When does Polearm Master deal more damage than an ASI?

Question

Would a fighter deal more expected damage by picking the Polearm Master feat at any point before they maxed their Strength, or is it better to first always increase your Strength in terms of damage? What other conditions will strongly influence this?

Answer 1

If you're making a lot of attacks, or are facing very high AC foes

We're trying to find expected damage when hit chance is relevant, so we need to look at actual expected damage: $$\text{Expected damage} = h(D+M) + cD$$ where \$0 ≤ h ≤ 1\$ is the hit chance, \$D\$ is the expected die damage (\$D = \frac{k+1}{2}\$ for a \$k\$-sided weapon die), \$M\$ is the ability modifier, and \$c\$ is the critical chance (normally \$c = \frac{1}{20} = 0.05\$).

We then want to find whether making \$n\$ attacks with a \$\frac{1}{20}\$ higher hit chance and a \$1\$ higher \$M\$ is better than making \$n\$ normal attacks and an additional attack with a d4 damage die (its expected damage denoted here as \$D_4 = 2.5\$).

Taking an ASI will thus be better when $$ n\left(h+\frac{1}{20}\right)(D+M+1) + ncD > nh(D+M) + ncD + h(D_4+M) + cD_4 $$ which, skipping a lot of algebra, simplifies to $$ h(D_4 + M - n) < \frac{1}{20}n(D+M+1) - cD_4. $$

If we assume that \$n < D_4 + M\$ (which will hold unless you're using some heavy homebrew, you're looking specifically at Action Surge, or you have some interestingly low values for \$M\$)^† we can further simplify this inequality to $$ h < \frac{\frac{1}{20}n(D+M+1)-cD_4}{D_4+M-n} $$ where the ASI will be the better choice if and only if the hit chance \$h\$ is less than the value of the expression on the right-hand side of the \$<\$ sign.

For example, for a single attack (\$n=1\$) with a d10 weapon (\$D=5.5\$), a +4 modifier (\$M=4\$) and normal crit chance (\$c=\frac{1}{20}\$), taking an ASI is only better if your hit chance is lower than \$0.073\$, which is probably not the case you want to optimize for.

From the inequality found above we can see that increasing \$c\$ or \$M\$ makes taking an ASI worse,^‡ while increasing \$D\$ or \$n\$ makes it better. However, there's very little room (outside of a few specific, and very powerful magic items) for increasing \$D\$ in practice.

Increasing \$n\$ has a more pronounced effect, with ASI being better for \$h<0.79\$ when \$n=4\$, and for all \$h ≤ 1\$ when \$n=5\$ (e.g. for a level 20 fighter under the effects of haste). This means that ASI isn't better until you're looking at very late progression (probably post level 20 progression).

Some calculated values:

n	D	M	c	h <
1	3.5	3	0.05	0.056
1	4.5	3	0.05	0.067
1	5.5	3	0.05	0.078
1	3.5	4	0.05	0.055
1	4.5	4	0.05	0.064
1	5.5	4	0.05	0.073
1	5.5	4	0.1	0.05
2	5.5	4	0.05	0.206
3	5.5	4	0.05	0.414
4	5.5	4	0.05	0.79
5	5.5	4	0.05	1.667

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† If \$n > D_4 + M\$, then the ASI will always be better unless \$\frac{1}{20}n(D+M+1) < cD_4\$. If both of these inequalities hold, then a lower \$h\$ can favor taking the feat over the ASI. However, in practice that's only possible if \$M\$ is very low or \$c\$ is absurdly high.

‡ To be specific, increasing \$M\$ decreases the threshold for \$h\$ only if \$\frac{1}{20}n(D-D_4+n+1) > cD_4\$. In practice, the only cases where this might not hold is when you're making a single attack (\$n=1\$) and you're either using a d4 weapon (\$D=D_4\$) or have an expanded crit range (\$c ≥ \frac{2}{20}\$).

Answer 2

We will assume that the attacks do not require more than a 20 to be successful or a 1 to miss, that is, they fall into the normal sort of AC ranges.

There are three variables at play here (lowercase letters are fixed variables, and uppercase are random variables):

$$ \begin{align} p &:\text{the probability of a successful hit}\\ &-\text{calculated from AC and attack bonuses including the strength bonus}\\ s &:\text{the damage bonus before taking the feat or ASI}\\ &-\text{this is the Strength bonus but also any other bonuses such as magic}\\ D &:\text{the primary damage done by the weapon}\\ &- \text{1d10 for glaives and halberds, 1d6 for quarterstaffs}\\ B &:\text{the butt-end damage done by the weapon using Polearm Master - 1d4}\\ \end{align} $$

Allowing for critical hits the base damage before the feat or ASI is:

$$0.05D+p(D+s)$$

With an ASI the chance to hit improved by \$0.05\$ and the damage on a successful hit by \$1\$ so this becomes:

$$0.05D+(p+0.05)(D+s+1)$$

an improvement of:

$$p+0.05(D+s+1)$$

With Polearm Master the improvement is solely the attack with the butt end. If we let the damage be \$B\$, a random variable representing the outcome of the d4 damage roll, the improvement is

$$0.05B+p(B+s)$$.

So, the answer to your question is Polearm master is better when:

$$ \begin{align} 0.05B+p(B+s)&\gt p+0.05(D+s+1)\\ 0.05B+pB+ps&\gt p+0.05D+0.05s+0.05\\ 0.05B+pB+ps-p-0.05D-0.05s-0.05&\gt0\\ (1+20p)B-D+20p(s-1)-s-1&\gt0\\ \end{align} $$

Now, there are only a limited number of values that the fixed variables can take and since we are only interested in averages, we can replace the random variables with their expected values. So we can just crunch the numbers in a spreadsheet.

It turns out, that if you are using a glaive or a halberd it is always better to take the feat so long as your total damage bonus is better than -5. For a quarterstaff, it needs to be better than -3.

Since we can make the reasonable assumption that someone contemplating the Polearm Master feat does not have a strength score with a negative bonus, the answer is:

It is always better to take the Polearm Master feat.