[RPG] At what point is Empower Spell better than Maximize Spell

metamagicoptimizationpathfinder-1espells

When I'm trying to increase the output of my spells such as Fireball, Cure Critical Wounds or Magic Missile, I can't decide whether Empower Spell or Maximize Spell is better.

I'd like to compare only the effects of the feats, without involving the difference in spell level adjustment for each one since I'm prioritizing the output of the spell without regards to the cost involved in to cast it. However, I only have enough resources to use one of the feats, therefore I need to figure out in what cases one is better than the other.

Best Answer

Empower Spell is better when your bonus per die is higher.

If we look at Maximize Spell, it says:

All variable, numeric effects of a spell modified by this feat are maximized. Saving throws and opposed rolls are not affected, nor are spells without random variables. A maximized spell uses up a spell slot three levels higher than the spell's actual level.

This means that any effects in the form of \$\text{x}\text{d}\text{y}+\text{z}\$ will instead be \$\text{x}*\text{y}+\text{z}\$. For example, a single magic missile deals \$1\text{d}4+1\$ damage normally, with maximize it instead deals \$1*4+1 = 5\$ damage.

If we looks at Empower Spell, we see:

All variable, numeric effects of an empowered spell are increased by half including bonuses to those dice rolls.

How the feat works is further clarified in this FAQ:

Empower Spell: If I use Empower Spell on a spell that has a die roll with a numerical bonus (such as cure moderate wounds), does the feat affect the numerical bonus?

Yes. For example, if you empower cure moderate wounds, the +50% from the feat applies to the 2d8 and to the level-based bonus.

From this we can see that this means any effects in the form of \$\text{x}\text{d}\text{y}+\text{z}\$ will instead be \$(\text{x}\text{d}\text{y}+\text{z})*1.5\$. For example, a single magic missile deals \$1\text{d}4+1\$ damage normally, with empower it instead deals \$(1\text{d}4+1)*1.5\$ damage.

To see how to two feats compare we're going to look over Cure Critical Wounds since it's the least complicated to explain. The effect of the spell states:

This spell functions like cure light wounds, except that it cures 4d8 points of damage +1 point per caster level (maximum +20).

This means that by base the spell heals for \$4\text{d}8 + \text{caster level}\$, if we empower this, it heals for \$(4\text{d}8 + \text{caster level})*1.5\$, and maximized it heals for \$4*8 + \text{caster level}\$. If we use the fact that the average value of a d8 is 4.5, we know that Maximize Spell only increases the output of the spell by 3.5 per die rolled and we know that the average of 4d8 is 18. Thus, we can get the following formulas for analyzing the spell:

\begin{array}{c|c} \text{maximize} & 32+\text{Caster Level} \\ \hline \text{empowered} & (18+\text{Caster Level})*1.5 \end{array}

We can then figure out values per die rolled using the following formulas:

\begin{array}{c|c|l} \text{Bonus Per Die Rolled} & \text{Caster Level}/{4}\\ \hline \text{Empowered Amount Per Die} & (\text{Caster Level}/{4}+4.5)*.5 \end{array}

We can then use these formulas to produce the following table. Note that technically caster levels 1-6 are impossible to cast Cure Critical Wounds at, but are merely included for the sake of doing so.

\begin{array}{c|c|c|c|c} \textbf{Caster} & \textbf{Mazimized} & \textbf{Empowered} & \textbf{Bonus Per} & \textbf{Empowered Amount} \\ \textbf{Level}& & & \textbf{Die Rolled} & \textbf{Per Die} \\ \hline 1 & 33 & 28.5 & 0.25 & 2.375 \\ 2 & 34 & 30 & 0.5 & 2.5\\ 3 & 35 & 31.5 & 0.75 & 2.625 \\ 4 & 36 & 33 & 1 & 2.75 \\ 5 & 37 & 34.5 & 1.25 & 2.875 \\ 6 & 38 & 36 & 1.5 & 3 \\ \hline \\ \hline 7 & 39 & 37.5 & 1.75 & 3.125 \\ 8 & 40 & 39 & 2 & 3.25 \\ 9 & 41 & 40.5 & 2.25 & 3.375 \\ \hline \color{red}{\textbf{10}} & \color{red}{\textbf{42}} & \color{red}{\textbf{42}} & \color{red}{\textbf{2.5}} & \color{red}{\textbf{3.5}}\\ \hline 11 & 43 & 43.5 & 2.75 & 3.625 \\ 12 & 44 & 45 & 3 & 3.75 \\ 13 & 45 & 46.5 & 3.25 & 3.875 \\ 14 & 46 & 48 & 3.5 & 4 \\ 15 & 47 & 49.5 & 3.75 & 4.125 \\ 16 & 48 & 51 & 4 & 4.25 \\ 17 & 49 & 52.5 & 4.25 & 4.375 \\ 18 & 50 & 54 & 4.5 & 4.5\\ 19 & 51 & 55.5 & 4.75 & 4.625 \\ 20 & 52 & 57 & 5 & 4.75 \end{array}

Looking at the table we can see that Caster Level 10 is where Empower Spell begins to increase the spell's output per die by the same value that Maximize Spell does so. Thus, at Caster Level 10, Empower Spell begins to become the optimal metamagic to apply to the spell. Using this table we can make the following statement then:

Empower Spell becomes as good or better than maximize when your bonus per die + the average value of the die equals twice the difference between the maximum and average of the die to be rolled.

As a formula this would be:

$$(\text{Average} + \text{Bonus}) = 2 * (\text{Max}-\text{Average})$$

Solving this for d4, d6, d8, d10, d12, and d20 we get the following bonuses where Empower Spell begins to be as effective or better than Maximize Spell:

\begin{array}{c|c} \text{Die Size} & \text{Bonus}\\ \hline 4 & .5\\ 6 & 1.5 \\ 8 & 2.5 \\ 10 & 3.5 \\ 12 & 4.5 \\ 20 & 8.5 \\ \end{array}