Rolling Stats and Point Buy equivalencies


Pathfinder First Edition General Discussion


The core rules suggest a 10 point buy is "Low Fantasy", 15pb is "Standard Fantasy", 20pb is "High Fantasy", and a 25pb is "Epic Fantasy". And it got me wondering... of you were doing dice rolls for stats, how would you differentiate?

Does 4d6 drop the lowest correspond to a 15 point buy or a 20 point buy? What about 5d6 drop the lowest two? 2d6+6 - the "Heroic" method from the PRD?

So then I thought, it should be possible to determine the probabilities on a given roll mechanic, and if you calculate the point buy needed to achieve those scores you could get a weighted value, an "Expected Point Buy" for a given roll system.

The problem is, the lowest you can buy down to is a 7, while with most rolling systems you can, possibly, get a 3. So with the method above you could get an Expected Point Buy for the 2d6+6, but the others would have invalid values that you can't just ignore.

Another alternative is to generate all possible ability score arrays for a given point buy (not *quite* as big as you might think, since order doesn't matter and so some possibilities are duplicates) and then determine the probability that a given roll mechanism will determine that set. Here you could at least see which rolls are more likely for given point buys, but it still doesn't really tell you what the point buy equivalency for a given roll is.

So, other than extending the point buy table down to three, is there any other way to get an expected point buy value for various rolling methods?


Not really answering your question, but I saw a guy who was thinking of playing a Duergar with stat-roll, and got a 3. I think he got to reroll, but that would have made for a very charismatic character.


I've seen a table comparing D&D point buy to dice rolls - it is definitely possible to take 3d6, 2d6+6, 4d6;drop one, 5d6;drop two, and even dice pool rolls and work out what they come out to, though even then there is the possibility of huge outliers - a lot of rolls don't guard against the possibility of rolling a 3, and we've seen a few threads of people asking for build advice on 16s and 17s or all 18s.

For the record - 4d6;drop one rerolling results less than 7 (My 'moderate fantasy' campaign roll) consistently scored around or above 15 points on the point buy system.


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Derek Vande Brake wrote:
Does 4d6 drop the lowest correspond to a 15 point buy or a 20 point buy? What about 5d6 drop the lowest two? 2d6+6 - the "Heroic" method from the PRD?

The standard method (4d6, discard lowest die) gives results that would on average cost 19,46 ability score points if you treat results in the range 3-7 as costing -4 points. If you instead reroll results less than 7, the average cost is 20,70 points. In either case, this is pretty close to a 20 point buy.

To check this, first work out that of the 1296 different outcomes of 4d6, the number of outcomes that give a result of 3, 4, 5, ..., 16, 17 and 18 are 1, 4, 10, 21, 38, 62, 91, 122, 148, 167, 172, 160, 131, 94, 54 and 21, respectively. Then do the sums.


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Generally, 4d6/drop lowest/no rerolls averages between 15 and 20 point buy (if you extend the "buy back" amounts for lower scores). However, because of the possibly of having lower stats than the "minimum" in point buy, it is considered roughly equivalent to 15 point buy. Any variations that reduce or eliminate the possibility of lower rolls (reroll ones, reroll "poor" rolls or sets of rolls, etc.) skews the point buy equivalent upward.

2d6+6 gives an average score of 13, but the "curve" is flatter (as there are only 36 possible variations): 1 in 36 for 8, 2 in 36 for 9, etc.; so, (-2 + -2 + 0 + 4 + 10 + 18 + 25 + 28 + 30 + 26 + 17) / 6 = 25.67 point buy equivalent on average.


This thread has some average point buy values four different stat rolling methods.


Thank you, Scythia! Following that link I also found this: AnyDice
Really helpful!


I looked into this a while ago.

Cevah wrote:

Did some spreadsheets with exhaustive stat selection.

3d6 average stat = 10.5
10 Point buy average stat = ~11.09
15 Point buy average stat = ~11.58
20 Point buy average stat = ~12.07
4d6 less one average stat = ~12.24
25 Point buy average stat = ~12.53

This is before racial modifiers.

/cevah


Derek Vande Brake wrote:

What about 5d6 drop the lowest two? 2d6+6 - the "Heroic" method from the PRD?

So then I thought, it should be possible to determine the probabilities on a given roll mechanic, and if you calculate the point buy needed to achieve those scores you could get a weighted value, an "Expected Point Buy" for a given roll system.

As indicated in my previous post, PB -> Average can be done. Took 2 hours for me to run the numbers for 20 points.

Derek Vande Brake wrote:
The problem is, the lowest you can buy down to is a 7, while with most rolling systems you can, possibly, get a 3. So with the method above you could get an Expected Point Buy for the 2d6+6, but the others would have invalid values that you can't just ignore.

True, which is why I do not have the other direction of die -> pb. You can calculate pb equivalent of any valid roll.

Derek Vande Brake wrote:
Another alternative is to generate all possible ability score arrays for a given point buy (not *quite* as big as you might think, since order doesn't matter and so some possibilities are duplicates) and then determine the probability that a given roll mechanism will determine that set. Here you could at least see which rolls are more likely for given point buys, but it still doesn't really tell you what the point buy equivalency for a given roll is.

Throwing out duplicates throws off the average. Same as throwing out anything with two 7s, 'cause who would use such?

Derek Vande Brake wrote:
So, other than extending the point buy table down to three, is there any other way to get an expected point buy value for various rolling methods?

If the rolling method generates point buy acceptable numbers, then you can convert it. Ex: 8,8,8,8,8,8+8d6 scattered as desired, max 18 can be calculated to an average point buy. 4d6*7,drop lowest,use 6 of 7, can also be made into an average, but not easily a point buy since you can get extremes. Also, the racial adjustments throw off the averages on both sides. You can get the same stats with different point buys.

/cevah


The racial adjustments, at least in this case, don't actually affect anything - whether using point buy or rolling, the stats are generated before racial modifiers.

Also, throwing out duplicate point buys won't affect averages, since point buys are not randomly generated. It's like if I was looking for the probability of rolling a 7 on 2d6. I know the values are 2 through 12; if I throw out an extra 7, the values are still 2 through 12. I only need one of each number to define the set of possible values.


I extrapolated the point buy table down to a score of 3.

Extrapolated Point Buy Values:

18 = 17
17 = 13
16 = 10
15 = 7
14 = 5
13 = 3
12 = 2
11 = 1
10 = 0
9 = -1
8 = -2
7 = -4
6 = -6
5 = -9
4 = -12
3 = -16

I used these values to evaluate the average equivalent point buy of different rolling methods in simulations of 10,000 rolls. Here is what I got:

Average Equivalent Point Buy (extrapolated):

Classic 3d6 = 3.1 points
Standard 3 of 4d6 = 19.0 points
Heroic 2d6 + 6 = 25.7 points
3 of 5d6 = 30.5 points

But, you are correct. We can't just ignore the method used to extrapolate scores below 7.

Alternatively we can drop arrays that have scores below 7 and calculate the average equivalent point buy from what is left over. But in that case we need to report the percentage of scores that were dropped because sometimes it is a significant proportion. Here is what I got:

Average Equivalent Point Buy (dropped):

Classic 3d6 = 8.4 points 44% dropped
Standard 3 of 4d6 = 20.8 points 15% dropped
Heroic 2d6 + 6 = 25.7 points 0% dropped
3 of 5d6 = 31.2 points 5% dropped

But again, we cannot just ignore this technique either, because dropping scores skews the results of some methods more than others and we're left without a clear comparison.

I suggest abandoning the average equivalent point buy as the statistic by which we compare various methods. Instead I suggest we use a something like average total score or average total modifier, whereby we sum the six scores or modifiers of each array and then use those sums to calculate the average over the entire set of arrays.

I like the average total modifier because the rerolling rules from Dungeons and Dragons 3.5 uses the total modifier of an array of scores as one of the criteria for determining whether one should reroll their scores or not.

Rerolling Rules:

If your scores are too low, you may scrap them and roll all six scores again. Your scores are considered too low if the sum of your
modifiers (before adjustments because of race) is 0 or lower, or if
your highest score is 13 or lower. (Player's Handbook page 8)

So, here is what I got comparing total average modifier among various rolling and buying methods:

Average Total Modifier:

Classic 3d6 = 0.0
Standard 3 of 4d6 = 5.3
Heroic 2d6 + 6 = 7.5
3 of 5d6 = 8.8

10 point buy = 1.7 (225 arrays)
15 point buy = 3.3 (262 arrays)
20 point buy = 4.8 (280 arrays)
25 point buy = 6.2 (272 arrays)

Note: There are 54,264 possible arrays with scores between 3 and 18. The numbers in parenthesis is the number of arrays that are equivalent to each particular point buy.

By using average total modifier as our statistic of comparison we can better compare rolling and buying methods because we don't have to consider the pesky scores below seven problem.

However this is not without its issues. For instance, of those 280 possible arrays that equal 20 points some are just not what a normal player would choose for their character, and the average total modifier climbs up closer to 7 when you look at a set of actual arrays chosen by real people.

Anyway, I hope this helps in your endeavours.


Average Total Modifier is a better method of comparison because of the even/odd problem (modifiers kick in with even values and odd ability scores are only slightly more valuable than the even score below them - a character who roles straight 13s has a PB equivalent of 18 but has inferior modifiers compared to a character with a 15PB of 5x 12s and a 14 in one attribute.

Even Average Total Modifier isn't a perfect method of comparing actual utility since not all statistics are equally useful to all characters and the SADder (single attribute dependent) a character is the more they benefit from an outlier score in their primary attribute and less they benefit from other attributes. This utility factor means that with a 10 PB about 80% of characters created by players will have one outlier attribute of 16+, while 4d6 drop lowest results in at least one attribute of 16+ about 56% of the time. Then again modifiers are only important to the utility in terms of die rolls, the more rolls a character makes using a modifier the more useful that modifier is and a fighter who rolled a 14 charisma (considered a wasted attribute for fighters)is going to benefit from that modifier any time he has to make a roll involving charisma.

Since it varies from person to person abd campaign to campaign, there is no way to mathematically calculate a comparison of the in-game utility of characters created using PB values and rolling methods. Effectively you're going to have to come up with a rule of thumb for comparing PB to roll methods based on the min/maxing ability of your players, their preferences for SAD classes, and what balance you have between roll-play and role-play.

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