MukkTB
2011-11-15, 04:27 PM
I've been trying to optimize my character using math to determine how much damage per turn I can produce. Mostly with melee. But this could apply to magic as well. The problem with determining an average is that in most games where you would attempt this you could separate your stats from the enemy's. I have 5 DPS (Damage Per Second) the Icky Gnoll has 60 EHP (Effective Hitpoints -> Which are raw HP modified by defenses) It will take me 12 seconds to kill the Icky Gnoll.
However I haven't found a satisfying way of simplifying it that much in D&D 3.x. The best I can do is create a DPT against AC graph. Find the enemy's AC then divide their HP by that value on the DPT graph. That's how long you can expect them to stand up against an attack. The DPT for a particular AC is the sum of all attack's damage where the damage for each attack is average weapon damage multiplied by chance to hit.
So an 18 STR lvl 1 Raging Barbarian Power Attacking with a Greatsword.
+5 hit STR +1 hit BS -1 hit PA = +5 hit
2D6(Average 7)+5*1.5 STR +2 PA = 16 average damage
AC____DMG
26____0.8
25____0.8
24____1.6
23____2.4
22____3.2
21____4
20____4.8
19____5.6
18____6.4
17____7.2
16____8
15____8.8
14____9.6
13____10.4
12____11.2
11____12
10____12.8
_9____13.6
_8____14.4
_7____15.2
_6____15.2
_5____15.2
_4____15.2
_3____15.2
_2____15.2
However for small creatures that die in one hit it makes more sense just to look at the hit chance of any given swing. So I made another field.
AC____DMG____Hit Chance
26____0.8_____0.05
25____0.8_____0.05
24____1.6_____0.1
23____2.4_____0.15
22____3.2_____0.2
21____4.0_____0.25
20____4.8_____0.3
19____5.6_____0.35
18____6.4_____0.4
17____7.2_____0.45
16____8.0_____0.5
15____8.8_____0.55
14____9.6_____0.6
13___10.4_____0.65
12___11.2_____0.7
11___12.0_____0.75
10___12.8_____0.8
_9___13.6_____0.85
_8___14.4_____0.9
_7___15.2_____0.95
_6___15.2_____0.95
_5___15.2_____0.95
_4___15.2_____0.95
_3___15.2_____0.95
_2___15.2_____0.95
Since I'm using probability math all probabilities are less than 1. A .95 chance is a 95% chance.
Just for comparison I'll put in a dual wielding raging barbarian and a wizard with a stick.
An 18 STR lvl 1 Raging Barbarian two weapon fighting with a longsword and a shortsword.
+5 hit STR +1 hit BA -2 hit TWF = +4 hit
1d8(Average 4.5) + 5 STR = 9.5 average damage longsword
1D6(Average 3.5) + 2 STR = 5.5 average damage shortsword
AC____DMG____Hit Chance
26____0.75____0.0975
25____0.75____0.0975
24____0.75____0.0975
23____1.50____0.19
22____2.25____0.2775
21____3.00____0.36
20____3.75____0.4375
19____4.50____0.51
18____5.25____0.5775
17____6.00____0.64
16____6.75____0.6975
15____7.50____0.75
14____8.25____0.7975
13____9.00____0.84
12____9.75____0.8775
11___10.50____0.91
10___11.25____0.9375
_9___12.00____0.96
_8___12.75____0.9775
_7___13.50____0.99
_6___14.25____0.9975
_5___14.25____0.9975
_4___14.25____0.9975
_3___14.25____0.9975
_2___14.25____0.9975
And a 10 STR Wizard with a staff.
+0 hit STR + 0 hit BA = +0 hit
1D6(Average 3.5) + 0 STR = +0 DMG
AC____DMG____Hit Chance
26____0.175____0.05
25____0.175____0.05
24____0.175____0.05
23____0.175____0.05
22____0.175____0.05
21____0.175____0.05
20____0.175____0.05
19____0.350____0.1
18____0.525____0.15
17____0.700____0.2
16____0.875____0.25
15____1.050____0.3
14____1.225____0.35
13____1.400____0.4
12____1.575____0.45
11____1.750____0.5
10____1.925____0.55
_9____2.100____0.6
_8____2.275____0.65
_7____2.450____0.7
_6____2.625____0.75
_5____2.800____0.8
_4____2.975____0.85
_3____3.150____0.9
_2____3.325____0.95
So if a 6 HP 17 AC armored Goblin is the target you look up 16 AC.
The THW Barbarian will kill it if he hits. His hit chance is .45. On the first round he has a 45% chance of killing it, the second round (1-(1-.45)^2)*100% chance of killing it and on the Xth round (1-(1-.45)^X)*100% chance of having killed it. After 2 rounds there is an 70% chance that goblin has gone down.
The TWF Barbarian will ill it if he hits with his main hand or gravely maim it if he hits with his off hand. So we will look at the hit chance which is .64 or 64%. We can fudge that a 6HP Goblin who just got hit with 1D6+2 damage is gonna bail from the fight. That gives a (1-(1-.64)^X)*100% chance of removing the goblin from the fight on the Xth round. After 2 rounds there is an 87% chance that goblin has gone down.
The wizard cannot expect to knock the Goblin out in one hit. He does .7 DPT. 6/.7 gives us 8.5 rounds on average to knock out the goblin in melee.
Let's pull out a bigger monster. A Worg with 15 AC and 26 HP.
The THW Barbarian will do 8.8 DPT against AC 15. At 26 HP it will take 3 rounds average to down it (26/8.8).
The THW Barbarian will do 7.5 DPT against AC 15. At 26 HP it will take 3.5 rounds average to down it (26/7.5).
Our puny wizard buddy has a measly 1.05 DPT. He needs 25 rounds.
If you examine the charts you will find that this mathematically proves the common knowledge that THF is better than TWF at all AC values although TWF can be nice in cases where you just need to score a hit.
Edit - I screwed up the math on my spreadsheet for the to hit values. Fixing now.
Edit 2 Fixed the numbers so the math works.
However I haven't found a satisfying way of simplifying it that much in D&D 3.x. The best I can do is create a DPT against AC graph. Find the enemy's AC then divide their HP by that value on the DPT graph. That's how long you can expect them to stand up against an attack. The DPT for a particular AC is the sum of all attack's damage where the damage for each attack is average weapon damage multiplied by chance to hit.
So an 18 STR lvl 1 Raging Barbarian Power Attacking with a Greatsword.
+5 hit STR +1 hit BS -1 hit PA = +5 hit
2D6(Average 7)+5*1.5 STR +2 PA = 16 average damage
AC____DMG
26____0.8
25____0.8
24____1.6
23____2.4
22____3.2
21____4
20____4.8
19____5.6
18____6.4
17____7.2
16____8
15____8.8
14____9.6
13____10.4
12____11.2
11____12
10____12.8
_9____13.6
_8____14.4
_7____15.2
_6____15.2
_5____15.2
_4____15.2
_3____15.2
_2____15.2
However for small creatures that die in one hit it makes more sense just to look at the hit chance of any given swing. So I made another field.
AC____DMG____Hit Chance
26____0.8_____0.05
25____0.8_____0.05
24____1.6_____0.1
23____2.4_____0.15
22____3.2_____0.2
21____4.0_____0.25
20____4.8_____0.3
19____5.6_____0.35
18____6.4_____0.4
17____7.2_____0.45
16____8.0_____0.5
15____8.8_____0.55
14____9.6_____0.6
13___10.4_____0.65
12___11.2_____0.7
11___12.0_____0.75
10___12.8_____0.8
_9___13.6_____0.85
_8___14.4_____0.9
_7___15.2_____0.95
_6___15.2_____0.95
_5___15.2_____0.95
_4___15.2_____0.95
_3___15.2_____0.95
_2___15.2_____0.95
Since I'm using probability math all probabilities are less than 1. A .95 chance is a 95% chance.
Just for comparison I'll put in a dual wielding raging barbarian and a wizard with a stick.
An 18 STR lvl 1 Raging Barbarian two weapon fighting with a longsword and a shortsword.
+5 hit STR +1 hit BA -2 hit TWF = +4 hit
1d8(Average 4.5) + 5 STR = 9.5 average damage longsword
1D6(Average 3.5) + 2 STR = 5.5 average damage shortsword
AC____DMG____Hit Chance
26____0.75____0.0975
25____0.75____0.0975
24____0.75____0.0975
23____1.50____0.19
22____2.25____0.2775
21____3.00____0.36
20____3.75____0.4375
19____4.50____0.51
18____5.25____0.5775
17____6.00____0.64
16____6.75____0.6975
15____7.50____0.75
14____8.25____0.7975
13____9.00____0.84
12____9.75____0.8775
11___10.50____0.91
10___11.25____0.9375
_9___12.00____0.96
_8___12.75____0.9775
_7___13.50____0.99
_6___14.25____0.9975
_5___14.25____0.9975
_4___14.25____0.9975
_3___14.25____0.9975
_2___14.25____0.9975
And a 10 STR Wizard with a staff.
+0 hit STR + 0 hit BA = +0 hit
1D6(Average 3.5) + 0 STR = +0 DMG
AC____DMG____Hit Chance
26____0.175____0.05
25____0.175____0.05
24____0.175____0.05
23____0.175____0.05
22____0.175____0.05
21____0.175____0.05
20____0.175____0.05
19____0.350____0.1
18____0.525____0.15
17____0.700____0.2
16____0.875____0.25
15____1.050____0.3
14____1.225____0.35
13____1.400____0.4
12____1.575____0.45
11____1.750____0.5
10____1.925____0.55
_9____2.100____0.6
_8____2.275____0.65
_7____2.450____0.7
_6____2.625____0.75
_5____2.800____0.8
_4____2.975____0.85
_3____3.150____0.9
_2____3.325____0.95
So if a 6 HP 17 AC armored Goblin is the target you look up 16 AC.
The THW Barbarian will kill it if he hits. His hit chance is .45. On the first round he has a 45% chance of killing it, the second round (1-(1-.45)^2)*100% chance of killing it and on the Xth round (1-(1-.45)^X)*100% chance of having killed it. After 2 rounds there is an 70% chance that goblin has gone down.
The TWF Barbarian will ill it if he hits with his main hand or gravely maim it if he hits with his off hand. So we will look at the hit chance which is .64 or 64%. We can fudge that a 6HP Goblin who just got hit with 1D6+2 damage is gonna bail from the fight. That gives a (1-(1-.64)^X)*100% chance of removing the goblin from the fight on the Xth round. After 2 rounds there is an 87% chance that goblin has gone down.
The wizard cannot expect to knock the Goblin out in one hit. He does .7 DPT. 6/.7 gives us 8.5 rounds on average to knock out the goblin in melee.
Let's pull out a bigger monster. A Worg with 15 AC and 26 HP.
The THW Barbarian will do 8.8 DPT against AC 15. At 26 HP it will take 3 rounds average to down it (26/8.8).
The THW Barbarian will do 7.5 DPT against AC 15. At 26 HP it will take 3.5 rounds average to down it (26/7.5).
Our puny wizard buddy has a measly 1.05 DPT. He needs 25 rounds.
If you examine the charts you will find that this mathematically proves the common knowledge that THF is better than TWF at all AC values although TWF can be nice in cases where you just need to score a hit.
Edit - I screwed up the math on my spreadsheet for the to hit values. Fixing now.
Edit 2 Fixed the numbers so the math works.