At what point does it become worth to take -2 to attacks to get a second one?

[SangoProduction]

I'm sure there's some kinda formula to work it out to find which option (1 attack at full bonus, or 2 attacks, each reduced by 2), where it's more likely for at least one to hit, or perhaps the marginal value of the second attack, given the penalties.

I just have no idea what it is.

Presumably, the more attacks you start off with, the less valuable that extra attack becomes, due to the total penalty you're taking being greater with more attacks you can do.

[CharonsHelper]

I'm sure there's some kinda formula to work it out to find which option (1 attack at full bonus, or 2 attacks, each reduced by 2), where it's more likely for at least one to hit, or perhaps the marginal value of the second attack, given the penalties.

I just have no idea what it is.

Presumably, the more attacks you start off with, the less valuable that extra attack becomes, due to the total penalty you're taking being greater with more attacks you can do.

What are you talking about? How are you getting the additional attack for -2 accuracy? TWF? Flurry? Are both attacks for the same damage that the single attack would be?

If they do the same damage and have no other penalties, then it's almost always worth it - so long as you hit on a 17+ with a single attack. (Because that single attack has a 20% chance, while each of the two attacks would have a 10% chance.)

[SangoProduction]

What are you talking about? How are you getting the additional attack for -2 accuracy? TWF? Flurry? Are both attacks for the same damage that the single attack would be?

If they do the same damage and have no other penalties, then it's almost always worth it - so long as you hit on a 17+ with a single attack. (Because that single attack has a 20% chance, while each of the two attacks would have a 10% chance.)

hmm. OK. Thanks.

Also...could have sworn I said "they are otherwise the same attack"...guess I forgot about it.

[Barstro]

If I recall the spreadsheet I did before correctly, it would be better to NOT use –2 for an extra attack ONLY if you need an 18 to hit.

16 and under you will hit more frequently with the extra attack.
17 is 20% on a single roll or 10% each on two rolls
18+ is 15% on one roll or 5% on each two rolls (must roll a 20)
19+ is 10% on one roll or 5% on each two rolls (must roll a 20)
20+ is 5% on one roll or 5% on each two rolls.

Needing to roll an 18 is the only time it would be better to stick with a single attack.

EDIT: and how does your character know he has exactly a 15% chance to hit, you power-gamer, you.

[MilleniaAntares]

If you simplify your assumptions, the math is simple:

Damage per round = chance of hitting * average damage per hit * number of attacks

Plug in the numbers and see what threshold works.

Of course, the reality is more complicated given that chance of hitting has a max of 95% and a min of 5%.... and the existence of critical strikes, critical damage multipliers, limited damage bonuses (sneak attack isn't affected by crits) miss chances, etc.

My Excel math indicates that it is always advantageous to attack twice with a -2 attack penalty than attack once with your full attack bonus, if your regular chance to hit is 5% and 25%+. It is a tie at 10% and 20%, and worse at 15%.

5% chance of hitting, 10 average damage...

Single attack: .05 * 10 = 0.5 DPR
TWFing: 0.5 * 10 * 2 = 1 DPR

An Excel or Google sheet will make calculations quicker.

[Cerefel]

Depending on how you're getting the extra attack it may limit what weapons and styles you're able to use which could definitely factor into whether it's better to do so or not.

[ExLibrisMortis]

In the general case, it's worth* taking a -2 penalty to get your [n + 1]th attack if and only if the lowest roll you hit on is a [19 - 2n]. This is without considering the auto-hit on a 20. Taking the auto-hit into account, you get a weird situation, where you either want dozens upon dozens of attacks, or just a handful. For example, if you can only hit on a 19 or 20, even on single attacks, it's always worth getting more attacks, because you'll just rely on natural twenties. That carries on to more accurate attacks: if your single attack can only hit on a [20 - m], it's worth getting m extra attacks, no matter the penalty.

For example, these are the (non-autohit) numbers for 0 < n < 5.

n = 1 : 20% chance to hit (17-20) becomes 10% chance to hit (19-20), average damage is 1 * (0.2 * 1) = 2 * (0.1 * 1).
n = 2 : 30% chance to hit (15-20) becomes 20% chance to hit (17-20), average damage is 2 * (0.3 * 1) = 3 * (0.2 * 1).
n = 3 : 40% chance to hit (13-20) becomes 30% chance to hit (15-20), average damage is 3 * (0.4 * 1) = 4 * (0.3 * 1).
n = 4 : 50% chance to hit (11-20) becomes 40% chance to hit (13-20), average damage is 4 * (0.5 * 1) = 5 * (0.4 * 1).

And so on.


*Dealing equal or more full attack damage, on average.

[Barstro]

My Excel math indicates that it is always advantageous to attack twice with a -2 attack penalty than attack once with your full damage, if your regular chance to hit is 5% and 25%+. It is a tie at 10% and 20%, and worse at 15%.

That 15% is only on needing an 18+ on a "regular" attack. Confirmed my spreadsheet.

[MilleniaAntares]

Did some more math:

In all cases, extra attacks with a -2 penalty are always better when you have a 5% chance of hitting.

Beyond that:

2 attacks are equal or better than 1 when you can hit on a 17 or lower.
3 vs 2 on a 15.
4 vs 3 on a 13
5 vs 4 on a 11.

Once you get into Improving TWF or similar:

4 vs 2: 18 or less.
5 vs 3: 16.
6 vs 3: 17
6 vs 4: 15
7 vs 4: 16.

Hope this helps!

[Ualaa]

I made a spreadsheet once, for average damage.

It took into consideration, your attack bonus.
The threat range and multiplier of your weapon.
Power attack hit penalty, and damage bonus.
The armor class of the opponent (which you frequently won't know).

There is a hit chance, critical chance, and a miss chance.
Each number of a d20 is equal to 5%.
Subtract the miss chance from '20' or '100%' first.
Then subtract the Critical chance from the remainder.
Anything after that is a hit, but not a critical hit.

If you need a 8 to hit a given AC.
And critical on a 19-20.
Then 01-07 = 0.35 miss chance.
Take away 19-20 (or 0.10) from the 0.65 (after taking away miss chance from 1.00).
So we have
Miss 01-07 = 0.35 x Average Damage x 0.
Hit...08-18 = 0.55 x Average Damage.
Crit*.19-20 = 0.10 x Average Damage x Chance to Confirm x Critical Multiplier

A Crit needs to confirm, so this is another roll, requiring above the miss chance.
And there is a critical multiplier, if that roll hits.

You need to figure out your average damage.
That is minimum plus maximum, divided by two.
Plus other bonuses.

Going with the above hit rate and a x3 multiplier...
If your average damage, on a hit is 10.50
We have 0.35 chance for 00.
We have 0.55 chance for 10.50
We have 0.10 (x 0.65) = 0.065 chance for 31.50

Expected damage is
00.00 x 0.350 = 00.000
10.50 x 0.550 = 05.775
31.50 x 0.065 = 02.0475
= 7.8225 damage per round.

If you adjust for a 2 point penalty...
00.00 x 0.450 = 00.000
10.50 x 0.450 = 04.725
31.50 x 0.065 = 02.0475
= 6.7725 damage x 2 attempts = 13.545 per round.

In this case, with a d8 weapon, +6 damage bonus, 19-20 threat range for x3...
Gaining a second attack, and having all attacks at -2, is worthwhile.



My spreadsheet was more for the effects of Power Attack & Deadly Aim, in Pathfinder where a fixed amount of penalty & bonus is applied.

But you can construct one fairly easily, then plug in your own numbers to test stuff out.
Start off with 1.00.
Subtract miss chance to get Hit or Crit chance.
Subtract crit chance to get Hit chance.
Multiply the chances by the average damage.
Crit chance is multiplied against chance to at least hit (Hit + Crit rates), and then the damage is multiplied against the crit multiplier.

If you want to do Power Attack, that penalizes attack bonus.
But increases average damage.

[OldTrees1]

Since the base case of "-2 attack penalty to gain a 2nd attack" is solved, why not generalize it?

When to take a -2 penalty to gain an extra attack:
Note: I am presuming all the attacks have the same attack bonus

Well generally a -2 penalty is a 10% cost per attack and you gain 1 attack at 10 less than the old rate:
Y% - (X+1) * 10% > 0 (where Y% is the old accuracy and X is the old number of attacks).

Edge case 1: If you only hit on a 20 or if you would also hit on a 0 if rolling a 0 were not impossible, congrats! The penalty means nothing to you. Accept the extra attack in a heartbeat.

Edge case 2: If you only hit on a 19-20 or if you only miss on a 1 because 1s always miss, great! The penalty is only half as strong for you so use Y% - (X+1) * 5% > 0 instead.

[Fouredged Sword]

The confusion is generally that TWFing is a horrible thing. It's not so much that getting a second attack is bad, it's that you are taking a penalty to deal the same amount of damage as you would if you did a single attack.

It's normally comparing (2d6+1.5str) VS (1d6+str) + (1d6+.5str) at a -2 penalty.

Note you roll the same number of dice and add the same strength bonus each time. Without a source of bonus damage TWFing is pointless.

Snap kick, for example, is a great feat. If I qualify for it I almost always take it even if I wield a greatsword.