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DarthVaader
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re: The Protection Paladin Explained With Economics

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you are welcome

Let’s use the parameter A to indicate the percentage of damage avoided per hit. Then A ranges from 0 (no damage avoided) to 1 (100% of damage avoided).

Then consider a function u(A) that produces a quantity indicating how “well off” we are given a certain level of A. We can normalize u(0) = 0 and u(1) = k, where k is some constant. Assume u(A) is strictly increasing in A; that is, the higher A is, the higher u(A) is - that is, the better off we are - for all possible values of 0 <= A <= 1. (Technically, u’ > 0.)

Consider a case where Player 1 has some chance p to completely avoid damage, but otherwise takes full damage. Then the expected value of u for Player 1 is:

p*u(1) + (1-p)*u(0) = p*u(1)

Now consider a case where Player 2 always blocks p damage on every hit, so that A = p. The expected value of u for Player 2 is then u(p). Note that expected (average) damage avoided in both cases is the same. For Player 1, expected damage avoided is p*1 + (1 – p)*0 = p, while for Player 2, it’s simply p (on every hit).

For the sake of example, let’s set p = 0.4 from now on.

If we look solely at average avoidance as what matters - as I think most of us have been wont to do pre-Cata - then u is linear in A (u'' = 0). (This is basically what we do when we treat effective health or average damage as the only thing that matters.) Here's an illustration, with p = 0.4.


In other words, for any value of p, u(p) = p*u(1). We call this risk-neutrality, because it doesn’t matter how damage is avoided – just how much.

Now consider the case when u has diminishing marginal returns from A (i.e., u’’ < 0). In other words, u still increases when we add avoidance, but the amount it increases per point of avoidance decreases when we have higher levels of avoidance. The graph below shows what happens:


0.4*u(1) is the expected value of u for Player 1 (who has a 40% chance to completely avoid damage and a 60% chance to take full damage). Meanwhile, u(0.4) is the value of u for Player 2, who always blocks 40% of damage per hit. Because of the shape of the function (u’ > 0, u’’ < 0 is all that’s required), u(0.4) > 0.4*u(1), even though expected damage is the same in both cases. The value at CE - the “certainty equivalent” - indicates the amount that would need to be blocked by Player 2 in order to have the same u as Player 1. Because p > CE, we call this risk-aversion.

With risk-aversion, how much damage is avoided is not the only thing that matters – how also matters. For a risk-averse player, it’s often better to block a low constant amount of damage than to avoid all damage some of the time and take full damage the rest of the time, and it’s always better to block a constant amount if expected damage is the same. In other words, the risk-averse player is willing to sacrifice effective health for “smoother” damage intake.

The argument in Cata is that risk-aversion is not some subjective thing that varies by a player's feelings, but that it has some actual measurable value - based on some measure of healer efficiency - that justifies the idea that it's worth pursing over dodge/parry even though it may not increase effective health as much, on a point-for-point basis.

In theory, u should be quantifiable in some way – we want to maximize healer efficiency, and healer efficiency is measurable. Coming up with some sort of suitable objective function for u(A) would then enable us to put an actual value on block relative to dodge/parry that might not be subject to the idiosyncrasies of any particular simulator or gear setup. We could then, in theory, come up with some basic guideline on how much effective health to sacrifice for a gain in “smoothing.”
~Holyrain
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re: The Protection Paladin Explained With Economics

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I want to point out that the amount of time it took you to do this, you could have geared for healing and saved us from me having to heal (or not heal) us from now on.

Just sayin.
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