The Myth that PRR has diminishing returns

[Ancient]

Redoubt stated a more accurate version of my thesis than the title of this thread:

"While PRR has diminishing returns in respect to damage reduction, it creates a linear affect on damage required to reduce your hit points to zero."

Chi_Ryu explains why it is problematic to refer to this as diminishing returns:

When viewed from the perspective of increased time to live under constant damage, PRR does not suffer from diminishing returns, unless the definition of diminishing returns has changed in the 20-odd years since I studied Economics, namely: "the decrease in the marginal output of production as the amount of a single factor of production is incrementally increased, whilst all other factors of production stay constant."

If you don't like economics, maybe Faltout's simplified explanation will help.

And why are you measuring the "via" part and not the actual outcome?

-------- original post follows ------
There seems to be some general confusion about the benefits of PRR. I often hear that PRR has diminishing returns. This is simply untrue. The reason that it is believed to be true is because the benefits of PRR are displayed as a percentage damage reduction.

In the table below, the first three columns on the left are taken straight from the DDOWIKI entry on PRR.

The fourth column is a simple calculation to display how many points of pre-prr damage is needed to kill a character with 1000 hit points. The simple formula is 1000 / the percentage of damage taken. The smaller the amount of damage taken gets, the larger that number will be.

While it can be argued that additional PRR is less useful... Each point of PRR gives almost exactly the same amount of effective HP. *EDIT For those who missed this, here is where I acknowledge there can be an opportunity cost for increaseing PRR *END EDIT*

PRR	Reduction	Damage Taken	Pre-Prr Damage to do 1000 Dmg
0	0.0%	100.0%	1000.0
10	9.1%	90.9%	1100.0
20	16.7%	83.3%	1200.0
30	23.1%	76.9%	1300.1
40	28.6%	71.4%	1400.0
50	33.3%	66.7%	1499.9
60	37.5%	62.5%	1600.0
70	41.2%	58.8%	1700.1
80	44.4%	55.6%	1799.9
90	47.4%	52.6%	1900.1
100	50.0%	50.0%	2000.0
110	52.4%	47.6%	2100.0
120	54.6%	45.5%	2200.2
130	56.5%	43.5%	2299.9
140	58.3%	41.7%	2399.8
150	60.0%	40.0%	2500.0
160	61.5%	38.5%	2600.1
170	63.0%	37.0%	2699.8
180	64.3%	35.7%	2800.3
190	65.5%	34.5%	2900.2
200	66.7%	33.3%	3000.3
210	67.7%	32.3%	3099.8

Here is another explanation which may be clearer:

Maybe this will help:

Most people think of PRR as non-linear, and the damage reduction part of that is true:


However, if you look at the EFFECT of that, you will see a linear relationship between the amount of PRR you have, and the damage required to kill you. The damage mitigation isn't what matters most. What matters most (to the monster trying to kill you) is how many HP of damage does it take to put you down?


So for every point in PRR you get, you are linearly increasing your chances of survival.

Say you go from 0 PRR to 20 PRR. Your damage mitigation is now 16.67%. Now it takes 1200 HP to kill you (if you had 1000hp to begin with). So that's an effective increase of 200hp.

Say you go from 20 PRR to 40 PRR. Your damage mitigation is now 28.57%. (Doubling PRR did NOT double your damage mitigation percentage). BUT WAIT. How much extra damage does it take to kill your 1000hp toon? 1400hp!!! That's an increase of 400, which is exactly double the amount from before.

The moral of the story is: Doubling your PRR doubles the amount of extra HP it takes to kill you, which is all that really matters. It does not double your damage mitigation percentage, but the survivability effect is purely linear.

So, every point of PRR counts, even at very, very high amounts of PRR.

[Chai]

Still parabolic - its just theres not enough practical PRR available to get to the point where the returns diminish at a level where it would be completely not worth it.

The better argument I believe is that it gets to a point where the character is better off going with some other form of defense which stacks with PRR rather than adding more PRR and just taking the hit - as this game now favors layered defenses rather than maxing out one stat.

[Ancient]

Still parabolic - its just theres not enough practical PRR available to get to the point where the returns diminish at a level where it would be completely not worth it.

Column 4 can be approximated by (10 X column one) + 1000. If that is "parabolic", then please provide your definition of parabolic.

The better argument I believe is that it gets to a point where the character is better off going with some other form of defense which stacks with PRR rather than adding more PRR and just taking the hit - as this game now favors layered defenses rather than maxing out one stat.

Here, I agree with you 100%

[Chai]

Column 4 can be approximated by (10 X column one) + 1000. If that is "parabolic", then please provide your definition of parabolic.

In this case parabolic would be that as PRR rises, the damage taken approaches zero, but never makes it to zero.

[Ancient]

In this case parabolic would be that as PRR rises, the damage taken approaches zero, but never makes it to zero.

And the entire point of my post is that percentage damage reduction appears to have diminishing returns, but in reality the effective HP gained by each point of PRR remains the same.... i.e. Column 4 is the new information.

The lower you go, the MORE each percentage point of damage reduction is worth, thus to achieve the same value of effective HP, you get less damage reduction. When I explained this with the example that going from 99% damage reduction to 100 was an infinite increase in effectiveness, I was told that I was picking an extreme value. This chart demonstrates that my statement holds true across all values of PRR.

[Chai]

And the entire point of my post is that percentage damage reduction appears to have diminishing returns, but in reality the effective HP gained by each point of PRR remains the same.... i.e. Column 4 is the new information.

The lower you go, the MORE each percentage point of damage reduction is worth, thus to achieve the same value of effective HP, you get less damage reduction. When I explained this with the example that going from 99% damage reduction to 100 was an infinite increase in effectiveness, I was told that I was picking an extreme value. This chart demonstrates that my statement holds true across all values of PRR.

Note that at 100 PRR, you have 50% reduction on the nail. On a linear system if you had 200 PRR you would suffer no damage.

Adding more PRR still means more reduction, adding the same number at different intervals =/= same gain due to the parabolic nature of the system.

Going from 50 to 100 PRR = 16.7% more reduction
Going from 100 to 150 PRR = 10% more reduction.
Going from 150 to 200 PRR = 6.7% more reduction.

You added 50 more PRR each time but got less reduction adding it the second time than the first time, and even less adding 50 more again a third time. In fact, you had to add 100 more just to get the same additional 16.7% reduction you got for adding the first 50. The higher you go, the less additional reduction you get for adding the same amount.

This is diminishing returns, which is fact, and no myth, in a parabolic system.

[Gremmlynn]

I see by that chart that the first 100 points of PRR reduce damage taken by 50%, the next 100 points by only 16.7% more. That seems pretty diminishing to me.

[Enoach]

Just a some points...

If you remove the rounding you are doing for the values you are using in the final formula you will actually see that for every 10 points of PRR it will in fact take +100 more of a SINGLE shot to reduce you from FULL to ZERO. Any rounding should always be done at the end not during the calculation, this can through calculations off by a significant amount. So on that point I agree with you every 10 Points means +100 more damage is needed to go from full to zero for a SINGLE shot. However, 100 Damage becomes less of a percentage of the WHOLE. So while 100 is 10% of 1000, it is only 3.33% of 3000.

Now while this works on a SINGLE shot, the actual diminish of return is because PRR is mostly used to reduce the incoming damage of multiple shots. These damage in epic levels will be in the 300 to 600 range on average where the amount of damage let through per hit starts to plateau where the difference between a 100 PRR and 200 PRR for a 300 Point incoming shot is 5 HP while the difference between a 10 PRR and 100 PRR is 19 HP.

Now I think the disconnect here is that its not as much a diminish return so much as it is an Opportunity Cost issue. In that any given build can have PRR or X without starting to significantly reduce other aspects of the build. We can't just say we are going to have 200+ PRR on a build we need to sacrifice gear slots/augments/type of gear, enhancements and possibly Feats to achieve this.

So then the thought goes to if I'm getting hit for 300 points of damage on average what would I need to do if I wanted this to be 1/2 (or 150)

Currently a PRR of 100 would do this

Or 90 PRR with 8 DR
Or 80 PRR with 17 DR
Or 70 PRR with 27 DR
Or 60 PRR with 38 DR
Or 50 PRR with 50 DR

Keeping in mind only the 100 PRR will Scale up as the damage increases while the DR needed would need to increase as the damage goes up since it is static.

Again this is more an issue of Opportunity Cost for a build than it is with diminishing returns.

[Kaytis]

Despite previous forum assertions that "Math never solved anything", PRR is based on a non-linear formula: 100/(100+PRR) -you simply can't get linear results out of it. Your assertion would have been somewhat defendable if you stated it approximates a linear function at low values of PRR, although your own data pretty much shows that it doesn't.

[IronClan]

What does it take to go from 100prr to 150 and then 210? Right it takes the sacrifices in opportunity cost (aka DPS lost) that are the diminishing returns stuff your "in a vacuum" math is ignoring. Ignoring something doesn't make it a myth.

A simple example.
Average KotC THF maybe 80-120 PRR no shield, without paying for more with lower DPS
Average vanguard 100-150 without paying DPS opportunity cost for more, already lower DPS than a KotC (loss of avenging cleave and holy retribution, and maybe could not afford exalted smites due to spending 25 points in defender T4 10% speed and stance)
Average full tropic thunder tank: 200+ much lower DPS than first two options for workable EE worthy AC (over 180ac)

Diminishing returns not a myth. Even if you choose to abstract it to a equivalent HP formula and ignore the costs.

[Vellrad]

If I got 1PRR, 1000 damage hit becomes 990 damage hit. 1st point of PRR protects me from 10 damage.
If I got 2PRR, 1000 damage hit becomes 980 damage hit. 2nd point of PRR protects me from 10 mode damage than first.
[...]
If I got 100PRR, 1000 damage hit becomes 500 damage hit. 100 points of PRR protects me from 500 damage.
If I got 101PRR, 1000 damage hit becomes 497 damage hit. 101st point of PRR protects me from 3 more damage than 100th point.

If 10=3 then sure, you're right.

[N-0cturn]

Note that at 100 PRR, you have 50% reduction on the nail. On a linear system if you had 200 PRR you would suffer no damage.

Adding more PRR still means more reduction, adding the same number at different intervals =/= same gain due to the parabolic nature of the system.

Going from 50 to 100 PRR = 16.7% more reduction
Going from 100 to 150 PRR = 10% more reduction.
Going from 150 to 200 PRR = 6.7% more reduction.

You added 50 more PRR each time but got less reduction adding it the second time than the first time, and even less adding 50 more again a third time. In fact, you had to add 100 more just to get the same additional 16.7% reduction you got for adding the first 50. The higher you go, the less additional reduction you get for adding the same amount.

This is diminishing returns, which is fact, and no myth, in a parabolic system.

Hm no that not really correct. At least not when you are trying to look at what you gain from the additional DR.

Going from 150 to 200 PRR gets you from 40.0% damage taken to 32.3% damage taken. This is an additional Reduction of 19,25 % (You take 323 damage on a Hit that would have hit you for 400 dmg before you got the additional PRR).

It still has a diminishing return. First 10 points of PPR are a 9.1 damage reduction. The 10 points from 200 to 210 PRR only give about 3% damage reduction.

[bsquishwizzy]

I should be noted that the term “diminishing returns” is wholly subjective. To a tank, there is no point of diminishing returns as every point of PRR counts. So a caster in the back, who has no intention of getting aggro, PRR vs. ASF (maybe) becomes something to consider.

You work with what you got and what you need. And with different builds, needs are different.

[Chai]

Hm no that not really correct. At least not when you are trying to look at what you gain from the additional DR.

Going from 150 to 200 PRR gets you from 40.0% damage taken to 32.3% damage taken. This is an additional Reduction of 19,25 % (You take 323 damage on a Hit that would have hit you for 400 dmg before you got the additional PRR).

It still has a diminishing return. First 10 points of PPR are a 9.1 damage reduction. The 10 points from 200 to 210 PRR only give about 3% damage reduction.

For one single hit that is correct.

For percentage variance in total damage absorbed from all hits it may or may not be depending on how many hits and at what damage.

[Chai]

I should be noted that the term “diminishing returns” is wholly subjective. To a tank, there is no point of diminishing returns as every point of PRR counts. So a caster in the back, who has no intention of getting aggro, PRR vs. ASF (maybe) becomes something to consider.

You work with what you got and what you need. And with different builds, needs are different.

For that tank, they would weigh the effect of adding more PRR to their total value versus adding something else, then make the decision according to the specific circumstances in which they play. The returns of adding more PRR do diminish in a system like DDO where at some point, adding a different type of defense, or even recovery, would be more benefit for that gear / feat / AP than adding more PRR for the same cost.

[IronClan]

What does it take to go from 100prr to 150 and then 210? Right it takes the sacrifices in opportunity cost (aka DPS lost) that are the diminishing returns stuff your "in a vacuum" math is ignoring. Ignoring something doesn't make it a myth.

A simple example.
Average KotC THF maybe 80-120 PRR no shield, without paying for more with lower DPS
Average vanguard 100-150 without paying DPS opportunity cost for more, already lower DPS than a KotC (loss of avenging cleave and holy retribution, and maybe could not afford exalted smites due to spending 25 points in defender T4 10% speed and stance)
Average full tropic thunder tank: 200+ much lower DPS than first two options for workable EE worthy AC (over 180ac)

Diminishing returns not a myth. Even if you choose to abstract it to a equivalent HP formula and ignore the costs.

[Connman]

Still hyperbolic - its just theres not enough practical PRR available to get to the point where the returns diminish at a level where it would be completely not worth it.

The better argument I believe is that it gets to a point where the character is better off going with some other form of defense which stacks with PRR rather than adding more PRR and just taking the hit - as this game now favors layered defenses rather than maxing out one stat.

FTFY

100/(100+x) if this is the true formula it results in a hyperbola not a parabola, other than that I agree with you 100% Chai, not that it matters math is not philosophy.

However to quote one of my math professors, "math is the lazy man's science," which is probably why I went that route in college.

[JOTMON]

Ancient and I have crossed paths on this before..
https://www.ddo.com/forums/showthrea...=1#post5520130


I think there is a conceptual issue here.

While I understand what you are trying to say...Each point invested into PRR is a benefit and every point you invest does give you better Damage Reduction..

but

It is still diminishing...not a myth... each point invested does not give you one point back
even your own chart shows this.

0PRR gets you 0 reduction in damage
50 PRR gets you 33.3% reduction in damage
100PRR gets you 50% reduction not 66.6% ...
150PRR gets you 60% reduction not 99.9% reduction
~as a point to point reduction would give you. therefore we have diminishing returns.


The devaluing benefit also becomes more apparent at higher investment
you would need a PRR investment of 19,900 to get 100% damage reduction.


A max invested PRR build is only hitting about 300PRR so a 75% reduction in damage is the most we can expect to achieve.
The difference between 240PRR with investment on a heavy armor build with a shield for 71% or gimp DPS to achieve 300PRR for 75%
is the 4% worth another 60 points in PRR at the cost of DPS or other sacrifices..

[Chi_Ryu]

This is not a new conversation.

Despite previous forum assertions that "Math never solved anything", PRR is based on a non-linear formula: 100/(100+PRR) -you simply can't get linear results

That depends on what you are considering the linearity of.

In terms of absolute damage reduction per hit it is indeed diminishing - no argument from me there.

However, in terms of raw damage needed to incapacitate a character at full HP, it is indeed linear. Severlin expressed this in a thread a few months back as "the time needed between heals".

[Connman]

http://dictionary.reference.com/brow...hing%20returns

Diminishing returns
noun
1.
any rate of profit, production, benefits, etc., that beyond a certain point fails to increase proportionately with added investment, effort, or skill.
2.
Also called law of diminishing returns. Economics. the fact, often stated as a law or principle, that when any factor of production, as labor, is increased while other factors, as capital and land, are held constant in amount, the output per unit of the variable factor will eventually diminish.

The confusion that some people have is they think that "diminishing returns = complete and total waste of time," this is where we have opinion coming in to the discussion, as it rightly should, because "waste of time" is an opinion and will vary from person to person and build to build.