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View Full Version : How Much Is A Reroll Worth?

DragonBaneDM
2015-03-12, 09:38 AM
I know that a reroll on a 1d20 and getting to use the higher result is roughly the equivalent of a +5 bonus, but what about on a d10? A d12? Is there a formula I could use to calculate it on 2d4 or 6d6, or any other random combination of dice?

Is it roughly a quarter of the highest possible result, or is that just how the math came out? I'm calculating DPR and use a d10, and I realized that I just have no idea how to value a possible reroll to my damage dice.

Mark Hall
2015-03-12, 10:41 AM
To extrapolate from a single example, a reroll-keep-better would be worth about 1/4th the value of the die. So, on a d10, the value would be about a +2; on 2d4, it would be about +1 (though it gets fuzzier with multiple dice making up the roll, and whether or not you have to choose to reroll the entire roll, or can just reroll the parts of the dice formula you want to).

On the roll of a single die, the chance of getting a better roll is (x-y)/x, where x is the number of sides and y is the amount you rolled. Roll a 13 on a d20, you'll get better about 35% of the time... 14-20. Roll a 1, and you get better 95% of the time. If you make it a d10, and you roll a 5, then you'll get a better roll 50% of the time. This is pretty straightforward. Turning that into an actual calculation of bonuses is beyond my math, but 1/4 the value die is consistent with the example, and seems intuitively right.

Ashtagon
2015-03-12, 01:43 PM
Mathematically, a single reroll is worth + ((min roll) - (max roll)) /4.

That is definitely true for any roll of a single die. I'm not sure it holds up for multiple dice (eg 3d6)

Galen
2015-03-12, 03:34 PM
I know that a reroll on a 1d20 and getting to use the higher result is roughly the equivalent of a +5 bonus, but what about on a d10? A d12? Is there a formula I could use to calculate it on 2d4 or 6d6, or any other random combination of dice?

Is it roughly a quarter of the highest possible result, or is that just how the math came out? I'm calculating DPR and use a d10, and I realized that I just have no idea how to value a possible reroll to my damage dice.

Average of d20: 10.5
Average of 2d20 take best: about 13.8

Here's how to calculate it for any die. I'll use d6 as an example, you can figure out the rest:

Step 1, perform those 6 multiplications:

1 * 1 = 1
2 * 3 = 6
3 * 5 = 15
4 * 7 = 28
5 * 9 = 45
6 * 11 = 66

As you can see, you start with 1*1, then the first number increases by 1 each repetition, and the 2nd number increases by 2.

Step 2, sum the results together:

1+6+15+28+45+66 = 161

Step 3, divide by 36 (6 squared):

161/36 = 4.472

So the average of 2d6-take-best is 4.472. The average of 1d6 is 3.5, so the value of a reroll here is +0.972.

For those who are into math, the formula for an N-sided die is:

Sigma(i=1; i=N)[i*(i*2-1)]/N^2

Coventry
2015-03-12, 03:39 PM
Actually, the average for rolling 2d20 and keeping the highest is not 15, but only 13.825. A reroll on a d20 is only worth +3.325.

The simple formula eludes me, but you can brute-force the calculation. There are 400 combinations of rolling 2d20. Only one (1,1) will give a maximum of 1. Three (1,2; 2,2; 2,1) will give a maximum of 2. Five (1,3; 2,3; 3,3; 3,2; 3,1) will give a maximum of three and so on.

Following that pattern all the way up to the 39 combinations that includes at least one "20" gives you this equation to get that 13.825 average: (1*1+2*3+3*5+4*7+5*9+6*11+7*13+8*15+9*17+10*19+11* 21+12*23+13*25+14*27+15*29+16*31+17*33+18*35+19*37 +20*39)/400

Rolling d6 will give you 3.5 on average. Rolling twice and keeping the best will average out to about 4.472, so at +0.972 it's not even worth a full +1 on a d6.

In effect, re-rolling a single die is worth about 1/6 of the maximum roll.

There is even less value when re-rolling something like 3d6. I brute-forced the calculation and got 12.176 as the average of 3d6 twice, keep the best, so only +1.676 above the usual average of 10.5.

Edit: Ah, drat. Too slow, am I.

endur
2015-03-12, 11:35 PM
It actually depends on the type of re-roll.

For example, one type of a re-roll is where you roll 4d6 and keep the highest 3d6 (essentially rerolling the lowest d6).

Another type of reroll is where the character can re-roll 1 twenty sided die roll per day.

These re-rolls vary vastly in their importance and the effect they have.

The re-roll 1d20 once per day could be worth as much as 9.5, but only if you use it after rolling a 1 on that particular day. If you never use it (saving it for the 1 that doesn't happen), then it is not worth anything.

Battlebooze
2015-03-12, 11:47 PM
I see much useful information, but everyone keeps missing the point.

I'll give you five dollars.

:wink:

Ashtagon
2015-03-13, 01:42 AM
It depends. Are we considering

Spend a daily re-roll. Roll 2d20. Pick the best one.
Roll 1d20. Don't like it? Choose to spend a daily re-roll and then roll again and choose the best.
Roll 1d20. Don't like it? Choose to spend a daily re-roll and then roll again and take the second roll, even if it is worse.

In each case, the maths and decision-tree are slightly different.

DragonBaneDM
2015-03-13, 07:13 AM
It depends. Are we considering

Spend a daily re-roll. Roll 2d20. Pick the best one.
Roll 1d20. Don't like it? Choose to spend a daily re-roll and then roll again and choose the best.
Roll 1d20. Don't like it? Choose to spend a daily re-roll and then roll again and take the second roll, even if it is worse.

In each case, the maths and decision-tree are slightly different.

The specific dice I'm trying to work with at the moment are a 1d10 roll and 2d10 roll. But I'm also trying to just trying to learn the formula for every type so I never have to ask this question again.

Battlebooze, deal.

I used Galen's formula and found the average reroll on a d10 to be worth 1.65, so our original estimate of 2 wasn't too far off.

Dimers
2015-03-13, 08:52 AM
The specific dice I'm trying to work with at the moment are a 1d10 roll and 2d10 roll. But I'm also trying to just trying to learn the formula for every type so I never have to ask this question again.

You already got accurate info about the average numeric improvement. The specifics of the game system make the practical answer more complex, though. For attack rolls in D&D, a 20 is much better than a 19, and a 1 can be much worse than a 2, and roll-twice-take-better makes 20s a lot more likely and 1s a lot less. It can more extreme in World Of Darkness rolling, where 1s actively take away from your success and 10s explode. In Shadowrun 4e, a 5 is very different from a 4, and a 6 is different from a 5 if you spend Edge a lot (so rerolls are worth more for characters with high Edge than those with low).

In addition, rolling two dice and taking the better result =/= rerolling a failure =/= rerolling an insufficient roll that might or might not have turned out to be a failure but you had to choose before you heard the results.

Sorry to muddy the issue.

Ashtagon
2015-03-13, 09:12 AM
Actually, it's even more complicated than I first thought...

Roll your die and keep it. No re-roll spent. This is the "base" value.
Spend a re-roll before any dice are rolled. Roll your die twice and keep the best result.
Roll your die. See the number of the die roll. Decide Whether to spend a re-roll. If you spend a re-roll, keep the best result.
Roll your die. See the number of the die roll. Decide Whether to spend a re-roll. If you spend a re-roll, keep the second result, even if it is worse.
Roll your die. The GM tells you what that die roll would mean. Decide Whether to spend a re-roll. If you spend a re-roll, keep the best result.
Roll your die. The GM tells you what that die roll would mean. Decide Whether to spend a re-roll. If you spend a re-roll, keep the second result, even if it is worse.

Letting players spend re-rolls after seeing the die roll allows for better dramatic tension moments, but conversely can lead to disputes about whether the GM allowed enough time for the player to declare a re-roll was being used. And waiting for that declaration each time can slow the game down.

Then there's the question of spending multiple re-rolls on a single roll.

Maybe...

Before you roll your die, decide if you want to spend one (or more) re-rolls. Roll one extra time for each re-roll you spend, and pick the best result. After that roll, you may spend re-rolls one at a time, but each time you spend a re-roll after the original result was rolled, you must keep the new roll, even if it is worse. You cannot spend re-rolls once the GM has begun describing the result of that die roll.

Segev
2015-03-13, 09:35 AM
"You must declare use of this re-roll before you know the results," is not the best of mechanics, anyway.

Consider the following:

Round 1: "I rolled a 19 to hit." "You hit!"
Round 2: "I rolled a 10 to hit." "You miss!"
Round 3: "I rolled a 9 to hit. I use a reroll."

You rolled a value you immediately knew was a miss based on prior results. Most DMs allow you to go ahead and use the reroll, of course, but it makes the requirement that you choose whether to use it before you know the results, if not meaningless, then at least a minor nuissance at best.

This is, in fact, similar in most circumstances to "if you use a re-roll, you must keep the second result, even if it's worse." Barring the DM house-ruling critical failures, a miss is a miss is a miss. Using your re-roll on that 9 and getting a 7 is no different if you have to keep the 7 or the 9.

Ashtagon
2015-03-13, 09:51 AM
"You must declare use of this re-roll before you know the results," is not the best of mechanics, anyway.

I was highlighting that there are several different stages of "knowing the result"...

Declare the re-roll before you roll the die.
Declare the re-roll after you see the die score, but before you know what implications that has.
Declare the re-roll after you see the die score, and after you know what implications that die result has.

#2 is more obvious during combat and other situations in which you are repeatedly aiming for the same target number. But it isn't always clear-cut, especially if the re-roll is used on a damage roll. ("I rolled 4 hp damage. I want to re-roll. Now I rolled 11 hp of damage." GM says that's cool, but the orc only had 2 hp left). And if it is your first roll or two against that target number, you might still not know what your target number is.

Letting the player spend a re-roll after knowing the implications of the first roll can cause problems in that it disrupts the narrative the group might be weaving in describing the results. That's not so much a problem if the group treats the action as a purely dice-rolling exercise, but if you try to describe the results in narrative / story-telling form, it can cause awkardness as lots of "take-backs" occur in the group's collective story-telling narrative.

The other confounding factor is whether you have to keep the re-roll even when it is worse than the original. On attack rolls, this can impose a fumble where the original roll was a simple miss, and on damage rolls it can result in doing less damage than originally planned; this can add more interesting results to the action. On the down side, this does amount to penalising the PC as a result of spending a limited resource.

DragonBaneDM
2015-03-15, 02:27 PM
So just to be 100% specific, the type of reroll I have is that I can see the original dice, choose to reroll it, and then take either result.

I know the math would be a little bit different every time, based on how the reroll's rules functions, but I feel like we've been doing things correctly thus far for my scenario.

Ashtagon
2015-03-15, 05:30 PM
So just to be 100% specific, the type of reroll I have is that I can see the original dice, choose to reroll it, and then take either result.

I know the math would be a little bit different every time, based on how the reroll's rules functions, but I feel like we've been doing things correctly thus far for my scenario.

What are you criteria for deciding whether to re-roll it or not? If you only ever choose to spend a re-roll if you roll less than half your maximum value, the benefit will be more than if you always choose to spend that re-roll.

hindsight
2015-03-15, 07:52 PM
The expected value for rolling two n sided dice and selecting the higher value is given by

{(2/3)n2 + (1/2)n -(1/6)}/n

The increase in expected value over a single roll is then just

{(2/3)n2 + (1/2)n -(1/6)}/n-(1+n)/2

For the standard dice:

n
Expected increase

4
0.6250

6
0.9722

8
1.3125

10
1.6500

12
1.9861

20
3.3250

Galen
2015-03-16, 12:09 AM
So just to be 100% specific, the type of reroll I have is that I can see the original dice, choose to reroll it, and then take either result.

I know the math would be a little bit different every time, based on how the reroll's rules functions, but I feel like we've been doing things correctly thus far for my scenario.
That's a bit different.

Since the average of 1d10 is 5.5, you obviously only reroll on a 5 or less. So rolls of 5 or less become 5.5, and rolls of 6 or higher are kept. On average:

(5.5+5.5+5.5+5.5+5.5+6+7+8+9+10)/10 = 6.75.

A reroll under those rules is worth 6.75 - 5.5 = +1.25

DragonBaneDM
2015-03-16, 08:01 AM
What are you criteria for deciding whether to re-roll it or not? If you only ever choose to spend a re-roll if you roll less than half your maximum value, the benefit will be more than if you always choose to spend that re-roll.

I can reroll lightning damage at-will with an item I'm going to try and obtain/craft, and I'm trying to figure out how that impacts my end game DPR so that I have something to shoot for. So I'll never really have to "spend" a reroll, it's just something that's going to happen each damage roll during a fight where I'm able to leave my Lightning Weapon turned on.

Since the average of 1d10 is 5.5, you obviously only reroll on a 5 or less. So rolls of 5 or less become 5.5, and rolls of 6 or higher are kept.

I mean if I roll a 9 I'm still going to reroll because there's a chance I can take a 10. I can take either result, so even if my second roll is a 2 I still keep my 9.

Segev
2015-03-16, 08:56 AM
If you are ALWAYS rerolling, it's just the situation of roll twice, keep higher, given as one of the earliest responses in this thread.

DragonBaneDM
2015-03-16, 09:57 AM
If you are ALWAYS rerolling, it's just the situation of roll twice, keep higher, given as one of the earliest responses in this thread.

Roger that.

AceOfFools
2015-03-16, 07:29 PM
Submitted without comment: http://anydice.com/