Need the Math Gurus: DR 1 vs AC +1

[Stalker0]

Alright I need all those math genuses out there to help me. Right now in Monte's Arcana Evolved Forums we are trying to figure out an approximate DR X to AC +X equivalent.

Now I know benefits of each varies wildly depending on attack bonuses and damages, which is why I need your help. However, there are some guidelines.

1) The exercise relates to levels 1-6.

2) The base AC value is for that of an oathsworn at those levels. (You can use the monk values if you want, as they are close). So for 1st level its AC 11 + dex + magic whatever. At 3rd, its AC is 12 + dex + whatever, and at 6th its AC 13 + dex + whatever. Most oathsworn I've seen usually have a dex mod about 1 or 2 lower than an equivalent monk, so around the 12 to 14 range.

3) Use attack values and damage for near equivalent CR encounters.

4) Crit inclusion would be very helpful.

Any help you guys could give would be appreciated.

 

[KarinsDad]

What matters is damage versus to hit chance.

Say the average damage is 4.5 (for a D8 weapon). You are dropping 22% of the damage with DR 1. In order for AC to match that, you would need to be able to hit on a 16 instead of a 17:

16 3.5
17 3.5 4.5
18 3.5 4.5
19 3.5 4.5
20 3.5 4.5

Total 17.5 versus 18

But, even at first level, the class you specified does not have an AC of 17. More likely, a 13. Plus, his opponents probably have +3 to hit (Str 14) or better and probably +2 damage. This means an equivalent AC of 10:

10 5.5
11 5.5 6.5
12 5.5 6.5
13 5.5 6.5
14 5.5 6.5
15 5.5 6.5
16 5.5 6.5
17 5.5 6.5
18 5.5 6.5
19 5.5 6.5
20 5.5 6.5 (criticals are basically white noise here)

Total 60.5 versus 65

So, the DR solution is stronger. This is still the case if your opponent has an 18 Strength:

08 7.5
09 7.5 8.5
10 7.5 8.5
11 7.5 8.5
12 7.5 8.5
13 7.5 8.5
14 7.5 8.5
15 7.5 8.5
16 7.5 8.5
17 7.5 8.5
18 7.5 8.5
19 7.5 8.5
20 7.5 8.5

Total 97.5 versus 102

In order for the +1 AC to help a lot more than than the +1 DR, you must have a very high AC to begin with. For example:

19 x
20 x x+1

It is always better to take x+1 amount of damage than 2x for x > 1.

Hence, 1 DR is basically stronger than 1 AC unless your AC is real high. For a first level character, you would need an AC of about 22 before the +1 AC case is stronger.

 

[Thanee]

The average damage per attack is the chance to hit (21-AC+attack modifier x 5%, min 5%, max 95%) times the average damage per hit (i.e. 10 on 2d6+3).

+1 AC removes 5% times the average damage per hit from this total for all reasonable cases (a natural d20 roll between 1 and 19 needed to hit with an attack, so the +1 AC actually changes the total).

DR 1/- offers the same benefit on every hit, that deals exactly 20 damage, since 5% of 20 is 1.

So, if the average damage per hit is exactly 20, then both benefits are equal.
If the average damage per hit is higher than 20, then +1 AC is better.
If the average damage per hit is lower than 20, then DR 1/- is better.

Attack modifiers are irrelevant as long as they are within reasonable limits (see above), which they typically are.

Bye
Thanee

 

[Stalker0]

Assuming Thanee's math is correct, when would you say the average damage a party encounters is greater than 20? When they hit level 4,5,6?

 

[KarinsDad]

Assuming Thanee's math is correct, when would you say the average damage a party encounters is greater than 20? When they hit level 4,5,6?

More like level 10.

A first level Fighter with a Strength of 18 and a Greatsword does an average of 13 points of damage. By level 10, assuming he bumped up his Strength by 2, he has Weapon Specialization, a +3 Weapon, and a +4 Girdle of Giant Strength, he is doing 23 average points of damage. But, this is a pretty much maxxed out damage Fighter. A two weapon fighter or one with weapon and shield will not be doing this much damage per successful attack.

Granted, some monsters could do more damage per hit at a lower CR than this, but not many.

+1 DR kicks butt on +1 AC.


However, Thanee's math is not quite correct. The chances of hitting someone is nowhere near 95% until much higher level.

Hence, you are not typically dropping 95% down to 90%. Most of the time, you are dropping 40% to 35% or 60% to 55% or some such.

So, depending on what your chance to hit is, the amount of damage you would need to do for +1 AC to be equal to +1 DR is:

95%: 19
90%: 18
85%: 17
80%: 16
75%: 15
70%: 14
65%: 13
60%: 12
55%: 11
50%: 10
45%: 9
40%: 8
35%: 7
30%: 6
25%: 5
20%: 4
15%: 3
10%: 2
5%: never

The equation is chance+5% of (damage-1) = chance% of damage. Or in the case of 95% at 18 damage = 90% chance at 19 damage.

In other words, if you have a high chance of hitting, then you have to hit for a lot of damage for that DR 1 to be negligible. If you have a low chance of hitting, then you do not need as much damage to negate it.

A first level Fighter with 18 Strength tends to have a +5 to hit (assuming no Weapon Focus or Masterwork weapon) and does about 8 to 11 points of damage depending on weapon. The first level oathsworn you talked about would have an AC of 13, so the Fighter would have a 65% chance of damaging him. He needs to do 13 or more points on average for +1 AC to be worthwhile.

A sixth level Fighter with 21 Strength (gauntlets +2) and Weapon Focus and Weapon Specialization and a +1 sword tends to have a +13 to hit and does about 12 to 17 points of damage depending on weapon. The sixth level oathsworn you talked about would have an AC of maybe 18, so the Fighter would have a 80% chance of hitting him. He needs to do 16 or more points on average for +1 AC to be worthwhile.

Eventually, the Fighter will get so good that the Oathsworn is screwed no matter which route he takes. But, once the Fighter gets the 95% chance to hit and does more than 19 points of damage, the DR is always the best choice because +25 to hit against AC 23 is the same 95% as +25 to hit against AC 24 (or AC 25 or AC 26).

So even if there is a level or so where the AC boost is better, eventually it goes back to the DR boost being better again.


For a different class that can have a very high AC (i.e. 5% to 25% chance of most opponents hitting), +1 to AC is always better since most opponents will be doing more than a few points of damage.

 

[Stalker0]

Good to know...so in general how much then is DR 1 worth? +1.5 AC, +2?

 

[Abraxas]

If the
average damage per attack is 5 points DR 1/?? is approximately equal to +4 AC
average damage per attack is 10 points DR 1/?? is approximately equal to +2 AC
average damage per attack is 20 points DR 1/?? is approximately equal to +1 AC

These are approximations over a wide range of attack bonuses and ACs.
Depending on the particulars (relative ACs and attack bonuses) this might be skewed one way or the other.

If you have a range of ACs and Attack bonuses that you are particularly interested in I have a spread sheet that does some basic calculations and shows the results in a simple graph. I have been doing a whole lot of number crunching comparing a warforged DR 2/adamantine vs another characters AC and have been trying to explain to a player that boosting his AC into the stratosphere is not equaled by the warforged DR 2.

 

[Thanee]

However, Thanee's math is not quite correct.

Yep, specifically I compared average damage per attack with average damage per hit!

Of course the damage reduction must be multiplied with the chance to hit...

5% times average damage per hit compared to chance to hit times 1

If the first part is higher, then an AC bonus is better, if the second part is higher, then DR is better.


But the result is the same, anyways, DR 1/- is usually better than AC +1 on average.

Bye
Thanee