DM vetoed a powerful ability, but is his new suggestion stronger?

[Naicz]

Backstory: We are creating characters for a campaign that got us to level 16 back in Pathfinder, and now are attempting to complete the campaign in 5e. We were extremely powerful with mythic abilities and were given free reign to make similar abilities for the 5e system.

The "too powerful" ability: Paladin: Maximize all radiant and smite damage dealt.

The "questionably weaker" compromise: Instead, reroll 1's and 8's 'explode' on smites/radiant dmg.

I'm looking at this, and I'm not great at math, but it seems like at some point (Xd8) the rerolling exploding d8s may end up dealing MORE damage than if it was just maximized. Am I crazy/stupid/correct?
Could someone smarter than me help me figure out the math?

Now it isn't clear at this point whether the DM is going to allow only ONE reroll (the initial rolled natural 1's and 8's only) or if he will allow UNLIMITED rerolls until no 1's or 8's appear on the dice. Now obviously the second option is stronger, but I would be curious to see just how much difference it would make.

Thanks in advance!

[Arenabait]

tbh that just seems like a hassle, but it checks out. most of the time it won't be a big deal, but it could potentially give you a ridiculous amount of damage

[GoodbyeSoberDay]

The "too powerful" ability: Paladin: Maximize all radiant and smite damage dealt.

The "questionably weaker" compromise: Instead, reroll 1's and 8's 'explode' on smites/radiant dmg.TL;DR version: Unless you like swinginess a lot, max smite damage is much better.

For simplicity I'm going to assume it's reroll until you get no 1s or 8s. Without the exploding feature, a d8 then just becomes 1d7+1, because the only actual end results are 2 to 8. It turns out that both exploding and maximizing add the same value to each smite die, so it doesn't matter how many dice there are (except how it changes variance, which I don't cover).

Anyway, the expected value of an exploding 1d7+1 is slightly more than 5.8 (work in spoiler), while for obvious reasons the expected value of a maximized d8 is 8. So unless you personally enjoy the fat right tail of the exploding dice (i.e., do significantly less damage more often but every once in a while getting ZOMGRIDICULOUS damage) it's a strict downgrade.

The value of 1d8, reroll 1s, explode on 8 = V(1d7+1e)=V

V=(6/7)E(1d6+1)+(1/7)(8+V)

E(1d6+1)=4.5

(6/7)V=(6/7)(4.5)+8/7

6V=6*4.5+8

V=4.5+8/6=~5.8

[Talamare]

3d8 Max = 24

3d8 Explode = ~18 without the reroll 1s

[Phoenix042]

If it's just a single reroll per die, it amounts to changing the average for each d8 of radiant damage from 4.5 to 5.6; a bump of about 1.1 damage per die of radiant damage you deal. In the end, only a small change, unlikely to cause damage to swing very far past what would otherwise be your maximum, and less likely to have big swingy effects the more dice you roll, since more dice means more average-ish results. This means that higher level smites will tend to be closer to the "extra 1.1 average damage" effect and further from the "sometimes almost doubling damage" effect that might happen with a few good rolls on a 1st level smite.


If it's just a single reroll per use of each ability (as in, one smite, one reroll) this amounts to about 1.1 extra damage per smite; occasionally nice when you roll a 1 I guess, but it'd slow down gameplay to very little effect. Not worth much.

If, on the other hand, you reroll every 1 or 8 until no more 1's or 8's show up, adding each roll to the total... Well, the theoretical damage maximum is now infinite, while the minimum is now 2 instead of 1. The number of d8s you end up rolling per original d8 is going to typically be 4/3. That is, if you use a second level smite, statistically speaking, you're usually going to be rolling 4 dice instead of 3 by the time you're done rolling. This means the average increases by about 4/3, to something like 6 per d8 instead of 4.5 This is only 0.4 higher than the average per die with the limited reroll, but has the added possibility of exploding well beyond your normal range for damage on a smite.

Lets take an extreme case and roll with it. You crit, and decide to do a 4th level smite for 10d8 radiant damage. Lets roll on rolz.org and follow it all the way down the rabbit hole:

8, 1, 8, 7, 8, 3, 3, 7, 6, 3 which would be 54 if we stopped there. Let's keep going.
| | | |
7, 6, 6, 4 which makes our total 77, or about 3 points shy of what would be max damage for you without this ability.


Lets try another
(6 +4 +6 +7 +8 +6 +4 +1 +8 +8) = 58
| | | |
8, 8 6 8 = 88 (now already 8 points above normal max smite damage)
| | |
4 8 7 = 107 (now 27 points above normal max, and 19 points higher than if we stopped at one reroll)
|
7 = 114


These are just two examples drawn from a literally infinite pool of possible rolls, but they demonstrate something important; even this crazy explodey dice thing will wind itself out eventually. From a whopping 10 starting rolls, and with some really good unlikely rolls to follow, we barely made it through 4 generations, each less impactful than the last, and our overall total was less than double the average result, about in line with the difference between the average roll and max roll for any die.

Basically, simply dealing max damage would be both faster for gameplay and less wildly swingy, but this isn't going to allow you to one-shot the Tarrasque, either. Not if we sat there jamming the reroll button on rolz.com for a million years would we get a set of rolls that unlikely.

[OldTrees1]

I tried this on anydice and got this function for a single 1d8 (reroll 1s & explode 8s): http://anydice.com/program/deaa

As expected, when I asked it to handle infinitesimal chances of infinite results, it exceeded the max depth and returned a truncated result. However since the chance decreases faster than the result -> the average of the truncated results is a good approximation for the average of the actual results.

Aka 1d8 (reroll 1s & explode 8s) is roughly 5.83 expected damage per 1d8.

Alternatively we can work it out as follows: (different math but same result as GoodbyeSoberDay)
F(1d8) = Reroll + Keep + Explode
F(1d8) = (1/8) * F(1d8) + 6/8 * (Average of 2-7) + (1/8) * (8+F(1d8))

F(1d8) = (1/8) * F(1d8) + 6/8 * (Average of 1d6+1) + (1/8) * (8+F(1d8))

F(1d8) = (1/8) * F(1d8) + 6/8 * (Average of 1d8) + (1/8) * (8) + (1/8) * F(1d8)

F(1d8) = (2/8) * F(1d8) + 6/8 * (Average of 1d8) + 1

(6/8) * F(1d8) = 6/8 * (Average of 1d8) + 1

F(1d8) = (Average of 1d8) + 8/6 = 5.833...

We can extend this to all dice:
The expected value of 1dX that rerolls 1s & explodes Xs => Average of 1dX + X/(X-2)

[Asmotherion]

I believe you can re-build something similar with core 5e material, and just re-fluff them Lore-Wise. Focus less on actual Mechanics, and more on the Aesthetics and Visual aspects.

You can give him the "exploding smite" visual whenever he rolls a natural 20 (thus a critical) without homebrewing anything.

Making up stuff into an eddition you have not fully understood yet (I'm assuming that much as you mention a transition from pathfinder) is bound to create balance issues; Unless you and your party don't care about this stuff at all (and making this post shows that you do to some level), I'd leave homebrewing out of 5e for a wile.

I'm suggesting all this in good faith, having experianced some pitfalls myself.

[Arkhios]

Wrath of the Righteous, eh? I've been thinking about converting it to 5th as well.

My opinion? I consider myself fairly well in touch with the rules, and a bit of swinginess isn't a bad thing. Even if it potentially (but not very likely) results in higher damage. The more dice, the more fun it is, right? I'd personally go with the RAF (Rules As Fun) here, and say go with the DM's suggestion.

[LeonBH]

Running a trial on my computer with 10 million rolls of the dice per trial, this is what you get:

1d8: 5.8315522
2d8: 11.6693232
3d8: 17.499536
4d8: 23.329869
5d8: 29.1688327
6d8: 34.9970144

The assumption is 1's are re-rolled until they are all gone, but 8's are accepted and not re-rolled (though a new dice is rolled, which can re-explode if an 8 or be re-rolled if a 1).

[Talamare]

I typed this up in Spreadsheet

https://docs.google.com/spreadsheets/d/1ZUkgrTs_gR_Oq5DJm8LnZ1-YUu4qNmsP9bInbRLqhqk/edit#gid=0

Assumes
Reroll 1, must take the new roll (pretty standard)
Original Explode, meaning only the original die can explode. Exploding dice can't Explode

Mathematically, chain explosions quickly become insignificant...
12.5% of 12.5% = 1.56% chance

[Malifice]

Ive actually considered running a game where ALL damage dice explode.

[Vaz]

How about somewhere in the middle, add your proficiency bonus to each radiant damage rolled. Allows for bonus on how much high rolls, and mitigates low rolls.

[Knaight]

The math is pretty simple. Any given roll has a final roll (F), plus some number of exploded rolls (E). The final roll has an average of 4.5 (2 to 7 linear distribution), each exploded roll an average of 8 (8 to 8).

So, how many rolls can be expected to explode? That's pretty straightforward. 1/8th of the first roll, then 1/8th of the next roll, then 1/8th of the third roll...

That's a geometric sum, which uses the formula S=(n)/(1-n) where n in this case is probability of rolling. That gets us 1/7, which when you multiply in the average gets us 8/7. Then you just add the 9/2 from earlier, and you can get exactly (16/14+63/14)=79/14.

Nice and easy, once you realize the trick behind it.

[Aelyn]

Ive actually considered running a game where ALL damage dice explode.

Savage Worlds does that - all rolls to hit explode (an attack can be made on anything from a d4 to a d12, and you get bonuses for exceeding their evasion by 4 or more) and damage rolls explode as well.

It's a small bonus from a hard mathematical perspective, but there is something very satisfying about firing two arrows for 6 and 8 damage, then getting a string of explosions and doing 34 on the next shot. It's good for cinematic moments.