Statistically, how high a roll should I use Empowered Spell to reroll for damage?

Question

Here's the language for the Metamagic Empowered Spell:

When you roll damage for a spell, you can spend 1 sorcery point to reroll a number of the damage dice up to your Charisma modifier (minimum of one). You must use the new rolls.

You can use Empowered Spell even if you have already used a different Metamagic option during the casting of the spell.

I'll be in my first combat as a sorcerer tomorrow, and I realized that I've been assuming that on e.g. a d6 I should reroll anything below a 4, because (as far as I can work out) statistically the chances that I'll roll 1-3 are equal to the chances that I'll roll 4-6.

Then I thought about the fact that I haven't studied statistics since 1987 and even then I didn't do well in the class, so I figured it would be better to ask here.

Where is the statistically optimal cutoff for rerolling damage dice?

P.S.: I know statistical language is super-specific, so if I've confused anything, please let me know and I'll fix it.

Answer 1

If you only care about maximizing your expected average damage, then yes, you should reroll all dice that rolled below average (or as many of your lowest rolls as your Charisma modifier allows). The average of a d6 roll is \$\frac{1+2+3+4+5+6}{6} = 3.5\$*, so this means rerolling any rolls of 1 to 3.

However, there may be situational reasons why you may not want to reroll that many dice, or any dice at all.

• Sorcery points are a limited resource. If you prefer to conserve them, and think your roll is already good enough, it may be better not to reroll.

• If you expect your initial roll to already do enough damage to kill the target, then there's very little to be gained from rerolling any dice. In the worst case, unnecessary rerolling even carries the risk of turning a lethal attack into a non-lethal one.

• In particular, if you only care about rolling at least some specific total (say, because you somehow know exactly how many HP your enemy has left), then you should only reroll enough of your lowest rolls to maximize the probability of getting a sufficiently high total. Note that, in this case, it might sometimes even be optimal to take a risk and reroll a 4 or a 5 on a d6, hoping for a 6, if that's the only way to have a chance of getting a high enough total.**

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^{*) A quick way to calculate this for normal dice, where the values of the sides form an arithmetic progression, is to simply take the average of the highest and the lowest side. If the lowest side is 1, this works out to half the highest side plus ½.}

^{**) I worked out the optimal strategy for this when rolling Fudge dice in an earlier answer, but I don't think it generalizes directly to d6 rolls, since having more sides per die makes things a bit more complicated. I might amend this answer later.}

Answer 2

Other ways of thinking about it

To add to Ilmari’s answer.

Expected improvement

For an \$n\$ sided die that is currently showing an \$x\$, the expected improvement in your roll is the expected value of the new roll minus the roll you already have: \${(n+1)\over 2} - x\$. So, for example, if you have a 6-sided die showing a 2, your expected improvement from a reroll is 1.5. If it shows a 4, your expected improvement is -0.5.

Chance of improvement

For an \$n\$ sided die that is currently showing an \$x\$, the chance that you will roll better is \$(n-x)\over n\$, the same is \$1\over n\$ and worse is \$(x-1)\over n\$.