bekeleven
2024-02-27, 11:42 PM
I've been goofing off with programatically answering some of these questions recently, and realized that the world gets better when everyone listens to what I say, so without further ado: I made a program to determine the average stat arrays using various stat generation methods. With it, we can perform activities like deriving the elite array, determining optimal point buy, and more.
Average Rolls and the Elite Array
The program I wrote perform the following: It generates a number of stat arrays based on the input stat generation algorithm, then orders the stats in each array from the highest to the lowest. Once it's done that, it extracts the top, middle, and lowest values in each position, then calculates the mean roll in that position.
I'll tell it to roll one million stat arrays with the rules of: Six values, 4d6b3 for each value. Here's the results:
4d6B3x6
Highest Value
18
18
18
17
17
16
Median Value
16
14
13
12
10
9
Lowest Value
8
7
6
4
4
3
Mean Value
15.66
14.17
12.96
11.76
10.41
8.50
This table has some interesting information for us. First of all, this is likely where the elite array (15, 14, 13, 12, 10, 8) was derived. However, we can see that by both mean and median, rolling dice has a higher expected output than the elite array. From the mean array, the elite array rounds 0.66, 0.17, 0.41 and 0.50 down, while rounding only 0.96 and 0.76 up.
We also learn that if you roll a million unique characters, you'll get at least one with three 18s, but four 18s is evidently asking too much.
Point Buy
The biggest hurdle between this data and data about point buy is fractional values. I solved this by estimating a regression line. After tinkering a little, I found one that looks ugly as hell, but gives a pretty close fit to the data at hand: You can see the match here. (https://docs.google.com/spreadsheets/d/11QvZfSpmUQH_7zL0IWPIhgCtTpgUbq-emEwvafeabeQ/edit?usp=sharing) It's least accurate at 14.
That said, if you plug in the above data, you can see that the mean roll is worth 27.69 points. This tracks, as the median roll is worth 28. From this we can conclude that, unless you are risk-averse, you should select an offer of rolled stats over an offer of 25 point buy. However, you should select 28 point buy over rolled stats regardless of your preferences because odds are you will get equal or lesser value from the rolls.
Rerolls
"But Bek!" I hear you shouting. (I have very good ears.) "What about rerolling if you have a negative total modifier? What about rerolling if you have no +2s?" Good point, forum-goer! You're so smart. These are both referenced in PHB page 8, under "rerolls." Basically, even a by-the-book DM has to let you try again if your rolls were so bad you ended up with 8,7,6,4,4,3.
Well, it just so happens, I can set a flag to toss score arrays with those characteristics and regenerate them. So, let's go a few more (millions of) times:
4d6B3x6, Highest Score 14+
Highest Value
18
18
18
18
17
16
Median Value
16
14
13
12
11
9
Lowest Value
14
7
6
4
3
3
Mean Value
15.89
14.34
13.08
11.86
10.49
8.57
Our highest array is higher! ...But our lowest array is lower, because we got at least one array with two 3s. Past that, we can see the two most important arrays have moved: Our median roll is up to 29 points, and all of the values in our mean roll moved up a small fraction, resulting in the fractional point buy value moving up to 28.86.
4d6B3x6, Modifiers sum to +1 or greater
Highest Value
18
18
18
18
17
16
Median Value
16
14
13
12
11
9
Lowest Value
12
10
8
6
4
3
Mean Value
15.84
14.38
13.18
12.00
10.65
8.73
This one was a larger bump. Our median array is still 29 points, and our mean array is up to 29.39 points. This means that even if offered a 29 point buy, you're slightly incentivized to roll under these conditions.
I should point out that the "lowest value" array isn't technically legal. This is due to it being made from the lowest values of each array, rather than a single array itself. For instance, the 6/4/3 at the end could've come from an array of 18/18/18/6/4/3.
Requiring a positive modifier will matter a lot more than requiring a 14 minimum, because most rolls with no +2s are going to end up without a positive modifier. Nonetheless, let's apply both at once to eliminate those all-12 results and fully simulate rolling by the book:
4d6B3x6, Highest Score 14+, Modifiers sum to +1 or greater
Highest Value
18
18
18
18
17
16
Median Value
16
14
13
12
11
9
Lowest Value
14
10
8
6
4
3
Mean Value
15.95
14.45
13.23
12.02
10.66
8.72
Everything went up just a tad (except the lowest mean ability; I assume that's noise), resulting in a fractional point buy value of 29.84.
Conclusion
How risk-averse are you? Because if your DM gives you the option to roll against a set point buy, you are technically advantaged to roll unless the point buy is 30 or greater. Given that my third-order polynomial regression line isn't a perfect fit, we can knock that down to 29 to be sure. But everybody's at least a little risk-averse, and I would take 28 point buy over rolling (Maybe even 27!).
Obviously, changing the roll parameters changes this result, as would changing the point buy values. Point buy is already designed to weight MAD over SAD, but perhaps not to the extent some tables prefer, since SAD classes (that typically need only one mental stat) start more inherently powerful than things like melee classes (that typically need all three physical stats in some measure).
I don't have a larger point to make. I just wanted to share some data that interested me.
Average Rolls and the Elite Array
The program I wrote perform the following: It generates a number of stat arrays based on the input stat generation algorithm, then orders the stats in each array from the highest to the lowest. Once it's done that, it extracts the top, middle, and lowest values in each position, then calculates the mean roll in that position.
I'll tell it to roll one million stat arrays with the rules of: Six values, 4d6b3 for each value. Here's the results:
4d6B3x6
Highest Value
18
18
18
17
17
16
Median Value
16
14
13
12
10
9
Lowest Value
8
7
6
4
4
3
Mean Value
15.66
14.17
12.96
11.76
10.41
8.50
This table has some interesting information for us. First of all, this is likely where the elite array (15, 14, 13, 12, 10, 8) was derived. However, we can see that by both mean and median, rolling dice has a higher expected output than the elite array. From the mean array, the elite array rounds 0.66, 0.17, 0.41 and 0.50 down, while rounding only 0.96 and 0.76 up.
We also learn that if you roll a million unique characters, you'll get at least one with three 18s, but four 18s is evidently asking too much.
Point Buy
The biggest hurdle between this data and data about point buy is fractional values. I solved this by estimating a regression line. After tinkering a little, I found one that looks ugly as hell, but gives a pretty close fit to the data at hand: You can see the match here. (https://docs.google.com/spreadsheets/d/11QvZfSpmUQH_7zL0IWPIhgCtTpgUbq-emEwvafeabeQ/edit?usp=sharing) It's least accurate at 14.
That said, if you plug in the above data, you can see that the mean roll is worth 27.69 points. This tracks, as the median roll is worth 28. From this we can conclude that, unless you are risk-averse, you should select an offer of rolled stats over an offer of 25 point buy. However, you should select 28 point buy over rolled stats regardless of your preferences because odds are you will get equal or lesser value from the rolls.
Rerolls
"But Bek!" I hear you shouting. (I have very good ears.) "What about rerolling if you have a negative total modifier? What about rerolling if you have no +2s?" Good point, forum-goer! You're so smart. These are both referenced in PHB page 8, under "rerolls." Basically, even a by-the-book DM has to let you try again if your rolls were so bad you ended up with 8,7,6,4,4,3.
Well, it just so happens, I can set a flag to toss score arrays with those characteristics and regenerate them. So, let's go a few more (millions of) times:
4d6B3x6, Highest Score 14+
Highest Value
18
18
18
18
17
16
Median Value
16
14
13
12
11
9
Lowest Value
14
7
6
4
3
3
Mean Value
15.89
14.34
13.08
11.86
10.49
8.57
Our highest array is higher! ...But our lowest array is lower, because we got at least one array with two 3s. Past that, we can see the two most important arrays have moved: Our median roll is up to 29 points, and all of the values in our mean roll moved up a small fraction, resulting in the fractional point buy value moving up to 28.86.
4d6B3x6, Modifiers sum to +1 or greater
Highest Value
18
18
18
18
17
16
Median Value
16
14
13
12
11
9
Lowest Value
12
10
8
6
4
3
Mean Value
15.84
14.38
13.18
12.00
10.65
8.73
This one was a larger bump. Our median array is still 29 points, and our mean array is up to 29.39 points. This means that even if offered a 29 point buy, you're slightly incentivized to roll under these conditions.
I should point out that the "lowest value" array isn't technically legal. This is due to it being made from the lowest values of each array, rather than a single array itself. For instance, the 6/4/3 at the end could've come from an array of 18/18/18/6/4/3.
Requiring a positive modifier will matter a lot more than requiring a 14 minimum, because most rolls with no +2s are going to end up without a positive modifier. Nonetheless, let's apply both at once to eliminate those all-12 results and fully simulate rolling by the book:
4d6B3x6, Highest Score 14+, Modifiers sum to +1 or greater
Highest Value
18
18
18
18
17
16
Median Value
16
14
13
12
11
9
Lowest Value
14
10
8
6
4
3
Mean Value
15.95
14.45
13.23
12.02
10.66
8.72
Everything went up just a tad (except the lowest mean ability; I assume that's noise), resulting in a fractional point buy value of 29.84.
Conclusion
How risk-averse are you? Because if your DM gives you the option to roll against a set point buy, you are technically advantaged to roll unless the point buy is 30 or greater. Given that my third-order polynomial regression line isn't a perfect fit, we can knock that down to 29 to be sure. But everybody's at least a little risk-averse, and I would take 28 point buy over rolling (Maybe even 27!).
Obviously, changing the roll parameters changes this result, as would changing the point buy values. Point buy is already designed to weight MAD over SAD, but perhaps not to the extent some tables prefer, since SAD classes (that typically need only one mental stat) start more inherently powerful than things like melee classes (that typically need all three physical stats in some measure).
I don't have a larger point to make. I just wanted to share some data that interested me.