] rounds

for the two options to be equal (break-even), the rounds you can survive would be the same, therefore:
HP/Damage * (1 - 0.167% * AC) = [(5 - 5 * x%)/(6 - 5 * x%)] * HP/ [Damage * (1 - 0.21% * AC)]

HP and Damage cancels:
1/(1 - 0.167% * AC) = [u/(1 - 0.21% * AC)], where u is the percentage of XP in HP, i.e. (5 - 5 * x%)/(6 - 5 * x%).
cross multiply:
1 - 0.21% * AC = u * (1 - 0.167% * AC)

expand and move AC to one side:
0.21% * AC - u * 0.167% * AC = 1 - u

AC = (1 - u)/(0.21%- u * 0.167%)

1 - u = 1 - (5 - 5 * x%)/(6 - 5 * x%)
=(6 - 5 * x ^ - 5 + 5 * x%)/(6 - 5 * x%)
=1/(6 - 5 * x%)

AC = 1/ (6 - 5 * x%) (0.21% - u * 0.167%)
AC = 1 / [(6 - 5 * x%) * 0.21% - (5 - 5 * x%) * 0.167%]
substitue x% = 0, the break-even point is 235 AC, at x% = 18%, the break-even point is 259 AC.
Q.E.D.

TchoobJune 20 2008 11:27 AM EDT

If you're depending on AS only as your source of HP, than SS is definitely the way to go - just add in basic SS and it will work at full effect. I'm too lazy to do the calcs for how it looks with different HP/AS values.

QBRangerJune 20 2008 11:28 AM EDT

Yes, this is a change that "helps" ROS teams.

Wizard'sFirstRuleJune 20 2008 11:30 AM EDT

Once you lose the link between HP and SS level, you have a lot more adding and subtracting to do to modify the number. At which point the actual level of HP and AS might become relevant. I really can't be bothered because I suspect that for each value of HP and AS, it would be different. What's wrong with a simple model of the relationship between HP and SS. :P

Wasp[C and S Forgery Lmtd.]June 20 2008 11:39 AM EDT

Great calculations. I'll be needing these : ). Cheers!!

Wizard'sFirstRuleJune 22 2008 6:00 AM EDT

Modifying formula to account for change to 1/8 of HP.

Notes:
If ToE cap is higher than damage, then divide your damage by 4, which still gets cancelled. It wouldn't matter if you have ToE.
Training AS for HP is annoying, but I suspect if AS is significantly higher than your trained HP, the break-even point would be close to 1 AC.

Under the same assumptions:
With 100% HP, you can survive:
HP/Damage * (1 - 0.167% * AC) rounds

To train to full SS with x% penalty:
HP = SS * (1 - x%) * 8
but SS = 1 - HP (you either train HP or SS as far as for survival goes)
HP = (1 - HP) * (1 - x%) * 8
simplifies to:
HP = (8 - 8 * x%)/(9 - 8 * x%)

With Max SS, you can survive:
[(8 - 8 * x%)/(9 - 8 * x%)] * HP/ [Damage * (1 - 0.21% * AC)] rounds

for the two options to be equal (break-even), the rounds you can survive would be the same, therefore:
HP/Damage * (1 - 0.167% * AC) = [(8 - 8 * x%)/(9 - 8 * x%)] * HP/ [Damage * (1 - 0.21% * AC)]

HP and Damage cancels:
1/(1 - 0.167% * AC) = [u/(1 - 0.21% * AC)], where u is the percentage of XP in HP, i.e. (8 - 8 * x%)/(9 - 8 * x%).
cross multiply:
1 - 0.21% * AC = u * (1 - 0.167% * AC)

expand and move AC to one side:
0.21% * AC - u * 0.167% * AC = 1 - u

AC = (1 - u)/(0.21%- u * 0.167%)

1 - u = 1 - (8 - 8 * x%)/(9 - 8 * x%)
=(9 - 8 * x ^ - 8 + 8 * x%)/(9 - 8 * x%)
=1/(9 - 8 * x%)

AC = 1/ (9 - 8 * x%) (0.21% - u * 0.167%)
AC = 1 / [(9 - 8 * x%) * 0.21% - (8 - 8 * x%) * 0.167%]
substitute x% = 0, the break-even point is 181 AC, at x% = 18%, the break-even point is 203 AC.
Q.E.D.

BluBBenJune 22 2008 3:58 PM EDT

Awesome!! You should put this somewhere in the Wiki!

DONE.