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Yakk
2008-06-30, 02:14 PM
This is an attempt to balance 4e encounters via a "damage budget" mechanism.

The theory goes as follows: Given a 6+ round fight in which players use no more than half of their Daily powers, the players will have probably consumed most of their non-at-will damage budget for the fight.

So you can work out, roughly, how much damage a party of 5 characters will do in ~6 rounds. This rate will be slightly higher at the start, and peter out towards the end.

We then work out roughly how long a group of 5 even-level monsters will live under this level of incoming damage, and work out how much damage the party will take, roughly, during those first 6 rounds.

Then, we work out how much longer the fight lasts, factor in at-will powers of the Party, and work out how long the clean-up takes.

This gives us a value: the damage budget of an even-level encounter: how much damage it should do, roughly before being defeated.

Then, we use this to build a solo monster from first principles. :) We can tweak HP and damage output in such a way that we can generate a fight that is about as dangerous to a party as a swarm of even-level minions, normal monsters, or elites.

This doesn't guarantee a fun encounter: but it does mean that we can see if the "solo monsters are not dangerous enough, and are too tough for their danger, for ideal fun".

...

A first-run approximation.
At level 1, each party member deals an average of 9 damage from at-will attacks that hit. Each attack has a 60% chance of hitting per round.

Per-encounter attack powers do an average of 14 damage, with a 65% chance of hitting. 1/4 of the party uses daily powers, which do an average of 19 damage, and hit 70% of the time. (Players are smart, and only use encounter/dailies when they are more likely to hit).

So, over 5 rounds, using half a Daily and an encounter power, each character dishes out (.7 * 19 + 3* 9*.6)/4 + (14*.65) + (9*.6)*3 damage, or about 37.7 damage. After that, they deal about 5.5 damage per round.

5 characters = 188.5 damage over 5 rounds, or ~37.7 damage per round.

Monsters, meanwhile, hit about 50% of the time. At level 1, they deal about 8 damage per hit, or 4 damage per round per monster. Each has a single per-encounter attack that does about 14 damage, hitting 60% of the time.

They have about 30 HP each. So ~1 monster dies per round. I'll assume 1 monster dies before it uses it's per encounter.

The party takes 4*(5+4+3+2+1 - 4) + 14*.6*4 = ~77.6 damage.

Now, let's build a solo monster from this.

We want it to last about 6 rounds. It should do about 80 damage to the party.

Let's assume it has 3 rounds of doing "double damage" via encounter powers. It also does +50% damage once bloodied.

If it has +4 defenses (this is to prevent stun-lock cheese -- misses rarely stun), and half of the players use a daily attack power, player damage output over 6 rounds becomes:
(.5 * 19 + 9*.4)/2 + (14*.45) + (9*.4)*4 = 27.25
or a total of 136.25 HP at level 1.

Supposing it is normal for 3 rounds, and bloodied for 3 rounds. It burns two damage-double encounter powers while normal, and one while bloodied.

If it has a 50% chance of hitting with it's powers, and one action point to burn at full and at bloodied, it ends up with:
(.5*X*2) + (.5*2X) + (.5*1.5*X*3) + (.5*2*X) = 5.25 X total damage.

It's damage budget is 80, giving us X = 15 damage.

So, a level 1 solo monster created based off of this (6 round expected lifespan):
Gibbering Ghost [Medium sized Solo Skirmisher, level 1, 400 XP]
136 HP.
1 Action Point, +5 Saves
Speed: 6 Alignment: Evil
Vunerable: 5 Radiant Resist: 5 Necrotic
18 AC 16 Fort 15 Reflex 18 Will
Str: 12 Con: 14 Dex: 12 Int: 8 Wis: 16 Cha: 18
Attacks:
Blade of Insanity (Basic Melee Attack, At-will): +4 vs Will, 1d8+4 Psychic Damage. Slide the target 1 square, then the target makes a basic attack against any target of the Gibbering Ghosts choice. (This basic attack does not provoke any opportunity attacks)

Mind Shards: (Basic Ranged Attack, At-will, Ranged 10) +4 vs Reflex. 1d6+1 Psychic Damage, and target takes an ongoing 6 Psychic Damage (save ends). Miss: Target is prone.

Illusions of Madness: (Burst 5, Recharge 4/5/6, Minor action) Exchange the places, via teleport, of any creatures in the Burst 5. New locations must be legal positions. This includes the Gibbering Ghost itself.

Winds of Hell: (Encounter, only after the Ghost takes damage, recharged when bloodied) Burst 1, +4 vs Fortitude, 1d10+4 Psychic Damage, target is dazed until the start of the Ghosts next turn.

Cage of Souls: (Move, Encounter, Only right after the Ghost has damaged a target within 1 square, recharge when bloodied). Ghost Teleports into the location of the target it hit. Target creature is pinned but cannot be directly attacked, and all damage to the Ghost is half-dealt to the Target. The target may attempt to break free: Charisma vs Will, hit and the creature escapes the Cage of Souls, miss and both Ghost and Victim take 1d10 psychic damage (this damage is not split). After the Ghost takes 15 damage, the Cage fails, and the Ghost teleports 3 squares.

Shadow Walk: (Move, at-will) Teleport 3 squares.

...

Damage done to party (about the same in bloodied and unbloodied phases)
Pre-bloodied: ~14 basic attack (ranged or melee) @ 50% chance
Scream of Hell: 9.5 burst 1 (2.5 targets) @ 40% chance
Cage of Souls: ~15 damage (pretty much guaranteed)

[14*.5*2.5 + 9.5*2.5*.4 + 15 = 39.5]*2 = 84 damage.

Right on the damage budget.

So the question is, what happens if you try this quick-attempt at a level 1 solo party? How does it compare to a fight against (say) 5 normal level 1 creatures, in actual play? How far off are my damage approximations for PCs/Monsters?

How hard would it be to reduce this to a formula, both on the player-damage-dealing side, and the monster-damage-dealing side?

JaxGaret
2008-06-30, 04:31 PM
I've been awaiting this thread, Yakk. :smallsmile: I'll come back later with a concrete reply.

Yakk
2008-07-01, 01:47 PM
Hmm. It appears as if the damage suggestions of the monster creation thing ... are junk. Do monster manual monsters line up with the suggestions?

Here is the averages of the various damages for a given level of monster:


Basic Limited
L M H L M H
1-3 6.5 8.5 10 13.5 14 16.5
4-6 7.5 9.5 13 14.5 17.5 20.5
7-9 9.5 12 14 18.5 21.5 23
10-12 9.5 12 15.5 18.5 23 27
13-15 11.5 15 16.5 22.5 24 28
16-18 12.5 16 20.5 22.5 29 33
19-21 14 18.5 20.5 25 29 33
22-24 15 18.5 22 26 34 35.5
25-27 18 22.5 23 31 36.5 41.5
28-30 19 23.5 28 31 36.5 41.5


You'll notice that they are on crack -- lots of flat parts, nearly random differences between low/medium/high damage in a category, etc.

Still, with a bit of effort, I might be able to generate a ... less ad-hoc damage curve, that doesn't have 6+ level damage plateau issues. And where the difference between low/high damage is less random. :)

Anyone see any errors in the above?

When I graph them, they honestly look like approximations of lines.

...

Ok, so if we take the average of the ratio of each category to the basic medium damage at that level, we get about:
BasicL: 80%
BasicM: 100%
BasicH: 120%
LimitedL: 145%
LimitedM: 170%
LimitedH: 195%

At level 2ish, basicM damage averages at 8.5.
At level 29ish, basicM damage averages at 23.5.

A2+B = 8.5
A29+B = 23.5
A27 = 15
A =~ 0.555, or 5/9
B =~ 7.4

So:
Basic Low Damage: Level*4/9 + 6
Basic Medium Damage: Level*5/9 + 7.5
Basic High Damage: Level*6/9 + 9

This doesn't actually work that well for Limited damage. Doing something similar for it, we get:
Limited Low Damage: Level*2/3 + 12
Limited Medium Damage: Level*4/5 + 13.5
Limited High Damage: Level*9/10 + 15.5

Hurm.

JaxGaret
2008-07-02, 10:42 PM
Argh, sorry Yakk, been super busy lately. I'll get to this ASAP.

Yakk
2008-07-09, 12:39 PM
Just adding a link to a thread related to this work here:
http://www.enworld.org/forum/showthread.php?t=229092

Thrawn183
2008-07-09, 02:28 PM
So just to clarify for for members of the playground who might not understand what you're trying to do (and to make sure that I'm understanding you, I'll be honest).

You are using HP and damage to balance encounters because there aren't many status effects around anymore. You decide how long a monster should survive and using the average damage output by the party per round you figure out how many HP the monster needs to survive that long. In addition, you decide how many of the party's resources you want consumed (in the form of hit points). Using the number of rounds the monster will survive and and the amount of damage it needs to inflict in that amount of time, you can come to an average damage per round (which I guess could be broken more finely into multiple attacks) so that it inflicts the right amount of damage.

I'm interested in seeing how the math works out with encounters against multiple monsters seeing as the average damage of the monsters as a group decreases as each monster is killed. I think you should pay particular attention to AoE's though when you get to this point because they could probably do screwy things with the party damage output when faced with too many monsters.

Sadly, I don't think this kind of thing can really be adapted to minions.

Yakk
2008-07-09, 04:22 PM
*nod* -- status effects that don't do damage or end the encounter (ie, pretty much all of them) don't change the length of the encounter.

4e has an implied and controlled pacing -- it takes 6 to 10 rounds to beat an encounter.

I'm taking the "5 normal monsters of the same level" as my base benchmark, and working out (roughly) how much damage they do to the party. This is a measure of how much "threat" they provide.

It has been observed that solo monsters don't have the same amount of "threat" as normal monsters -- that they reduce to a punching-bag of HP that takes a long time to take out, but they don't threaten the party's survival.

In an encounter with "normal" monsters, you are threatened early on because the enemy damage output is high (lots of living creatures). On the other hand, you have lots of per-encounter powers to use to wipe them out quickly. As you kill off the "normal" monsters, their threat per-round to you is reduced -- but you have used up defensive and offensive powers, so you are less able to deal with that threat. Finally, often you'll be reduced to swatting the last few creatures.

If solo monster damage output isn't tuned right, or solo defense isn't tuned right, you can easily end up reaching a point where solo damage output isn't enough to threaten the party seriously at any point, and the monster has an ass-load of HP to work through. Zzzz.

So... I'm attempting to approach the HP and Damage output of Solo monsters from first principles.

---

I'm ignoring AOEs for now, because while they do tend to do more damage than single attacks, they also don't kill a monster dead ASAP. Similarly, debuffs will almost certainly be better against a solo monster (because a debuff on a single solo monster is roughly the same as debuffing every normal monster in the encounter).

---

I have to keep track of how much damage a solo monster will deal the party over an encounter -- otherwise, the party could be flattened or not even blink. This is a function of how long the creature lives, and it's per-round damage output (and, how easy it is to daze/stun/debuff/etc).

---

What I really need to do is work out the normal monster damage budget as a function of level. Hence the thread link. :)

Starbuck_II
2008-07-09, 07:02 PM
:smallmad:

A first-run approximation.
At level 1, each party member deals an average of 9 damage from at-will attacks that hit. Each attack has a 60% chance of hitting per round.



Here is where you lost me.

What party members did you use for this average?

Thrawn183
2008-07-09, 10:08 PM
Well, if you're interested, I do have an example.

My friend and I built 5 level 5 PC's and sent them up against the early EL 6 encounter from the preview and then an EL 7 Red Dragon. That dragon would have taken PC's down very quickly had we not decided after about 1and 1/2 rounds to have it divvy up its attacks evenly among the party. In addition the dragon took a really long time to go down even though we intentionally metagamed the parties actions (to simulate a party that has worked together for a long time) and assumed that they'd saved every daily for the fight.

Trying to balance encounters to the difficulty of the encounters listed in the Monster Manual might not be the best idea, because I'm starting to think some of them are way too hard, and long. But kudos, I'll be watching where this goes with interest.

Conners
2008-07-10, 01:01 AM
You should look at this thread, it's also on the topic of "Boss Fighters" (aka: Solo-Monster fights): http://www.giantitp.com/forums/showthread.php?p=4553749

Yakk
2008-07-22, 04:23 PM
9 damage at level 1? That's a level 1 character with 18 in their stat, attacking with a power that does ~5 damage on average from dice.

Magic Missile, Long Sword, Eldrich Blast, etc etc.

It might be a little low.

...

I'm going to attack this from the other end.

Monster DEF:~ 14+Level (-2 for non-AC).
Player ATK:~ 6+Level*0.9 (-2 for non-Weapon).
Monster HP:~ 8.5*L + 20

If we presume that a monster takes on average (6+level/10) rounds to beat into a pulp, we can produce an average damage-per-round damage rate for players. The chart resulting does not look that ridiculous.

A simpler model is to presume players deal 8+Level*2 damage per hit on average. This gives something similar to the previous model, but with simpler math. :-)

Damage-per-action (including accuracy) then works out to something like 5+5*level/4, or (1+level/4)*5, which might work out to be easier.

(This would mean that some pretty insane combos are expected to be used at level 30.)

Hurm.

5 Monsters HP @ standard Defence should die in, say (6+level/10) rounds.

ATK is~ 6+.9*level (-2 for non-AC)
Base DEF is~ 14+Level (-2 for non-AC)

Debuff (until next round) powers, normal monsters: ATK vs DEF, one monster-round DPS reduced.

Debuff (until save) powers, normal monsters: ATK vs DEF, ~1.8 monster-round DPS reduced.

Debuff (until next round) powers, solos: ATK vs DEF, one solo-round DPS reduced.

Debuff (until save) powers, solos: ATK vs DEF, 20/(21-SAVE_TARGET) solo-round DPS reduced.

With "smooth" monster counts and (6+Level/10) rounds of combat, and (bad assumption) even kill rate over the period, the players take ~(6+Level/10)*(6+Level/10)/2 monster-rounds of damage.

With a solo, the players take one solo-monsters worth of damage per round for the entire (6+level/10) rounds.

So:
S * (6+level/10) = M * (6+level/10)*(6+level/10)/2
S =~ M * (6+Level/10)/2
S =~ M * (3 + Level/20)

(Interestingly, the DMG tells you to boost solo monster HP more at higher levels!)

Now, we do not want stun-lock to kick ass too much on solo monsters.
6+.9*level vs 14+level gives us:

(12-Level/10)/20 chance to hit a normal monster.

Let X be the Defense boost of a solo monster.

(12-Level/10)/20 * M is the damage reduced by a one-round debuff on a normal.

(12-Level/10-X)/20 * S is the damage reduced by a one-round debuff on a Solo.

Set them equal:
(12-Level/10)/20 * M = (12-Level/10-X)/20 * (3 + Level/20)*M
(12-Level/10) = (12-Level/10-X) * (3 + Level/20)
0 = [(12-Level/10)+(-X)] * (3 + Level/20) - (12-Level/10)
X(3 + Level/20) = [(12-Level/10)] *(3 + Level/20) - (12-Level/10)
X = [(12-Level/10)(3 + Level/20)-(12-Level/10)]/(3 + Level/20)
X = (12-Level/10) * [1 - 1/(3+Level/20)]

Crunch crunch... Holy crap. +8 down to +7 defenses! That's pretty extreme.

At +4 to all defenses, we R as the ratio between stunning a solo and a normal of:
R := (8-Level/10)/20 * (3+Level/20) / (12-Level/10)/20
which goes from 2x at level 1 to 2.5x at level 30.

At +5 to all defenses, we get 1.76 at level 1, up to 2x at level 30.

I like the number +5 to all defenses. It means "until next round" powers are still damn good on solos, but not crazy-good.

Now, for Solo HP (SHP). It starts out at 5*MHP =~ 5*(8.5*L+20) at standard (12-Level/10)/20 hit rate.

But we are boosting solo defenses by +5 over standard creatures. So we claw back their HP.

SHP / (7-Level/10)/20 = 5*MHP / (12-Level/10)/20
SHP / (7-Level/10) = 5*MHP / (12-Level/10)
SHP = MHP*5*(7-Level/10)/(12-Level/10)

Hurm. SHP ends up being 2.9 down to 2.2 times standard monster HP.

Then again, "damage increasing debuffs" like AC debuffs are way more effective on solo monsters. But AOE damage is less effective. Hmm.

Let's make SHP = 3*MHP, and SDEF = MDEF+5 (crap, Solo Soldiers have 21+Level AC!)

This requires us to go back and calculate how much damage solos should do. :-) (which ends up being ... -3% at level 1, up to -14% at level 30, compared to what we had before -- which happens to make stunning worth ~75% more on both solo and normal creatures).

This changes our solo damage-per-round to be about 3+Level/30 (down from 3+level/20) in the end.

...

The next thing to look at is "save powers". There is some problem with "perma-stun" type effects, where a debuff to save chance causes problems.

What if solos had a +5 to their save but it increased each time they failed the save. Ie, you have a +5 round 1, then a +10 round 2, then a +15 round 3, etc?

That means a massive debuff on save chance is still very useful, but it doesn't break the game when used on a solo.

...

From this approach:
Solo HP is 25*Level+50

Solo Defense is +5 over standard monsters of that type.

Solo Saves are +5, but they get a +5 cumulative bonus every time they fail a save.

Solo damage output is 3x normal at level 1+, 3.5x normal at level 11+, 4x normal at 21+, and 4.5x normal at level 30.

...

Here is another question: what is normal damage output for monsters?

How about we say 60% of what a player can do. PCs are about as tough as monsters, and at 60% of the player does, we have a creature that could threaten a party, but would find it hard to kill them all. (and players have healing powers).

Monster ATK 5+Level (-2 vs non-AC), with some monsters being more/less accurate who do different damage amounts, or who have less / more defenses and/or HP.

I'll use a Hide armored Rogue for PC AC (it is shockingly close to Plate AC).

At level 1, it is 18 AC (3 Hide, 4-5 Dex, 0-1 other).
At level 30, it is 10+([email protected]+5)+([email protected]+8-9)+([email protected])+([email protected]+2-3) =~ 41 AC

Or, about 17+Level*.8.

Monster Hit Chance is ~ [(5+Level)-(17+Level*.8)+21]/20 = [9+Level/5]/20

Before we worked out that a PC should do about 5+1.25*L damage per round of attempting to hit. 80% of this is 3+.75*L.

This works out to a monster damage-per-round-of-hits of about 7.5+0.9*L.

Solo damage-per-round ends up being about 20+L*4.

...

This post's mathematical model is:
Normal Monster:
HP: 20+8.5*L
Dam/Round of Hits: 7.5+0.9*L
AC: 14+L (Other def: -2)
ATK (vs AC): 5+L (other target: -2)
Saves: On a 10+

Solo Monster:
HP: 50+25*L
Dam/Round of Hits: 20+L*4
AC: 19+L (Other def: -2)
ATK (vs AC): 5+L (other target -2 -- no change?)
Saves: On a 5+, gains a +5 bonus cumulative against powers it fails a save against until it wins.

Different subtypes of monsters would vary:

Brutes have -2 AC, -2 ATK, +25% HP, and do more damage (how much?)
Soldiers have +2 AC, +2 ATK, and do less damage (how much?)
Artillery have -2 AC, +2 ATK, and 25% less HP.
Lurkers have -25% HP.
Controllers have +1 vs non-AC defenses ATK.

I presumed that Solo monster stats don't vary -- but maybe they should. That would grant them extra ATK, which would mean we would have to drop their damage output per hit.

...

Now, how do you use this?

Well, we look at the "all hit" damage budget, and we break things down.

In the ~(6+level/7) rounds that a solo monster is active (call it 6 low heroic, 7 heroic, 8 paragon, 9 high paragon/low epic, 10 high epic), it gets 6+level/7+2 standard actions (action points!).

I'd advise delaying the use of the 2nd action point until the creature is bloodied (ie, give it a 2nd action point when it gets bloodied) to prevent alpha-strike instant-dead PCs.

Here is a damage-budget table for solos:


Budget At-Will Cost
1 55 8
2 75 8
3 95 8
4 115 8
5 135 9
6 160 9
7 180 9
8 205 9
9 230 9
10 255 9
11 285 9
12 310 10
13 340 10
14 370 10
15 400 10
16 430 10
17 460 10
18 490 10
19 525 11
20 560 11
21 595 11
22 630 11
23 665 11
24 705 11
25 745 11
26 780 12
27 820 12
28 865 12
29 905 12
30 945 12


This presumes an at-will damage of (10+Level). Additional at-will damage can be purchased at the At-will cost column cost (which is based off of the estimated number of standard actions the creature gets in a fight).

The main purpose of the Budget is used to pay for additional per-encounter type attacks.

A per-encounter standard action attack gets the "at will" damage free. Further damage must be paid for from the budget. (If you boost "at will" damage, your encounter power also gets that boost).

Each recharge of such powers gets charged as well. Things that have a chance to recharge, estimate how often they will recharge, and charge that. (each pip on a random is about 1+Level/50 recharges).

Per-encounter powers that do not use standard actions, or otherwise are free, do not get the at-will damage free.

AOE "infinite target" attacks get -2 to hit, and cost ~2.5x as much (less if small target region, more if large target region, etc).

Multi-target powers cost per-target. Single target powers should be weaker.

The at-will attack should generally be split between ~two targets, especially at higher levels, or you will kill players.

The At-will cost column is what you pay for +1 or -1 at-will damage. Don't change it too far, or the rounding in the table will screw up.

...

I wonder if that will work. :-)