A statistics question

[Dannyalcatraz]

Actually, we work together pretty well...3 of us have been gaming together for going on 20 years, so we know pretty well how each of us thinks...and those who haven't been a part of that triad have all known each other about the same length of time, (with about 10 years of
shared gaming experience).

As a result, we generally have DMs who set up wicked encounters.

Thanks for the crunch, Agback. Pretty much what I suspected in terms of "it depends" with an edge for the Str based monk overall.

 

[Agback]

Is it worth looking at a strong monk fighting a dextrous monk? Let's try the Level 6 NPC specimens out of the DMG.

Dextrous monk: hp 36, AC 16, AB +8/+5, damage d6+3 (kama)
Strong monk: hp 36, AC 15, AB +9/+6, damage d6+4 (kama)

Dextrous monk hits on a 7 and a 10, a total of 25 times in 20 turns, for 6.5 hp/hit: average damage output = 8.13 hp/turn.

Strong monk hits on a 7 and a 10, a total of 25 times in 20 turns, for 7.5 hp/hit: average damage output = 9.38 hp/turn.

The fight goes to the strong monk in 3.84 turns, with 4.8 hp to spare.

Let's put each of these 6th-level monks against a troglodyte {hp 13, AC 15, AB +1, damage d8 (longspear)}.

Dextrous Monk hits on a 7 and a 10, a total of 25 times in 20 turns, for 6.5 hp/hit: average damage output = 8.13 hp/turn. Trog hits Dextrous Monk on a 15 (6 times in 20 turns) for 4.5 hp/hit: 1.35 hp/turn. Trog goes down in 1.6 turns on average, dishing out an expectation of 2.16 hit points of damage.

Strong monk hits on a 6 and a 9, a total of 27 times in 20 turns, for 7.5 hp/hit: average damage output = 10.13 hp/turn. Trog goes down in 1.28 turns on average. But in that interval the trog hits on a 14! Seven times in twenty turns for 4.5 per hit, an average of 1.58 hp/turn. Expected damage delivered in 1.28 turns: 2.01 hp. Strong Monk still outclasses Dextrous Monk, even with five challenge levels of easybeat.

We might have to go to 9th level monks against CR 1 esaybeats to show what DEXy monks are better at than strong monks: winning against vastly inferior mêlée combatants.


The basic principle is this: +1 to AC makes a big difference to the outcome of a fight when the opponent's chance to hit is small, ie. in an easy fight. The +1 to AB makes a big difference when the monk's chance to hit is low, ie. in a hard fight. The +1 to damage makes a big difference when damage is low, ie. at low levels. So DEX is better than STR for high-level monks fighting low-CR opponents. STR is better than DEX for low-level monks and for any monk who fights challenging opponents.

 

[Agback]

Let's pit those two 6th-level monks against level-appropriate challenges, such as an average xorn. Xorns take half damage from slashing weapons, so the monks won't use their kamas.

Dextrous monk: hp 36, AC 16, AB +6/+3, damage d8+2 (bare hand).
Strong monk: hp 36, AC 15, AB +7/+4, damage d8+3 (bare hand).
Average xorn: hp 45, AC 22, AB +10/+8/+8/+8, 4d6+3/d4+1/d4+1/d4+1.

Dextrous Monk hits on a 16 and a 19, a total of 7 times per 20 turns, for 6.5 hp/hit: average damage output = 2.28 hp/turn. The xorn bites on a 6, 15 times in 20 turns for 17 hp/hit, 12.25 hp/turn, and claws on an 8, 39 times in 20 turns at 3.5 hp/claw: another 6.83, for a total of 19.08 hp/turn. Dextrous monk lasts 1.89 turns and delivers 4.29 hit points to the xorn.

Strong monk hits on a 15 and an 18, a total of 9 times per 20 turns, and does 7.5 hp/hit, for an average of 3.38 hp/turn. The xorn bites on a 5, 16 times in 20 turns for 17 hp/hit, 13.6 hp/turn, and claws on an 7, 42 times in 20 turns at 3.5 hp/claw: another 7.35, for a total of 20.95 hp/turn. Dextrous monk lasts 1.72 turns and delivers 5.8 hit points to the xorn. That isn't encouraging, but it is 35% better than Dextrous monk.

For a comparison, a 6th level fighter with a masterwork bastard sword, +1 full plate and a large metal shield is

Fighter: hp 49, AC 22, +11/+6, d10+6 (halved against the xorn).
Average xorn: hp 45, AC 22, AB +10/+8/+8/+8, 4d6+3/d4+1/d4+1/d4+1.

He hits on an 11 and a 16, 15 times in 20 turns, for 5.75 hit points per hit. Average damage output: 4.31 hp/turn. The xorn bites on a 12, 9 times in 20 turns for 17 hp/hit, 7.65 hp/turn, and claws on an 14, 21 times in 20 turns at 3.5 hp/claw: another 3.68, for a total of 11.33 hp/turn. The fighter lasts 4.32 turns and delivers 18.65 hp to the xorn. Witness the firepower of this fully-operational battle station!

 

[Agback]

At low levels, the Str Monk's bonus damage is going to be 1/6 to 2/3 of his base damage- lets assume the best case. He successfully strikes 2 times and he does a total of 2d6 + 8 points of damage in 2 rounds...

But there are damage-dealing specialists out there, like Fighters and Barbarians. Odds are good that his alpha strike is going to be significantly smaller than the true Warrior classes- he tangles with one of those and he might not live 2 rounds to deal damage.

By way of contrast, the Dex Monk may be able to win by inflicting the "death of 1000 cuts" because the warrior has difficulties hitting him and even d6's add up over time.

I'm glad that you are ready for disappointment, because it doesn't work out that way. Moving 2 characteristic points from DEX to STR gives both the monk and his/her opponent one extra hit per twenty turns. The primary effect is to speed the fights up, the secondary effect is an advantage to the monk if he/she hits less often than his/her opponents and to the opponent if the monk hits more often. That is, the effect on number of hits of increasing STR at the cost of DEX advantages the monk in difficult fights and disadvantages him in easy ones. The effect in damage bonus is larger, and consistently in the monk's favour.

Let's look at a 3rd level monk against a 3rd level fighter (and assume that the monk doesn't simply run away). (As always, I use the NPCs from the DMG for characteristics and equipment.)

Fighter: hp 27, AC 21, AB +7, d10+2
Monk: hp 20, AC 13, AB +6, d6+3 (kama)
Strong monk: hp 20, AC 12, AB +7, d6+4 (kama)

The fighter hits the standard monk in 15 turns out of twenty, and the strong monk in 16. Thats a 7% advantage to the standard monk.

The standard monk hits the fighter in 6 turns out of 20, and the strong monk hits the fighter in 7 turns out of 20. That a 17% advantage to the strong monk. The standard monk does an average of 6.5 hp/hit with his or her kama, and the strong monk does 7.5. That's an advantage of 15% to the strong monk. Combining the two effects, the standard monk inflicts an average of 1.95 hp/turn, while the strong monk inflicts an average of 2.63 hp/turn, an advantage of 35% to the strong monk.

The fighter does an average of 7.5 hp per hit, so the standard monk indeed lasts longer: 3.56 rounds instead of 3.33. 7% longer.

But in the time the fight lasts, the strong monk hits 1.17 times, and the standard monk hits only 1.07 times. That's actually fewer hits than the strong monk, and a long way from the 'death of a thousand cuts'.

Combine the extra hits with the extra damage per hit: the standard monk delivers only 6.94 hp, while the strong monk delivers 8.78.

 

[Dannyalcatraz]

I'm not dissapointed, but I've noticed one thing in your examples that does sidestep something I brought up, which is that each of your examples is a one-on-one combat, which is where I expect the Str build to shine.

But as I pointed out in the first post, until the PCs reach advanced levels, the majority of encounters (IME, at least) feature the PCs being outnumbered, where the AC benefits should favor the Dex build.

It isn't unusual in our campaigns for PCs being outnumbered 2-5 to 1 for large portions of the combat. Typically, this results in a (melee) PCs being in melee combat with 2 or 3 opponents at a time.

There is also the aspect of the monk changing some of his combat emphasis to ranged attacks.

For example, in a 3rd level campaign, my Dex/Wis/Str build monk had the best AC in the party- over the course of 3 sessions, he took 1 hit while dishing out several successful strikes per combat. At least one of those combats was against a higher HD creature (a large ooze)- the only damage it did to him was when he struck it with his bare hands (it sounds like player error, but the PC had never faced such a creature). Furthermore, he accentuated his combat impact by raining javelins and shooting crossbow bolts into approaching foes (his sole magic item was a Quiver of Ehlonna).

My point? You have to take into account all aspects of a build, including build-appropriate play, before you can baldly assert that one build is "better" than another.

So, you've shown that in one-on-one pure melee combat, the Str build is superior to the Dex build at dishing out damage. The next step is comparing them in cases of multiple opponents and factoring in ranged combat.

 

[MatthewJHanson]

Some other things to consider: A Dex monk will have a better Reflex Save. He will also have a better Tumble check, and thus be able to flank more easily. A Dex based monk will probably Weapon Finnesse.

 

[Victim]

Other abilities to factor in are grapples or trips (bonus to strong monk).

That combat comparison also ignores Flurry of Blows. If the monk is using Flurry, then:

Dex monk: Att: +4/+4. hit chance: .2. Hits per round: .4. DpH: 6.5. DpR: 2.6.
Time to kill: 10.38.

Strong monk: Att: +5/+5. hit: .25. HpR: .5. DpH: 7.5. RpR: 3.75
Time to Kill: 7.2

The strong monk's advantage in damage per round with Flurry becomes 44.2%. As flurry of blows improves (greater flurry, mostly), I'd expect strength's advantage in this area to be compounded. Unlike TWF, Flurry allows full STR bonus with the extra attacks.

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Of course, both of those monks are insanely weak. Their ACs are 12 and 13! If they're swarmed by goblins or something, both of them will go down like chumps. We could slap a mage armor on them to get 16 and 17 - certainly not the highest in the party, but at least not total crap. Those spells and/or potions/scrolls/wands are still pretty easy to get even at 3rd level.

Defensively with mage armor:

Dex monk: AC 17. .55 chance to be hit. DpH: 7.5. DpR: 4.125.
Time to die: 4.85

Strong monk: AC 16. .6 hit chance. DpH: 7.5. DpR: 4.5
Time to die: 4.44

Both monks still lose to the fighter pretty handily without using their other abilities or combat options. The strong monk can trip with a higher bonus, has a higher grapple bonus, and his improved hit chance even makes stunning fist more effective. Even so, the strong monk still has a significant edge in damage dealt, while the difference in damage taken is rather small.

In terms of ranged options, the strong monk will do more damage with javelins, although his hit chance is lower. Overall, I'd still give the ranged advantage to the Dex monk, unless DR is common. But against oozes and the like, the strong monk can just bash with a staff or other cheap weapon anyway. Quarterstaves can get 1.5x STR to damage.

 

[Agback]

But as I pointed out in the first post, until the PCs reach advanced levels, the majority of encounters (IME, at least) feature the PCs being outnumbered, where the AC benefits should favor the Dex build.

It doesn't. I thought it did, but I worked through fights between Strong and Dextrous 6th-level monks and three troglodytes each, and it didn't make any difference. Strong Monk outclassed Dextrous Monk by the exactly same percentage in the three-on-one fight as in the one-one one. The reason is inescapable once you think of it. A +1 to AC reduces incoming damage by the same percentage regardless of the number of the attacks. The percentage is small if the attacks hit a lot, it is high if the attacks hit rarely, but it does not depend in any way on the number of attacks. The +1 to AB likewise increases damage output by a large proportion against targets with high AC and a small proportion against targets with low AC, But it makes no difference whether there are several targets or only one with more hit points. The percentage increase in damage output is the same. And then, the damage bonus increases damage output by a fixed percentage tht depends only on the averge amount of the damage roll. It is a high percentage if the average roll is low (ie. if the monk is low-level) and it is a low percentage if the damage roll is high (ie. if the monk is high-level). The ratio of damge Strong Monk dishes out to what Dextrous Monk dishes out is unaffected by the number of foes. and the ratio of damage Stong Monk suffers to what Dexrous Monk suffers likewise is unaffected by the number of attacks incoming.

If you put a character up against three foes in mêlée, it takes him or her three times as low to kill them and he or she takes six times as much damge doing so. On average, those figures are exact, and they do not depend on the the attributes of the character.

Now, in fighting multiple foes there is a chance of manoeuvring for position. But the strong monk can do that just as well as the dextrous monk. And there is a chance of being flanked, which hurts the two monks just the same in total, but the dextrous monk more proportionately.

There's no getting around it. In any sort of mêlée, the strong monk is a better build than the dextrous monk, except when the monks are high level and they are fighting opponents that are much, much weaker. The advantages of the dextrous monk are his slightly better reflex save and his slightly more accurate shooting with a sling.

 

[Dannyalcatraz]

I'm not sure I agree with the analysis of facing multiple foes (without seeing a true proof).

While the AC & to hit bonuses are still affected the same way as in the single foe problem, when you face multiple opponents, the monk is still using that single limited pool of HP.

So if the 2 monks are facing the same # of identical foes, arguendo a Trog with the same stats as the one you posted originally, each monk must has to concentrate his strikes on one, then the other then the next foe, while facing multiple attacks- more than the monk himself has.

Example: 1st-level monk with AB +2, AC 13, d6+2, 9 hp fights a CR 1 monster, say, a troglodyte with a longspear AB +1, AC 15, d8, 13 hp. Monk hits on a roll of 13 or better, ie. 8 times out of twenty and does 5.5 hp per hit for an average of 2.2 hp per turn, and the trog hits on a 12 or better (9 times per twenty turns) for 3.5 hp per hit, which comes out to 1.575 hp/turn. It will take the monk on average of 5.91 round to down the trog, in which time he or she will suffer 9.31 hp damage. In short, it's close, but we expect the troglodyte to win.

Now suppose that we swap the monk's 16 DEX for his 14 STR. His AB becomes +3, AC 12, d6+3, still 9 hp.The monk now hits on an 12 (9 rolls out of 20) for an average of 6.5 hp per hit. Average damage output: 2.93 hp per turn, The trog will go down in 4.44 combat turns. The trog now hits on 10/20 turns, and still does 3.5, for an average of 1.75 hp per turn. The monk now expects to survive the fight taking 7.78 hit points damage instead of 9.31.

That Dex based monk is facintg 3x 1.575hp/rd (or 4.725hp/rd), while the Str based monk is facing 3x 1.75 hp/turn (5.25hp/rd). (Of course, that isn't technically correct if/when trogs start dropping, thus changing average damage output/round.) Unfortunately, with the HP posited, that means both monks fall in 2 rounds, having taken 9.45hp for the Dex monk and 10.5hp for the Str monk.

If each monk has 18HP (ignoring increased damage output or ACs) the Dex monk drops on round 4 by .9 points), the Str monk drops the same round by 3hp. With 27HP, still ignoring increased damage output or AC changes, both monks last to Round 6, but the Dex monk is 1.35 in the hole, as compared to the Str monk being 4.5 in the hole.

Are there factors that would alter this? Of course- the Str monk could have feats or items to enhance his melee damage...the Dex monk could have feats or items that could enhance his ranged damage or AoO's.

(Forgive me if I ramble- tonight was a 1+ bottle of wine evening! )

 

[Agback]

I'm not sure I agree with the analysis of facing multiple foes (without seeing a true proof).

Okay, here we go. Suppose that combatant is facing n similar foes in mêlée. Obviously, his best strategy is to concentrate on one foe until that goes down, then switch to another, then to the next, and then to the next, until they are all down. His chance of hitting with any attack is not affected by the number of foes, and the damage he does with any hit is not affected either, so his average damage output is not affected by the number of foes. That means that the average number of rounds he takes to put any single foe down is not affected by the number of foes facing him at the time he is concentrating on that foe.

(Note: in the following algebra, capaital letters denote values pertaining to the combatant of interest, and lower-case letters denote values pertain to his foes. AB is attack bonus (including STR bonus). AC is armour class (including DEX bonus). HP is hit points, SB is strength bonus to damage, R is the average value of the character's damage die roll.

Ignoring for the moment cases in which the combatant either always hits or never hits, his chance of hitting is (21+AB-ac)/20, and it doesn't depend on number of foes. The damage he does on a successful hit is R+SB, where R is the average value of hit damage die roll and SB is his STR bonus. So the average damage output is (R+SB)(21+AB-ac)/20. If his opponents have hp hit points he will put the first down after 20*hp/{(R+SB)(21+AB-ac)}, and then the next after as long again, and so on until all n of them are down, which will take n*20*hp/{(R+SB)(21+AB-ac)}. Define T = 20*hp/{(R+SB)(21+AB-ac)}. The combatant will face n opponents for T turns, then n-1 for another T turns, then n-2 for another T turns, and so on for nT turns. He will face n attackers for T turns, then (n-1) for T turns, and so on.

During this process, each foe still standing will have a chance of hitting of (21+ab-AC)/20, and will produce r+sb hp per successful hit. So each foe the combatant faces will produce on average (r+sb)(21+ab-AC)/20.

So for T turns the combatant will face n foes and take n*(r+sb)(21+ab-AC)/20 hit points per turn. a total of T*n*(r+sb)(21+ab-AC)/20.

Then for another T turns the combatant will face n-1 foes and take (n-1)*(r+sb)(21+ab-AC)/20 hit points per turn. a total of T*(n-1)*(r+sb)(21+ab-AC)/20.

And so on for nT turns in total, taking a total of T*{n*(n+1)/2)*(r+sb)(21+ab-AC)/20. Substituting the value of T bck in to the equation we get

[20*hp/{(R+SB)(21+AB-ac)}]*{n*(n+1)/2)*(r+sb)(21+ab-AC)/20

={n*(n+1)/2} * (r+sb)(21+ab-AC)/{(R+SB)(21+AB-ac)}

That is exactly n*(n+1)/2 times the amount of damage the combatant takes fighting one such foe until it drops. There is no interaction term between n and AC.

The argument is not affected if the monk uses 'flurry of blows'. You get a more complicated expression for the monk's average damage output, but the number of foes does not enter into the expression for damage output. The number of hit points the monk suffers might be lower but it will still scale strictly with the nth triangular number. And in fact, the strong monk's advantage is increased when he uses flurry of blows, because his attacks are made at reduced AB, and the STR bonus to hit is proportinately more valuable when the chance to hit is low.

Dextrous monk has the advantage when the foes have attacks with low attack bonus, ones where a -5% chance of hitting chews up a large proportion of their chance to hit. It tried to show that with the xorn, a monster I chose because it had a lot of innaccurate secondary attacks. The problem is that in D&D low-accuracy attacks are nearly always weak, so that avoiding them is not critical.