A bit of neat 4e math

[Yakk]

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:

Spoiler

Show

-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:

Spoiler

Show

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.

[Random NPC]

Uhm... tl;dr?

[Asbestos]

Uhm... tl;dr?

I think Yakk the Mathmage sums it up pretty well after the last spoiler.

Now, Yakk, am I misinterpreting this (if I am... I blame the getting out of class at 9:40pm part of today) or are you saying that encounters need to be 'harder' to actually be balanced XP-wise?

[Gralamin]

That is neat. Too bad its difficult to equate mathematically how much each effect is worth. Its clear some such as Daze are worth a lot more then others such as weakened.

I think Yakk the Mathmage sums it up pretty well after the last spoiler.

Now, Yakk, am I misinterpreting this (if I am... I blame the getting out of class at 9:40pm part of today) or are you saying that encounters need to be 'harder' to actually be balanced XP-wise?

Assuming they do not have any tricks, yes, see:

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".

[elliott20]

dude, last time I saw an analysis like this, I was reading about a debate on Marines vs. Hydralisks in starcraft.

[Dhavaer]

Now, Yakk, am I misinterpreting this (if I am... I blame the getting out of class at 9:40pm part of today) or are you saying that encounters need to be 'harder' to actually be balanced XP-wise?

I think he's saying they need to be easier. The power growth is too fast compared to the XP.

[Ka'ladun]

Where did the 1.68 and the 0.6 come from. I see that 1/1.68 ~ .6, but how did you derive 1.68 in the first place?

[JaxGaret]

dude, last time I saw an analysis like this, I was reading about a debate on Marines vs. Hydralisks in starcraft.

Did they do the analysis with Stims and without?

[elliott20]

first without then with. Though, in the conclusion of the analysis, even without stims, Marines had better bang for the buck due to faster weapon cool down and cheaper price. (they ended up with some wierd damage per crystal value.)

With stims? marines blow hydralisks out of the water.

this is, of course, assuming that both sides spend equal amount of resources on the units. (But it neglected to factor in the infrastructural and time costs)

[Yakk]

Where did the 1.68 and the 0.6 come from. I see that 1/1.68 ~ .6, but how did you derive 1.68 in the first place?

5 monsters deal 15 (N(N+1)/2) times the damage that 1 monster deals when up against a damage source that kills monsters one at a time.

5^1.68 =~ 15. 5 is the "standard" number of monsters an encounter has.

If you graph [(X)(X+1)/2]/[X^1.68] in the range X = 1 ... 9, you get values from 94% to 112%. That's close to unity. And near 5 opponents, it is really close to 100%.

We have 5^1.68 * X = Y is the 'real equation' for how tough 5 monsters are, compared to fighting the monsters one-at-a-time.

Take the ^.6 root:
5 * X^.6 = Y^.6

So by taking the 0.6th power of the XP values, we allow them to be added up roughly linearly. :-)

...

I'm saying that power growth at low levels in monsters outpaces the rate of XP growth significantly, assuming the many assumptions I made.

By mid levels the power growth rate of monsters has slowed down to a being just a bit faster than "it is worth".

And by epic tier, the power growth rate of monsters is really close to spot on.

The assumptions I made, however, may or may not be accurate.

Monster damage output should, in theory, be based off of PC HP increases (which is similar to monster HP increase rate), and based off of PC ability to stop monsters from attacking (which goes up), and based off of PC ability to heal themselves as a fraction of their HP (which I think goes up).

So ... monster damage output might be faster than I modeled. Or slower. I need to work that out.

The balance I implemented was based on monster-vs-monster numbers. Possibly WotC took into account the fact that a low level character actually hits more accurately than a typical monster, and a high level character less accurately. This ... would change some of the curves, I don't know how much.

[Edge of Dreams]

What makes this even harder to deal with is that player damage output does not grow anywhere near linearly. Instead, it goes in spurts and jumps, combined with long periods of no change.

If I get a +3 weapon at level 11, and nothing better for a while, at level 14 the stat change might give me +1 more damage over those three levels. Let's say then that I also suddenly decide to take Dwarven Weapon thingy and happen to get the item from Adventurer's vault that grants +4 to all melee attack damage. Suddenly the difference over three levels is +6.

*IF* you could work out a respectable approximation of the rate of growth of player attack and damage boni, I'd love to see it. The difference between player attack/damamge growth and monster is huge.

This also completely ignores the effect the number of powers you have has on battle. During heroic tier, you keep getting more new powers, so you can spend more powers per fight after each level or two. In paragon, you still get two new attack powers, but mostly it's swaps. In epic, you get no new attack powers, just swaps. That could account for the different monster growth rates you found.

[Yakk]

I'm basically presuming that:
1> Your hit rate remains basically unchanged. With reasonably optimized characters, this actually works.

2> The number of hits it takes to take a target down remains unchanged.

#2 is "on average" over the day, and should include using action points, magic item uses, daily powers, encounter powers, and at-wills.

That is the idea that in 4e, combats should take about the same amount of time at all levels.

Or, to be more accurate, I'm presuming this is true _on the part of the monsters_, because I'm trying to measure how hard monsters should be. Characters can have strong and weak levels, and the above analysis holds fine.

[Tengu_temp]

While I trust Yakk's math...

I'm basically presuming that:
1> Your hit rate remains basically unchanged. With reasonably optimized characters, this actually works.

As the table I cooked up some time ago shows, this is very hard to achieve - it is possible if you're using weapons of your level+4, but remember that, by default, a 5-man group should only have one such item at a time.

[CabbageTheif]

this is reassuring. my dm for 4th was all worried that he was reading the rules wrong... the party was divided, and we had 2 3rd levels and a 1st level fighting 6 dwarves, and 2 3rd levels fighting 2 bats and 2 skeletons... and the ones fighting the dwarves had it easy.

also, where do they find bats with a tail attack?! that really confuses me*grumbles about a bat killing a dragonborn paladin with a stupid stub of a bat tail*

[Yakk]

Player to hit:
+4 (attribute) + 15/29 + 6/29 (enchantment) + 4/29*L (attribute-ups) + 1/29 (Demigod) + 1/29 (Paragon paths that offer +1 to hit, +1 crit width, or the like) + Proficiency + Other
= +4 + L * (15+6+4+2)/29
= +4 + Prof + Other + L * (27/29)
Now, Prof stays relatively constant from level 1 to level 30.

"Other", however, can go up. In particular, bonuses to being able to hit from debuffing someone's AC (which tends to scale from -4 at level 1 to -8 or -9 at level 30), boosting your own to-hit chance (which scales from +4 at level 1 to +8 to +9 at level 30), getting a superior weapon, gaining magic items that grant you bonuses to hit (BZR weapons for example), etc etc.

And even without that other category, you gain +27 to hit over the 29 levels, which is pretty close to even-steven.

Yes, some paragon paths/epic destinies don't offer bonuses to hit or to attributes: but, in theory, they should offer something as good. So I don't feel that including it is wrong -- rather, failing to take into account such bonuses means that you end up under-estimating the power of high level characters.

I expect there to be dips and lulls, and other times when you are ahead, and the like as you get upgrades and gain levels.

[Tengu_temp]

(Paragon paths that offer +1 to hit, +1 crit width, or the like)

Does anyone have a list of those? I'm curious, but too lazy to dig them up myself.

[Hal]

I was told there wouldn't be any math in this course . . .

[The New Bruceski]

I was told there wouldn't be any math in this course . . .

We like to say that to identify the gullible students.