Yakk
2008-10-20, 09:02 PM
We have Bobs and Alices. Each has 100 HP, and deals 10 points of damage on a hit.
Alice has 21 in all defenses, and +10 to hit.
Bob has 21+X in all defenses, and +10+X to hit.
We arrange Alices and Bobs in a line, and the first ones fight to the death. Then the next one. In the limit, what is the ratio of Alices killed to Bobs killed?
Quite clearly, it is the ratio of Bob hits to Alice hits.
Alice hits Bob 50%-5%*X percent of the time.
Bob hits Alice 50%+5%*X percent of the time.
Ratio := Bob hits / Alice hits = [(10+X)/20] / [(10-X)/20]
This generates this table:
-9 0.052631579
-8 0.111111111
-7 0.176470588
-6 0.25
-5 0.333333333
-4 0.428571429
-3 0.538461538
-2 0.666666667
-1 0.818181818
0 1
1 1.222222222
2 1.5
3 1.857142857
4 2.333333333
5 3
6 4
7 5.666666667
8 9
9 19
which is reasonably close to 3^(x/5) for values near 0 (ie, values within +/- 5).
Now, player and monster HP goes up linearly with level, as should damage-per-attack-on-average. Well, note quite linearly -- a level 1 monster has about 20 HP, and it gains about 8 HP per level. Monster HP should go up (roughly) at the same rate that monster damage goes up.
We care about the number of swings it takes for Alice to kill Bob, compared to the number of swings it takes Bob to kill Alice. If it takes each creature (say) 3 hits to kill itself, then we end up with:
AliceHitsToKill = (BobL+1.5)/((AliceL+1.5)/3)
BobHitsToKill = (AliceL+1.5)/((BobL+1.5)/3)
BobHitsToKill/AliceHitsToKill = [(AliceL+1.5)/((BobL+1.5)/3)] / (BobL+1.5)/((AliceL+1.5)/3)
= (AliceL+1.5)^2 / (BobL+1.5)^2
BobHitsToKill * (BobL+1.5)^2 = AliceHitsToKill * (AliceL+1.5)^2
Or, in short, your "raw HP advantage" and "raw damage advantage" goes up roughly with (L+1.5)^2.
So we have our "one-on-one power equation" of:
3^(L/5) * (L+1.5)^2
However, the XP tables in 4e are built around building an encounter. And a monster that could take on 5 other monsters one-at-a-time would die horribly to 5 of those monsters at once.
However, if that monster could take on 15 of the monsters one-at-a-time, it would probably also be able to take on 5 at once.
In the area in question (1 to 9 creatures), # of creatures^1.68 is a reasonable approximation of the effects of numbers on a fight.
So if we want to turn a "one on one" power ratio into a "account for number of creatures" ratio, we should raise the "one on one" ratio to the power 0.6.
(3^(L/5) * (L+1.5)^2)^0.6 =~ 3^(L/8) * (L+1.5)^1.2
Now what we really care about is the ratio in XP between adjacent levels (for encounter building). If you compare the ratios between adjacent levels from the above equation to the 4e standard table, we get:
1 1
2 1.374307089
3 1.292505126
4 1.251046862
5 1.226617088
6 1.089701709
7 1.110933609
8 1.123722257
9 1.131897348
10 1.023623843
11 1.056607084
12 1.078454616
13 1.093677532
14 0.994229349
15 1.030484838
16 1.055257124
17 1.073021998
18 0.977609928
19 1.015130813
20 1.041153794
21 1.060085953
22 0.966924483
23 1.005024797
24 1.031673517
25 1.051223454
26 0.959476468
27 0.997868952
28 1.02486333
29 1.044772176
30 0.953988196
31 0.992536026
32 1.019734476
33 1.039865945
34 0.949776136
35 0.988408165
In effect, from level 1 to 5 the power growth of monsters is probably 25% too fast per level, from 6 to 10 it is probably 10% too fast per level, from 11 to 20 it is probably 5% too fast per level, and from 21+ it is about 1% too fast per level, given the XP values of the monsters.
This doesn't take into account "more tricks" on the part of monster or players that isn't "increase in average damage done", "increase in average HP", "increase in defenses/AC", "increase in to-hit rolls".
Practically, players ability to self-heal and stun monsters goes up as they gain levels. That means an even-level challenge should probably be doing more damage per-attack at higher levels than at lower levels.
A more complete model could take this into account.
However, this was amusing. I find it interesting that this "first principles" generation of an XP table ended up generating nearly exactly the XP ratio of 4e D&D between monster levels from level 11 to 30, off only by a few percentage points per level.
Alice has 21 in all defenses, and +10 to hit.
Bob has 21+X in all defenses, and +10+X to hit.
We arrange Alices and Bobs in a line, and the first ones fight to the death. Then the next one. In the limit, what is the ratio of Alices killed to Bobs killed?
Quite clearly, it is the ratio of Bob hits to Alice hits.
Alice hits Bob 50%-5%*X percent of the time.
Bob hits Alice 50%+5%*X percent of the time.
Ratio := Bob hits / Alice hits = [(10+X)/20] / [(10-X)/20]
This generates this table:
-9 0.052631579
-8 0.111111111
-7 0.176470588
-6 0.25
-5 0.333333333
-4 0.428571429
-3 0.538461538
-2 0.666666667
-1 0.818181818
0 1
1 1.222222222
2 1.5
3 1.857142857
4 2.333333333
5 3
6 4
7 5.666666667
8 9
9 19
which is reasonably close to 3^(x/5) for values near 0 (ie, values within +/- 5).
Now, player and monster HP goes up linearly with level, as should damage-per-attack-on-average. Well, note quite linearly -- a level 1 monster has about 20 HP, and it gains about 8 HP per level. Monster HP should go up (roughly) at the same rate that monster damage goes up.
We care about the number of swings it takes for Alice to kill Bob, compared to the number of swings it takes Bob to kill Alice. If it takes each creature (say) 3 hits to kill itself, then we end up with:
AliceHitsToKill = (BobL+1.5)/((AliceL+1.5)/3)
BobHitsToKill = (AliceL+1.5)/((BobL+1.5)/3)
BobHitsToKill/AliceHitsToKill = [(AliceL+1.5)/((BobL+1.5)/3)] / (BobL+1.5)/((AliceL+1.5)/3)
= (AliceL+1.5)^2 / (BobL+1.5)^2
BobHitsToKill * (BobL+1.5)^2 = AliceHitsToKill * (AliceL+1.5)^2
Or, in short, your "raw HP advantage" and "raw damage advantage" goes up roughly with (L+1.5)^2.
So we have our "one-on-one power equation" of:
3^(L/5) * (L+1.5)^2
However, the XP tables in 4e are built around building an encounter. And a monster that could take on 5 other monsters one-at-a-time would die horribly to 5 of those monsters at once.
However, if that monster could take on 15 of the monsters one-at-a-time, it would probably also be able to take on 5 at once.
In the area in question (1 to 9 creatures), # of creatures^1.68 is a reasonable approximation of the effects of numbers on a fight.
So if we want to turn a "one on one" power ratio into a "account for number of creatures" ratio, we should raise the "one on one" ratio to the power 0.6.
(3^(L/5) * (L+1.5)^2)^0.6 =~ 3^(L/8) * (L+1.5)^1.2
Now what we really care about is the ratio in XP between adjacent levels (for encounter building). If you compare the ratios between adjacent levels from the above equation to the 4e standard table, we get:
1 1
2 1.374307089
3 1.292505126
4 1.251046862
5 1.226617088
6 1.089701709
7 1.110933609
8 1.123722257
9 1.131897348
10 1.023623843
11 1.056607084
12 1.078454616
13 1.093677532
14 0.994229349
15 1.030484838
16 1.055257124
17 1.073021998
18 0.977609928
19 1.015130813
20 1.041153794
21 1.060085953
22 0.966924483
23 1.005024797
24 1.031673517
25 1.051223454
26 0.959476468
27 0.997868952
28 1.02486333
29 1.044772176
30 0.953988196
31 0.992536026
32 1.019734476
33 1.039865945
34 0.949776136
35 0.988408165
In effect, from level 1 to 5 the power growth of monsters is probably 25% too fast per level, from 6 to 10 it is probably 10% too fast per level, from 11 to 20 it is probably 5% too fast per level, and from 21+ it is about 1% too fast per level, given the XP values of the monsters.
This doesn't take into account "more tricks" on the part of monster or players that isn't "increase in average damage done", "increase in average HP", "increase in defenses/AC", "increase in to-hit rolls".
Practically, players ability to self-heal and stun monsters goes up as they gain levels. That means an even-level challenge should probably be doing more damage per-attack at higher levels than at lower levels.
A more complete model could take this into account.
However, this was amusing. I find it interesting that this "first principles" generation of an XP table ended up generating nearly exactly the XP ratio of 4e D&D between monster levels from level 11 to 30, off only by a few percentage points per level.