In a future screed, I'll be talking about the Summon Monster / Summon Nature's Ally spells. At third level and higher, you have the option of summoning one max-power monster, 1d3 sub-max monsters, or 1d4+1 sub-sub-max monsters. The reason I mention this that I am wondering which is better with this spell, Empower or Maximize?

Let's look at possible outcomes. Rolling a d3, there are three possible outcomes: 1, 2, or 3. Empowered, those outcomes become 1, 3, or 4 -- or, on average, 2.67. Maximized, the possible outcomes are 3, 3, and 3 -- or on average, 3. The maximized roll does give a better result, on average, than the empowered roll, as you would expect.

But if the range of outcomes is 1d4+1, the possible outcomes are 2, 3, 4, and 5. Empowered, that's 3, 4, 6, and 7 -- average 5. Maximized, the outcomes are 5, 5, 5, and 5 -- also an average of 5! But using maximize is much more expensive, either in requiring a higher spell slot with the feat, or buying the more expensive metamagic rod. Interpreting the rules in this way, there is no advantage, on average, in using maximize over empower -- in fact, it is dumb to do so.

And consider the Spiritual Weapon spell. This spell does "1d8 force damage per hit, +1 point per three caster levels (maximum +5 at 15th level)." That means damage outcomes at 15th level are 6 to 13 per hit, average 9.5. Empowered, the 6 rises to 9, the 7 to 10, the 8 to 12, etc., for a range from 9 to 19, average 14. Maximized, that damage is 13. So once again, empower beats maximize, at a lower cost.

The same holds true for magic missile. At 1d4+1, the average empowered damage is 5, same as the maximized damage. At 2d4+2, the average empowered damage is 10.29 vs. 10 maximized. And at 3d4+3, it's empowered 15.5 vs. maximized 15.

The thing is, in the case of empowering 1d4+1, or 1d8+5, rounding up the added part -- the +1 or the +5 -- skews the system. If you only empower the portion of the results that comes directly from the die roll, maximize wins. For example, that magic missile goes from (1 + 1, 2, 3, or 4) to (1 + 1, 3, 4, or 6), or (2, 4, 5, or 7) -- an average of 4.5. That's less than the average of 5 under the previous interpretation. The d8+5 becomes (5 + 1, 3, 4, 6, 7, 9, 10, 12), or 6 to 17, average 11.5, and much less than the average 14 from the first interpretation of empower.

Empowering only dice rolls, not add-ons, gives the advantage to maximized magic missiles. For one missile, empower averages 4.5 vs. 5 for maximized; for two, 9.29 vs. 10; for three, 14 vs. 15. It's still debatable that maximize is not **that** much better than empower, but it is clearly better.

If you can convince your DM to multiply the added bonuses by 1.5 when empowering a spell, good for you. But it is entirely fair -- and perhaps a better interpretation of the rules -- to apply the 1.5 multiplier only to actual die rolls, before adding bonus numbers.

But even under the more conservative interpretation, is maximize worth it? A maximized magic missile does an average of 5 hp damage; an empowered magic missile, 4.5 hp. Is the extra half a point of damage -- or the extra half a summoned creature -- worth a higher spell slot or the more expensive rod? Judgement call -- it may be worth it to you or not, but under the conservative interpretation, at least there is some advantage to maximize over empower.

**Update [28 August 05]** I noticed this text in the Player's Handbook, page 93: *For example, an empowered magic missile deals 1-1/2 times its normal damage (roll 1d4+1 and multiply the result by 1-1/2 for each missile).*

That means on a d4 roll, an empowered magic missile does 3, 4, 6, or 7 hp damage, averaging 5 hp damage. A maximized magic missile does 5 too. Clearly the PHB favors the more liberal interpretation, and adds half-again to the +1 as well as to the die roll. So I'll take empower over maximize, any day!