Tuesday, July 22, 2014

Simulating Variable Weapon Damage

Exploring Weapon Damage Part II

Here are some results of all that number-crunching that went into last post, a hit table that simulates using “variable” weapon damage by altering the hit chances for AC 9 to 2 but rolling 1d6 for damage on all successful hits.

Chart 9a: Simulating Variable Damage by Hit Chance
Simulated
Die
Armor Class
9[10]
8[11]
7[12]
6[13]
5[14]
4[15]
3[16]
2[17]
1d4
12
13
14
15
16
17
18
19
1d6
10
11
12
13
14
15
16
17
1d8
8
9
10
11
11
13
14
15
1d10
6
7
8
8
9
11
12
13
Note: the 1d6 isn’t simulated but is included on the chart for reference. 

Chart 9b: Comparing true damage dice to simulated damage dice
Damage
Die
Average Likely Damage with True Roll
Average Likely Damage with Simulated Roll
d4
0.9375
0.9625
d6
1.3125
1.3125
d8
1.6875
1.6843
d10
2.0625
2.056

Example of use:  using a “d4” weapon to attack roll a d20 to hit on the row for d20 and when you hit roll 1d6 for damage (do the same with the other types of notational damage but only rolling 1d6 for actual damage). Over the course of multiple combat rounds the different hit chances will come very close to producing the same average likely damage  (as per Chart 9b).

Charts 9a and 9b show how significant higher damage weapons are in the common variable weapon damage used these days. We also see what variable weapon damage means, when a weapon does 1d4 points of damage or 1d8 points of damage (when playing with such notation) due to the inherent abstract nature of classic RPG combat we aren’t just saying “Weapon A does this much damage when it strikes a target” what we are actually say is  “Weapon A will provide the opportunity to inflict this much damage on a target during combat”.  So a d8 weapon isn’t necessarily inflicting more damage because it is inherently more awful in the types of wounds it inflicts but it offers the user more opportunity to inflict wounds.

Here’s a cleaner version of attack matrix for damage by hit chances that just replaces the dice notation for damage with relative verbiage should anyone want to make use of such a mechanic but doesn’t want it cluttered up with dice notation for dice not being used.

Damage Class Attack Table
Damage
Class
Armor Class
9[10]
8[11]
7[12]
6[13]
5[14]
4[15]
3[16]
2[17]
Light
12
13
14
15
16
17
18
19
Medium
10
11
12
13
14
15
16
17
Heavy
8
9
10
11
11
13
14
15
X-Heavy
6
7
8
8
9
11
12
13


Monday, July 21, 2014

Exploring Weapon Damage

I've done a fair bit of number crunching the past few days, some of it was on the likely damage charts I posted last week. I've been working out what a bonus to hit or damage actually does in the game. I've bee using the S&W hit chances as a base for my math and AC values used on the charts, if one considers the odds to hit the tables are applicable across editions and versions of the game.

I've cut the charts to a smaller AC range this time about to have charts that fit on this blog on most browsers and because a vast majority of predictable combats would likely be occurring in the AC range used of 9[10] for unarmored on to 2[17] for Plate and Shield.
 



Chart 1: Likely Weapon Damage vs AC
Damage
Die
Armor Class
9[10]
8[11]
7[12]
6[13]
5[14]
4[15]
3[16]
2[17]
d4 (2.5)
1.375
1.25
1.125
1.0
0.875
0.75
0.625
0.5
d6 (3.5)
1.925
1.75
1.575
1.4
1.225
1.05
0.875
0.7
d8 (4.5)
2.475
2.25
2.025
1.8
1.575
1.35
1.125
0.9
d10 (5.5)
3.025
2.75
2.475
2.2
1.925
1.65
1.375
1.1

From the above table you can see how much damage an attacker with no adjustment to hit is likely to inflict upon a foe each round based on the foes armor class . A 1d6 weapon with no adjustments to hit is likely to inflict 1.4 points of damage per round vs an AC 6 opponent (14 points in 10 rounds). 
The charts above do not say a d4 weapon would inflict but 1/2 a point of damage per round (for example) but they do indicate the likely damage  that would be inflicted each round by a weapon vs a specific AC across many combats.

Chart 2:Likely damage by die as a % of damage on a d6.
Damage
Die
% of Damage Inflicted
Compared to 1d6
Average Likely Damage vs AC 9 to 2
d4
71%
0.9375
d6
100%
1.3125
d8
128%
1.6875
d10
157%
2.0625

Take a look at that, eyeballing the raw dice roll tells us a d4 is 2/3rds as effective as a d6, and a d10 is 167% as effective as a d6 but, when we factor in hit probability the numbers shift and damage based on the average damage likely to be inflicted the %’s in the chart above bear out across the board.
So what does a +1 to hit and a +1 to damage actually  compared to a d6 in combat? Traditional tells us each +1 to hit is a 5% improvement in hit chances but the impact on likely damage is more significant.

I'm using a d6 here in this chart and for further analysis as a baseline to compare weapon damage as the d6 was the only die type used for damage in the earliest version of the rules and is still used by some today instead of the more popular variable damage ranges.

Chart 3a: Likely weapon damage with a d6 weapon and a bonus to hit only.
D6 with
Hit Bonus
Armor Class
Average
Likely
Damage
9[10]
8[11]
7[12]
6[13]
5[14]
4[15]
3[16]
2[17]
+1
2.1
1.925
1.75
1.575
1.4
1.225
1.05
0.875
1.487
+2
2.275
2.1
1.925
1.75
1.575
1.4
1.225
1.05
1.662
+3
2.45
2.275
2.1
1.925
1.75
1.575
1.4
1.225
1.837
+4
2.625
2.45
2.275
2.1
1.925
1.75
1.575
1.4
2.0125

Chart 3b
D6 with
Hit Bonus
Average
Likely
Damage
Percent of damage compared to 1d6 alone
+1
1.487
113%
+2
1.662
126%
+3
1.837
142%
+4
2.0125
153%

A +1 to hit means a lot more than a simple bump in chances to hit of 5%. A character with a 1d6 weapon and but a +1 to hit chance bonus will be dishing out 113% the damage of someone with no hit bonus.

Chart 4a: Likley damage with 1d6 and damage bonus alone.
Dmg.
Bonus
Armor Class
9[10]
8[11]
7[12]
6[13]
5[14]
4[15]
3[16]
2[17]
+1
2.475
2.25
2.025
1.8
1.575
1.35
1.125
0.9
+2
3.025
2.75
2.475
2.2
1.925
1.65
1.375
1.1
+3
3.575
 3.25
 2.925
 2.6
 2.275
 1.95
 1.625
 1.3
+4
4.125
 3.75
 3.375
 3
 2.625
 2.25
 1.875
 1.5

Chart 4b:
D6 with
Damage Bonus
of:
Average
Likely
Damage
Percent of damage compared to 1d6 alone
+1
1.6875
 128%
+2
2.0625
 157%
+3
2.4375
 185%
+4
2.8125
 214%

A +1 to damage is pretty telling, with 1d6 each successive +1 in damage is like shifting up 2 in total damage range (from 1d6 to 1d8 for example).

Chart 5a: Likely damage with 1d6 including Hit and Damage bonus
Bonus
Armor Class
9[10]
8[11]
7[12]
6[13]
5[14]
4[15]
3[16]
2[17]
+1
2.7
2.475
2.25
2.025
1.8
1.575
1.35
1.125
+2
3.575
3.3
3.025
2.75
2.475
2.2
1.925
1.65
+3
4.55
4.225
3.9
3.575
3.25
2.925
2.6
2.275
+4
5.625
5.25
4.875
4.5
4.125
3.75
3.375
3

Chart 5b:
D6 with
Hit and
Damage
Bonus
Average
Likely
Damage
Percent of damage compared to 1d6 alone
+1
 1.9125
  145%
+2
2.6125
 199%
+3
 3.4125
 260%
+4
 4.3125
 328%

A +1 to hit and damage with a 1d6 weapon likely produces 145% of the damage of a straight 1d6, pretty sweet.  A +3 to hit and damage is better than 2 normal attacks a round, a +4 hit and damage is better than 3 normal attacks.  Level, ability bonuses, magical bonuses and situational bonuses likely have much more impact then typically expected.

Chart 6a: Average likely damage by die with a hit bonus only vs AC 9[10] to 2[17]
Damage
Die
Hit Bonus Only
+1
+2
+3
+4
1d4
1.0625
1.1875
1.3125
1.4375
1d6
1.4875
1.6625
1.8375
2.0125
1d8
1.9125
2.1375
2.3625
2.5875
1d10
2.3375
2.6125
2.8875
3.1625

Chart 6b: above as % of unmodified d6  (see chart 2)
Damage
Die
Hit Bonus Only
+1
+2
+3
+4
1d4
81
90
100
109
1d6
113
127
143
153
1d8
145
163
180
197
1d10
178
199
220
241
The above shows that hit bonus alone while beneficial is not a significant bonus in regards to likely damage inflicted upon opponents. A weapon that does 1d4 damage would need a +3 bonus to hit to do the same damage as a 1d6 with no bonus to hit. A d10 weapon essentially doubles vs a d6 with a +2 to hit bonus.  

Chart 7a: Average likely damage by die with damage bonus only vs AC 9[10] to 2[17]
Damage
Die
Damage Bonus Only
+1
+2
+3
+4
1d4
1.3125
1.6875
2.0625
2.4375
1d6
1.6875
2.0625
2.4375
2.8125
1d8
2.0625
2.4375
2.8125
3.1875
1d10
2.4375
2.8125
3.1875
3.5625

Chart7b:  above as % of unmodified d6  (see chart 2)
Damage
Die
Damage Bonus Only
+1
+2
+3
+4
1d4
100
129
157
186
1d6
129
157
186
214
1d8
157
186
214
243
1d10
186
213
243
271
A damage bonus alone is clearly more significant than a hit bonus alone. A 1d4 weapon with a +1 to damage done is likely to do as much damage as an unmodified d6 is on average.  Damage bonus is also more significant relative to the unmodified roll of the same dice than one finds with a hit bonus (but no damage bonus).

Chart 8a: Average likely damage by die with hit bonus and damage bonus vs AC 9[10] to 2[17]
Damage
Die
Hit and Damage Bonus
+1
+2
+3
+4
1d4
1.4875
2.1375
2.8875
3.7375
1d6
1.9125
2.6125
3.4125
4.3125
1d8
2.3375
3.0875
3.9375
4.8875
1d10
2.7625
3.5625
4.4625
5.4625

Chart 8b: above as % of unmodified d6  (see chart 2)
Damage
Die
Hit and Damage Bonus
+1
+2
+3
+4
1d4
113
163
220
285
1d6
146
199
260
329
1d8
178
235
300
373
1d10
210
271
340
416
Having modifiers applied to hit and damage is obviously a significant bonus, even a meager +1 to hit and damage bonus would have a 1d6 weapon putting out more damage on average than an unmodified d8. Even a +1 to a d10 weapon allows that weapon to inflict damage above that of 2 attacks with an unmodified d6.  

A whole lot of data, I'm going somewhere with this in a future post but felt it was worth sharing the above before I got there.