Thursday, 20 March 2014

Vertical Asymptotes

Nowhere are the bonuses for paragon levels so directly comparable as between armor and resist all.  Armor and resist have mathematically identical effects on your Toughness - formerly called effective health by people who analysed such things - except that resist all points are the equivalent of ten armor points.  That doesn't mean that one resist all and ten armor will have the same impact on your Toughness, but they would if your armor happened to be exactly ten times your resist all.  Because they have a synergistic effect, having a lot of one makes the other better.

But the paragon options are not at all equivalent.  One gives 0.5% more armor per point, the other 5 more maximum resist.  If you are level 4 with 40 armor, then the resist all option is transparently superior.  Picking armor will reduce incoming damage by a little under tenth of a percent, resist all by twenty percent.

At level 60 it actually continues to be rather obvious.  My wizard is now using all patch 2.0 equipment, the majority of which is unique with a few rares - particularly crafted rares of intelligence.  I play on Torment III because I can easily butcher things on that difficulty with very little slowdown, but I'm sure I could handle a higher difficulty if I wanted to slog instead of slaughter.  I'll admit I'm a little bit glass-cannony in my item selection, but I have nearly a million toughness so I'm not all that flimsy.  At this gear level I have 3755 armor and 580 resist all.  So increasing my own armor by 0.5% would reduce damage taken by 0.3% while 5 maximum resist would reduce damage taken by 0.6%.  It's not as ludicrous as it was in the low level example, but it's still a pretty big difference.

Of course the break even point is very easy to calculate.  We know that if your armor is ten times your resist all then they ten armor is the equivalent of one resist all.  If either the armor option does not provide ten times as much armor as the resist all provides resist all, or if the armor total is higher than ten times the resist all total then resist all will tend to be better.  That's a pretty fine line to walk.

A half of a percent of your armor is 50 - that is, ten times five - when your armor is ten thousand.  Thus if you have 10,000 armor and 1,000 resist all, the two are equivalent options.  Of course that's not a realistic break even point with current gear levels so let's look at the graph:


That vertical line coming up from nowhere is a sign that excel doesn't deal too well with functions that have a vertical asymptote where they approach negative infinity from the left and positive infinity from the right.  At 700 resist or lower, at level 60, there is no armor value which will make the armor better.  But forget any amount, lets talk about realistic amounts.  With 750 resist all and 60000 armor, you would be better off boosting armor at level 60, but you can't have 60000 armor.  You can't really have 12000 armor either, so 950 resist is out.  Basically, armor is never going to be the right choice.  I'm not that familiar with the best possible gear, but right now it seems unlikely to me that there is anyone who should invest the first point in armor before the 50th point in resist all.

Assuming the formulae don't change, level 70 doesn't change the numbers that much.  You need at least 650 resist all to go through negative infinity into the possible range, and we know that 1000 resist all and 10000 armor will always be a break even point at every level.  The question is, can we read anything into the choices of these numbers?  Should we hazard a guess that we will have about 1000 resist all and 10000 armor is we are tough at level 70?

That's well within the realm of possibility.  For the value of affixes to double seems plausible, and that would put us into the approximate range where we have to do math to make a choice rather than knowing it is resist all every time.

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