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!