The question is simple: I saw the above equivalence really often. But never saw some expansion. I think it's about statistic and probability, but I like to see the explanation for that. Could someone help me?
Basically after a certain point, mostly with multiple attacks, you simply do more damage landing more hits more often than dealing more damage.
EXAMPLE!
Fighterbro1 needs to hit an AC of 26. Hes got a +12 to hit with 2 attacks. He deals 30-40ish damage and drops the enemies relatively quickly.
Fightersis1 needs to do the same as well. She gets a +4 to damage and has a +10 to hit. Her attacks deal 34-44ish damage and the extra damage is relatively meaningless in her attacks. Shes 10% less likely to hit as well.
Point to remember, Damage is only meaningful so far as dropping the enemy in one turn. Anymore past that is useless.
It is the ratio that power attack let you change the two. But it is not the same thing.
The game is fortunaly a bit more complex than that:)
It's a good rule of thumb. If you do not hit, you cannot do damage so accuracy is always more important - especially once you start factoring in iterative attacks.
For feats n stuff, that ratio is what you will find the game considers roughly equal.
Hitting your opponent is more important than doing damage.
Because +2 damage is nice, but in the long run doing 0 damage is a lot worse than doing X-2.
Without getting into the statisticals there is power attack which gives a -1 to hit for +2 damage. Which would lend itself to the suggestion that 2 damage is worth +2 to hit.
Let's say you're a fighter with an 18 STR and a greatsword fighting a simple AC 16 Goblin. You have no other factors (e.g. weapon focus) that affect your to-hit and damage, so you're swinging at a +5 bonus to deal 2d6+6 damage. You need an 11 or higher to hit and crit on a 19 or 20 for double damage, so that means your average DPR is 7.15. An additional +1 to hit improves our DPR to 7.865. An additional +2 to damage improves our DPR to 8.25. So in this case, the +2 to damage is superior.
Let's say instead we're playing a race with a +4 strength bonus (so we have a 22 base), we have an Enlarge Person on ourselves (+2 size bonus and increases our weapon damage dice to 3d6), and a monty haul DM who gave us a +4 Belt of Giant Strength for god knows what reason. We fight the same goblin, but this time we have an attack bonus of +9 and we're swinging for 3d6+13. Our DPR is now 18.095. Adding another +1 to hit improves our DPR to 19.3875. Adding +2 to damage instead increases our DPR to 19.635. So the +2 to damage is still superior, but gives less of an advantage than it did in the first case.
Finally, let's say our DM is even more monty haul, and gives us the Weapon of Ridiculous Awesomeness that gives a +1 enhancement bonus to hit and does 100 additional damage whenever it hits an enemy. Our DPR is now 101.8875. A +1 to hit improves our DPR to 108.68. A +2 to damage increases our DPR to 103.5375. The +1 to hit finally wins out, though this is an extreme example.
The basic idea is that improving damage is constant, but improving to-hit is multiplicative: However, the scaling factor is pretty tiny. So the +2 to damage is better unless your base damage is already really, really high: This is why Power Attack is a good feat, especially with two-handing when it adds +3 instead.
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Do not forget that at 1st and second level, even just a l1d8 ong sword and 18 str will often be enough to one shot kill your average 1st and 2nd level opponents. In this case, I believe the +1 to hit is more valuable. This also gives you better survivability at low levels through use of a shield. The opponents who will need more damage to kill at these levels are ment to stick around a while anyway, such as a boss fight for example. The 2d6+6 wound only kill them on a crit anyway so you still need more than one hit against them, maybe it goes from 2 to 3 hits in such a case. While the 2d6+6 is so often over kill at level one and two, you are still more dependent on the luck of the dice for the attack roll. I strongly advise in favor of attack bonus in early career, you can wait for the damage bonus later. So many people focus on power attack at level one because it can be taken at level one and those people want what they consider a sexier feat at later levels, something with an attack bonus prerequisite, improved critical for example. Yet meanwhile, all that power attacking at level one and two should result in more misses and the extra damage really meant nothing most of the time as the regular 2d6+6 was already enough to kill the opponent anyway.
As a rule of thumb, +1 to hit, is worth about 5% of your average damage in dpr. +1 damage is worth an amount of dpr equal to your chance to hit. This does not take criticals into account.
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Well... it is not a hard and fast equivalency...
In fact, there are a lot of cases where power attacking is a terrible idea. Two weapon fighting rogues know this to be true.
ANd it all comes down to the same things it came down to in 3e... too much accuracy is a waste, so dropping some of that for damage is a great option.
Example:
CR 12 monsters have an average AC of 28
If you are a level 14 fighter, you will likely see a lot of these, plus some CR 14 monsters that are meant to be solo according to the CR tables.
So example level 14 fighter has:
Str 24
Weapon Focus
Greater Weapon Focus
Weapon Training +3, plus gloves of dueling for another +2
A +3 Greatsword
BAB +14
So a total bonus to hit of +31
Against a 28 AC, this is just wasted bonus.
If they power attack, then their attack bonus drop to +27, but they get +12 damage with their two-hander. So its a great trade off.
Adding some more feats (W.Spec., G.W.Spec just to show static damage)
Full attack without power attack (+31/+26/+21), 2d6+22
With power attack (+27/+22/+17), 2d6+34
Against 28 AC, he will average 2.6 hits per full attack without power attack, and 2.2 hits per full attack with power attack.
That means, that before crits, the DPR without power attack is 75.4
With power attack it is 90.2
So obviously a good trade... crits will also fall in favor of power attacking, especially since critical focus can pump up the confirm roll, and a fighter is pretty much always going to hit with the threat roll anyway.
And this all makes sense... the fighter goes from an average damage per hit of 29 to an average hit of 41 when power attacking. That means that his attacks are dealing 41% more damage. But he is only taking a -4 to attack rolls... Going from a +31 to a +27 is a difference of about 13%.
You can't just look at the flat 5% per -1 concept here, because everything is multiplicative... so you need to think about the ratios.
Effectively DPR = Accuracy X Damage.
Attack bonus = A
Damage = D
Power attack accuracy reduction = X
Power attack damage increase = Y
Ratio of Damage Change = (D+Y)/D
Ratio of accuracy Change = (A-X)/A
To get the DPR change, just multiply these two results.
If these ratios multiplied are greater than 1, then power attacking is a good idea... but as you can see, the higher D is before adding Y, and the lower A is before subtracting X, the worse power attack will be.
In the fighter's case, we see an actual DPR change of +23%
So plug in some other numbers... lets say that you have a +10 to hit, and deal an average of 40 damage per hit. You can power attack for -3/+6.
Damage Change = (40+6)/40 = 1.15
Accuracy Change = (10-3)/10 = 0.7
Multiplied we get 0.805, so in this case, power attacking will reduce the character's damage output by about 20%.
Let's say you're a fighter with an 18 STR and a greatsword fighting a simple AC 16 Goblin. You have no other factors (e.g. weapon focus) that affect your to-hit and damage, so you're swinging at a +5 bonus to deal 2d6+6 damage. You need an 11 or higher to hit and crit on a 19 or 20 for double damage, so that means your average DPR is 7.15. An additional +1 to hit improves our DPR to 7.865. An additional +2 to damage improves our DPR to 8.25. So in this case, the +2 to damage is superior.
Let's say instead we're playing a race with a +4 strength bonus (so we have a 22 base), we have an Enlarge Person on ourselves (+2 size bonus and increases our weapon damage dice to 3d6), and a monty haul DM who gave us a +4 Belt of Giant Strength for god knows what reason. We fight the same goblin, but this time we have an attack bonus of +9 and we're swinging for 3d6+13. Our DPR is now 18.095. Adding another +1 to hit improves our DPR to 19.3875. Adding +2 to damage instead increases our DPR to 19.635. So the +2 to damage is still superior, but gives less of an advantage than it did in the first case.
Finally, let's say our DM is even more monty haul, and gives us the Weapon of Ridiculous Awesomeness that gives a +1 enhancement bonus to hit and does 100 additional damage whenever it hits an enemy. Our DPR is now 101.8875. A +1 to hit improves our DPR to 108.68. A +2 to damage increases our DPR to 103.5375. The +1 to hit finally wins out, though this is an extreme example.
The basic idea is that improving damage is constant, but improving to-hit is multiplicative: However, the scaling factor is pretty tiny. So the +2 to damage is better unless your base damage is already really, really high: This is why Power Attack is a good feat, especially with two-handing when it adds +3 instead.
That's an outstanding breakdown and presents an excellent picture of the discussion - but I do think one thing isn't taken into account - that targets have a finite number of hit points.
Sure a Fighter hitting his foe for 18 hit points is better than for 16, unless the target only has 12 to begin with. A great deal of 'bonus' damage is wasted by overflow, even if it takes 5 hits to kill a foe, and with that understanding I feel its always more important in my mind to be able to hit than to deal more damage - damage can be wasted, but a hit never is.
Wow guys. You're awesome. This thread is helping me to figure what's a good trade between to hit and damage. Mr thoughts for now are that even if the to hit scales faster than damage, the accuracy tend to pay more than damage, inside certain limits. For sure it's better hit and deal less damage than don't hit and don't do any damage. But please, keep on the thread. I'm finding it amusing, and more info you give, better become my understanding of this thing.
Actually, Kimera, melees are supposed to be hitting at every level, and monsters are supposed to hit frequently, too.
The reason for the former is that monster AC scales to match 3/4 BAB classes. Full BAB classes are supposed to have bonuses so that they always hit on first or second iterative, and frequently with their later ones.
PC's are supposed to be getting hit to keep the risk factor high. It's the reason why PC's have so many hit points...so the risk is there, so healing is valuable, and they aren't invulnerable or unkillable.
3.5 P-A isn't that overpowered unless you have monstrously abusable to hit numbers. Statistically, PF Power Attack is actually stronger, - for dmg...but is also more limited. It was Shock Trooper that made 3.5 PA a monster, as your to hit penalty was reduced to 0 in return for an AC penalty that didn't matter if the enemy was dead from your Pouncing Ubercharger.
==Aelryinth
I've actually instituted a houserule for hit points - originally I'd have the players reroll if hit points were below half (I've players who are very unlucky - and they'd have to reroll three or four times before their hit points would be above half - at which point they'd roll near-max).
Now I have the rule that you roll for hit points. If it is below or at half, you get half hit points for the die, plus one. (Thus 4 for wizardly types, 5 for rogues and clerics, 6 for fighters, and 7 for barbarians.) It helps with unlucky rolls... while also keeping from rolling a half dozen times and then lucking out with max hit points (as happened several times in the past).
The basic idea is that the more you hit the more damage you do over the course of time. You can have a +8000 damage modifier, but if you never hit it does not matter. There is also a DPR formula which backs up that equal modifiers to attack and damage favor attack modifiers over damage modifiers.
I've actually instituted a houserule for hit points - originally I'd have the players reroll if hit points were below half (I've players who are very unlucky - and they'd have to reroll three or four times before their hit points would be above half - at which point they'd roll near-max).
Now I have the rule that you roll for hit points. If it is below or at half, you get half hit points for the die, plus one. (Thus 4 for wizardly types, 5 for rogues and clerics, 6 for fighters, and 7 for barbarians.) It helps with unlucky rolls... while also keeping from rolling a half dozen times and then lucking out with max hit points (as happened several times in the past).
We max out the first HD and thereafter the player gains half the hit die +1 + CON. Meaning that a 2nd level Wizard with a 12 Constitution has 12 (6 + 4 + 1x2) hit points while a 2nd level Fighter with a 16 Constitution has 22 (10 + 6 + 3x2) hit points. Its worked out pretty well and avoids the feeling that we had to 'save' a particular PC via handwave.
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Surprised more people don't use the d6 rule.
d8 hd = d6+2.
D10= d6+4
d12= d6+6
Pretty much takes care of most of the low hp problem, and pretty much guarantees that high HD classes WILL have more hp then low HD classes.
==Aelryinth
As a general rule, I don't think you can say that +1 hit ~= +2 damage. I would say that it depends on which is scaling up faster for your character between your current hit percentage against your average enemy versus your current damage output. If your (current% + hit%)/current% - 1 > (currentDamage + damage)/currentDamage - current% then the +hit will give your more average damage. If the inequality is the other way around, +damage will win out.
Truth.
But again, if one is looking for a hard and fast general rule of thumb, I say it has to be taken into account that your foe isn't a single wall of limitless hit points - much more often its a group of numerous foes all with a very limited supply... and every time you kill a foe, unless you hit the number exactly right, you lose whatever benefit the bonus damage provided.
The advantage of dealing 2d6+12 damage against a foe instead of 2d6+10 when all of your foes only have 12 hit points is no advantage at all whereas a bonus to hit is never wasted.
As a general rule, I don't think you can say that +1 hit ~= +2 damage. I would say that it depends on which is scaling up faster for your character between your current hit percentage against your average enemy versus your current damage output. If your (current% + hit%)/current% - 1 > (currentDamage + damage)/currentDamage - current% then the +hit will give your more average damage. If the inequality is the other way around, +damage will win out.Truth.
But again, if one is looking for a hard and fast general rule of thumb, I say it has to be taken into account that your foe isn't a single wall of limitless hit points - much more often its a group of numerous foes all with a very limited supply... and every time you kill a foe, unless you hit the number exactly right, you lose whatever benefit the bonus damage provided.
The advantage of dealing 2d6+12 damage against a foe instead of 2d6+10 when all of your foes only have 12 hit points is no advantage at all whereas a bonus to hit is never wasted.
If all enemies had the same AC that would be true, but having a higher to hit means you can hit those with higher target AC's, and it means that even if you are debuffed you can hit AC's you could not hit otherwise due to the debuff.
As am example if you normally need a 10 to hit and you are debuffed by -2 then that extra +2 is not wasted and it matters so more is better.
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As a general rule, I don't think you can say that +1 hit ~= +2 damage. I would say that it depends on which is scaling up faster for your character between your current hit percentage against your average enemy versus your current damage output. If your (current% + hit%)/current% - 1 > (currentDamage + damage)/currentDamage - current% then the +hit will give your more average damage. If the inequality is the other way around, +damage will win out.Truth.
But again, if one is looking for a hard and fast general rule of thumb, I say it has to be taken into account that your foe isn't a single wall of limitless hit points - much more often its a group of numerous foes all with a very limited supply... and every time you kill a foe, unless you hit the number exactly right, you lose whatever benefit the bonus damage provided.
The advantage of dealing 2d6+12 damage against a foe instead of 2d6+10 when all of your foes only have 12 hit points is no advantage at all whereas a bonus to hit is never wasted.
By the same token, if you manage to exceed your foes AC by an amount equal to or greater than the bonus to hit that was traded for dmg then you have effectively lost dmg (i.e., you would have hit without the bonus to hit and the dmg would have been more beneficial). For example, a fighter with +8 BAB (power attack at -3 att and +6 dmg) beats his targets AC by 3 and he did not power attack. In this situation the fighter could have traded the hit bonus for more dmg.
However, these arguments rely on specific outcomes. The statistical analyses works based on average case to determine what is superior. This guarantees the highest rate of return in the largest number of cases. Many people prefer to boost to hit over dmg especially in 3/4 and 1/2 BAB classes to get that sense of consistency, and some prefer to go with the X4 crit weapons and boost dmg as much as possible to have those "epic" hits. These are all valid ways to play. The "+1 att = +2 dmg" rule of thumb is just meant is a basic means to quickly evaluate two options (and is pretty close to accurate in the average case).
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That's an outstanding breakdown and presents an excellent picture of the discussion - but I do think one thing isn't taken into account - that targets have a finite number of hit points.
Sure a Fighter hitting his foe for 18 hit points is better than for 16, unless the target only has 12 to begin with. A great deal of 'bonus' damage is wasted by overflow, even if it takes 5 hits to kill a foe, and with that understanding I feel its always more important in my mind to be able to hit than to deal more damage - damage can be wasted, but a hit never is.
I have to disagree strongly with this statement. Overkill is much more preferable than overhit.
Pathfinder makes no concession for combat effectiveness of badly injured creatures. A monster at 1 hp is every bit as effective as he was at full hitpoints.
If you beat his AC by 10, but leave him alive, you are going to have to endure his attack this round. It is exactly the same as if you had missed. If you meet his AC, and beat his hp by 25, he is dead, and you take no damage from him this round. (There are worse things than damage, too. Dead is all around better.)
One should also observe that a hit might do things other than mere damage, especially in the case of a magus, poisoned weapons and the like.
Conversely, DR. That +2 damage is more significant if the opponent is soaking 10 of it on each hit than if not. That's taken into account in the calculations above, though it is naturally situational.
The question is simple: I saw the above equivalence really often. But never saw some expansion. I think it's about statistic and probability, but I like to see the explanation for that. Could someone help me?
In addition to Power Attack you see the same math at play in other feats. E.g. Weapon Focus (one feat) adds +1 to hit, and Weapon Specialization (one feat) adds +2 to damage. It's more of a rule of thumb for game design than it is an rule of thumb for play, however, since the question in play always depends on the situation. The damage may be more valuable than usual if a foe has damage reduction, for example, because in that case a lower damage hit may be effectively a miss.
Typically, the more damage you are doing per hit, the more valuable attack bonuses become, and the less valuable damage bonuses become. The less damage you do per hit, the more valuable increasing your damage becomes, compared to attack.
So for example, this is why Power Attack gives a +3 damage per -1 attack, when you use it with a two handed weapon. Since a character swinging a two handed weapon is probably doing significantly more damage per hit already, the attack penalty will probably be hurting them more than another character, so they get extra damage per penalty, in an attempt to make up for it and keep the feat from being worse for them than for others. A game design consideration.
The statistical analyses works based on average case to determine what is superior.
It's worth noting that figuring out the average damage can't tell you what damage-dealing option is superior in terms of in game power to deal damage. It can only tell you what has the highest resulting average damage. A higher average damage does not actually reliably indicate damage dealing superiority, it merely correlates with it most of the time. While correlation is useful, it's good to be aware of its limits.
If Power Attacking is a roughly break even trade or, even, a small to moderate increase in your average damage, it can be an actually superior choice to not use Power Attack and lower your average damage per round. In general the concept of "average" damage becomes an increasingly poorer mathematical approximation of how Pathfinder combat behaves, the lower the chances to hit of the character in question, and thus average damage becomes an increasingly worse guide to evaluating your character's options.
Mostly this comes into play at higher levels. Because adding a full extra hit onto your damage is better than 2 damage each. Now you can't get "a full extra hit" most of the time. In terms of efficiency though, a simple +1 to hit at high levels with haste is a +.25 attacks a round in dpr. Why?Because you're getting an extra 5% hit chance on every attack and thus you end with about 25% bonus damage of a hit.
The singular question becomes, when is a flat damage bonus to all attacks bigger than a percentage of a single attacks damage? That ratio becomes quite large when you're rocking 2d4+50 (55) with a 15-20 crit ration (mult 1.3) for 1.3*55=71.5. That damage isn't hard to get at high levels and 25% of that is just around 18 points of damage. So adding 1 point of to hit with that kind of attack bonus, with your high to hit not being capped out and 5 attacks, is a +18 dpr. A +2 to damage with an assumed 4 hits per round (High, Haste, medium, and third attack assumed to hit so pretty good) would give you +8 damage instead. Even all 5 would give you +10.
Some commentary.
It is true that some of higher level combat's characteristics make using Power Attack become less and less attractive, because of the factors you mention (increasing numbers of attacks dealing higher damage). Power Attack's benefit remains fixed relative to its penalty - you're getting the same damage bonus per attack point penalty at 20th level as you did at first level - but the relative cost of a point of attack bonus is becoming higher and higher due to these factors outside of the feat itself, so the feat's trend is to become less and less good.
However, it is possible to beat this equation and keep Power Attack worthwhile. The way you do this is to achieve a starting attack bonus of higher than a typical AC you will face, such that you can use Power Attack without actually taking a real penalty on at least some of your attacks (because even with a penalty you can still hit on any roll but a natural 1). If you're only actually paying for Power Attack on two of four iterative attacks, say, then that offsets the diminishing ratio of what you get for what you pay.
Thus, by getting your attack bonus high enough to significantly exceed target ACs, you can avoid Power Attack shrinking into a trap option at high levels.
Fighters, barbarians, rangers, and all their other high base attack bonus buddies can often achieve this.
One should also observe that a hit might do things other than mere damage, especially in the case of a magus, poisoned weapons and the like.
Conversely, DR. That +2 damage is more significant if the opponent is soaking 10 of it on each hit than if not. That's taken into account in the calculations above, though it is naturally situational.
Incidentally I play a magus. I land a lot of frostbite, that help me to debuff enemies. Our GM give us a free teamwork feat, and we're toward outflank, as I'm a 3/4 bab, the paladin use power attack a lot and the rogue has gang up as feat, so we should be able to get the +4 nearly every combat.