First off, this neglects the fact that the number of points "earned" for a roll goes up non-linearly, with good rolls being rewarded more than bad rolls are punished. The second factor that the PHB and DMG neglect is the reroll rule: If the sum of the ability modifiers is less than or equal to 0, or the highest score is a 13 or lower, you get a free reroll. This can happen any number of times, reducing the chance of getting a catastrophically low set of rolls.
Using a Perl script, I have simulated 1,000,000 rolls with this system, both without rerolls and with rerolls. The script also has produced histograms showing the probability of each result for each die and for point buy.
Note on point-buy scores: Since standard point buy does not deal with scores lower than 8, I set 7 to be -1 point, 6 to be -2, 5 to be -3, 4 to be -4, and 3 to be -5.
RAW DATA:
Without Rerolls
Points: Mean = 28.5307, standard deviation = 9.10989
15.6649
14.175
12.9556
11.7608
10.4083
8.50104
A simple rounding give a set of scores 16,14,13,12,10,9, which does amount to a 28 point buy. But rounding .66 down instead of up gives the standard array. Thus the question becomes: how important are those 3 points?
Also, consider a probabilistic approach rather than a statistical one. Of the 1296 possible rolls with 4d6, the probability of each score (3 to 18) is:
3=7.716049382716049E-4,
4=0.0030864197530864196,
5=0.007716049382716049,
6=0.016203703703703703,
7=0.029320987654320986,
8=0.047839506172839504,
9=0.07021604938271606,
10=0.0941358024691358,
11=0.11419753086419752,
12=0.12885802469135801,
13=0.13271604938271606,
14=0.12345679012345678,
15=0.10108024691358025,
16=0.07253086419753087,
17=0.041666666666666664,
18=0.016203703703703703,
(I'm not sure whether these numbers correspond with your test. It's understandable if they don't--after all, computer-generated numbers are not random)
These probabilities have an expected value (the weighted average) of 12.2445987654321. Thus, a character should have an average of 12.24 in every stat. Your array (16,15,14,13,12,10,9) has an average of 12.33, while the elite array has an average of 12.0. So the elite array does follow the expected value, just rounding down (though perhaps too much).
Also, simple math shows that scores of 13,12,12,12,12,12 (in points 5+4+4+4+4+4=25) gives an average of 12.16, while scores of 13,13,12,12,12,12 (in points 5+5+4+4+4+4=26) gives an average of 12.33. In other words, simply looking at the average cases, the system should use either a 25 (or maybe 26) point buy.
But part of the problem with a probabilistic analysis is that it is hard to account for the sudden increase of stats costing 1 point apiece to costing 2 points apiece (changing at 15). Thus perhaps to allow the 25-point buy to continue to work at non-average values would require making a 16 cost 8 points rather than 10.
Thus any problem seems to be not in the number of points required for the point-buy, but in the spread of the point costs. Though I will admit, an elite array of 16,14,13,12,10,8 would have fit more with the average.
My conclusion: using the 25point buy is in fact balanced with using the 4d6 rolling method, though rounded down (perhaps the floor function of balanced?). There is nothing wrong with Wizards simply rounding down on rolls.
* Note: Allowing rerolls increases the power level of the game--"average" characters are no longer allowed (since the average character doesn't have a score greater than 13). Thus, allowing rerolls is equivalent to increasing the value of the point buy system--apparently to 29.
Rolling four six-sided dice has 1,296 possible outcomes, and these outcomes fall on a nice symmetrical distribution. If you drop the lowest die from the four, you affect the mean, standard deviation, kurtosis, and skew of that distribution. For all 1,296 outcomes, each minus the lowest die for each 4d6 outcome, the distribution has these statistics:
mean = 12.24, stdev = 2.85, kurtosis = -0.34, skew = -0.28
A "typical" array of 6 abilities, then, would have similar statistics in terms of mean, standard deviation, kurtosis, and skew. You can use a Monte Carlo technique or a nonlinear solver to find the closest possible integer-value match, which is:
16, 14, 13, 12, 10, 8
(mean = 12.17, stdev = 2.86, kurtosis = -0.47, skew = -0.25)
That's a +6 in total modifiers. It also corresponds to a 27-point buy.
The next closest integer-value match is:
16, 14, 14, 11, 11, 8
(mean = 12.33, stdev = 2.88, kurtosis = -0.56, skew = -0.35)
That's also +6 in total modifiers, corresponding to a 28-point buy.
I'll point out that the analysis above comes out slightly different than "typical" array I derived elsewhere in this forum by analyzing histograms of each ability score in a +6 array from lowest to highest:
16, 14, 13, 12, 11, 9
That's also a +6 array, but a 29-point buy. I like this array, actually. It's challenging to design a character around it, but the character you get is quite playable.
The point here is, no matter how I calculate it, it looks to me like the DMG's 25-point buy is unbalanced to the low side. It's a less-than-average array. So one has to ask, is a 25-point buy appropriate for characters meant to be heroes in a game designed around a 4d6-drop-lowest method to generate abilities?
-Alex
It is neat to see that the standard rolling for stats is actually statistically equivalent (even superior) to the point buy. For some reason I feel like D&D stats are meant to be rolled. Point buy is for stuff like white wolf and hero system, but then that’s me.
I see we have some math majors in the room... ;) Curious - what’s the spread on the "organic" method (strait roll 3d6 no drops, no rerolls)?
Straight 3d6 has the following stats:
mean: 10.5 (exactly)
stdev: 2.96
kurtosis: -0.40
skew: 0 (it's a perfectly symmetric distribution)
The best match of six abilities to these stats requires the lowest ability to be 6, which you can't get with a point buy that has a minimum of 8. The array is 15, 12, 12, 9, 9, 6. There isn't a good match if you constrain the minimum ability to be 8. The best I can find is 15, 12, 12, 11, 8, 8 -- which corresponds to a 19-point buy.
How about the roll 4d6 7 times (drop lowest) reroll 1s and 2s (aka the superhero) method?
That's a lot harder to figure out, because you don't have an easily-determined of total possible outcomes if you re-roll selected values. At the moment I can't think of an elegant way to do it.
I can say that if you do a 4d6-drop-lowest, 7 times, and drop the lowest ability, then it's easy to calculate a "typical" array. You just solve for the array of seven abilities that best fits the 4d6-drop-lowest stats I posted earlier in this thread, then drop the lowest of the seven. The typical profile is 16, 15, 13, 12, 12, 10, corresponding to a 33-point buy.
-Alex
I grabbed my dice and threw.
Allow me to explain why. Our current DM and the DM that is to follow use the point buy system. The current one using the 32 point buy. I personally do not like point buy systems because I like to roll the dice. Point buy deprives a player of the opportunity to get great scores at the same time negating the chance for crappy scores. I hate trading down. There are very few things in D&D that are a sure thing. Most things that are not worth much.
My results in 9 sets of ability scores using the PHB method and converting it to a point buy turned out like this.
I had 7 out of nine sets beat come in between a 33point buy equivalent and a 49 point buy. Most falling between 36 and 40.
The other two results came out so low as to be discarded by the guidelines given in the PHB.
Even considering those two terrible sets 8 and 16 respectively the average was 33.
If those to are discarded as they should be the average jumps to 39 points.
I will be trying to convince the up and coming DM the dice are the way to go. :D
I disagree with the claim that 25 pt buy is therefore too low. This is my understanding of the logic for that claim:
- Standard rolling rules generate scores of about 29 pts.
- 25 pt buy generates scores below 29 pts.
- 25 pt buy is therefore substandard.
I contend that the prior logic is no different from the following:
- Standard rolling rules generate scores of about 29 pts.
- 29 pt buy generates scores of 29 pts.
- 29 pt buy is therefore as good as standard rolling rules.
Clearly false. Of course it is better to choose how to spend your 29 points than have them randomly assigned.
The ability to decide whether you have an 18 or not is worth something. The ability to choose the number of dump stats you have is worth something. No form of random score generation can provide these abilities.
The only thing one can really prove in looking at random distribution point values is that 30 pt buy is better than standard rolling. You can't claim 25 pt buy is underpowered without taking into account the value of cherry picking hi/low stats.
The game is designed around 25 pt buy. 25 pt buy keeps all the CRs appropriate challenges for well-made characters. That is the true test of whether 25 pt buy is appropriate.
How about the roll 4d6 7 times (drop lowest) reroll 1s and 2s (aka the superhero) method?
re-roll 1s and 2s simply turns the d6 into a d4+2, giving:
mean: 14.6171875
stdev: 1.891658029
kirtosis: -0.342627475
skew: -0.316302666
I don’t know how to get an integer best fit array with that, how do you do it, amatulic?
Example: I flip a coin with a slight bias towards heads. If I flip it a million times, the most likely outcome is all heads.
There’s a smaller chance of rolling straight 3s. Specifically that you roll four 1s. This gives us one possible way to roll a 3 compared to 24 for an 18 *due to the drop the lowest rule*.
the probability of each score (3 to 18) is:
3=7.716049382716049E-4,
4=0.0030864197530864196,
5=0.007716049382716049,
6=0.016203703703703703,
7=0.029320987654320986,
8=0.047839506172839504,
9=0.07021604938271606,
10=0.0941358024691358,
11=0.11419753086419752,
12=0.12885802469135801,
13=0.13271604938271606,
14=0.12345679012345678,
15=0.10108024691358025,
16=0.07253086419753087,
17=0.041666666666666664,
18=0.016203703703703703,
Why is 18 harder to get than a 3 on this chart here?
I am no hardcore programmer so forgive me if this is stupid, but couldn't you drop the lowest die roll in an analysis just by doing it one at a time? I know that with xml editing you can just script in a sort by numerical order, and you can choose to analyse only six cells.
I don't know how you are doing the analysis, but if it anything like xml stuff then (writing this out so it is easier to see)
Store random integer between 3-6 in cells D1, D2, D3, D4, D5, D6, D7
Sort in descending order D1, D2, D3, D4, D5, D6, D7
Move D1 to list X1
Move D2 to list X2
Move D3 to list X3
Move D4 to list X4
Move D5 to list X5
Move D6 to list X6
and then loop it.
Once you have everything in your lists, run whatever you see fit to analyse the stuff.
4d6 rerolling 1's and 2's dropping lowest roll (aka Superhero)
Treat it as rolling 4d4's drop lowest and add 6 to find the total.
Lowest possible total is 9 (3,3,3,3)
The distribution is the same as 3d4 skewed right (because of the extra d4) then the whole curve is shifted 6 units to the right.
Rolling 7 sets and dropping lowest results is more skewing to the right.
I have never really understood the attraction to a point-buy system. Perhaps one of you could tell me what you think the benefits are. I like to look at all sides of something, so let me also say what I think about it.
While it is true that DMs can get quite a headache dealing with players that aren't satisfied with a character's stats that they have rolled, I prefer one of the rolling methods. The inherent weakness that I see with a point-buy system is that if a character wants a very strong attribute, they will need to have a weakness in another attribute. Now while that seems all nice and balanced, there are people in the world who have been born with a silver spoon in theirs mouths, so to speak, and who have no attribute problems at all, and there are people who are truly unfortunate in there base stats (both make for good characters). But the point-buy system seems to exclude these possibilities. I don't use a point-buy system with my players because I think my players would feel limited by it (not numerically limited), but conceptually limited. What do you all think?
My rule for character generation is this: I allow my players to roll 3 unique sets of attributes, which I have found gives them a very good chance to get numbers they are at content with. A player then chooses one of the sets, and without modifying the numbers, chooses which roll goes to which attribute (high roll to int for a mage, for example). Then, after the entire character process is complete, I let each player roll 1d6 (1 for strength, 2 for dex, etc.). For whichever stat they rolled, they add 1d6 to that attribute, to a max of 20.
It may seem strange or overpowered, but what I have found is that it is actually very nice. If a player's main attribute gets rolled, it usually only increases a small amount, since it's already usually ~16, so the increase is nice, but not broken. But what I enjoy about it is that it allows players to have charismatic wizards, intelligent fighters, strong rogues, etc., which adds an nice depth to roleplaying, while not limiting the main attributes of a class. What usually comes out of the process is a character that someone would look at and say "Pretty nice stats...ah, a weakness here....wait, this is a fighter? Interesting."
I'm pretty sure this wouldn't work for some DMs, but since I like to make gold and magic items more difficult to attain than many DMs, slightly higher stats are quite useful in surviving the increasingly difficult monsters that Wizards is adding to the game.
Anyway, to ask again, why do you like a point-buy system?
~Falidamari
Cookies for all those that actually took the time to run the numbers and share them with others.
I've think it's pretty well known that rolling for stats is more powerful than point buy. And in a game where players roll for their stats it's generally accepted that your playing in a "more powerful than the average commoner" setting. As such, the DM may allow stats to be rerolled that exceed the PHB guidelines for rerolling stats, in order to keep a low roller in line with the player who rolled two 18s and nothing else below a 12.
However, when this is not the case and the player is given the choice of rolling for his stats(with no option of rerolling except as specified by the PHB) or using a point-buy system, many players will take advantage of the inherent certainty of point-buy for fear of getting no exceptionally high rolls. For example, six 14s would be 36 points on point-buy, and slightly above average overall for rolled stats, but be disastrous for the player. Players thrive on playing to their strengths and burying their weaknesses, but you can't do that very well when all you have is mediocrity. And only a point-buy system can insure that you have both strengths and weaknesses. Dice are more than capable of delivering both extremes, but it is in their nature to seek the average ground.
Look at the end: e-4
Actually, according to Matlab, the most likely set of rolls is all thirteens. When you roll four dice and drop the lowest, the most likely result is a 13. If you have a slight bias towards one thing (Say, heads) and you do something where that is one of the options (flipping a coin) a lot of times, the most likely outcome is all things you have a bias towards (all heads). The dice have a slight bias towards 13, so the most likely score is 13, 13, 13, 13, 13, 13.
Example: I flip a coin with a slight bias towards heads. If I flip it a million times, the most likely outcome is all heads.
Only if you care about order. All 13s is more likely than, say, a 12 followed by 5 13s in that order, but it's not as likely as a 12 and 5 13s in any order.
Now, maybe I slept through my statistics course but I’m a little confused. Now on a 4d6 meth there are a number of ways to roll an 18, being 3 6's and one other number, giving us a total of 24 different ways to roll an 18.
There’s a smaller chance of rolling straight 3s. Specifically that you roll four 1s. This gives us one possible way to roll a 3 compared to 24 for an 18 *due to the drop the lowest rule*.
Why is 18 harder to get than a 3 on this chart here?
The E-4 at the end means you multiply the number by .0001, or move the decimal over four spaces, making the probability of a 3 0.0007716, significantly less than the .01620370 of the 18. On a side note, I believe there are only 21 chances that you get an 18, as you can only count straight sixes once.
I've been reading through all this math, and it's quite interesting, as I too like to do numerical analyses for different types of stat generation, but I have a more basic question for point-buy fans.
I have never really understood the attraction to a point-buy system. Perhaps one of you could tell me what you think the benefits are. I like to look at all sides of something, so let me also say what I think about it.
While it is true that DMs can get quite a headache dealing with players that aren't satisfied with a character's stats that they have rolled, I prefer one of the rolling methods. The inherent weakness that I see with a point-buy system is that if a character wants a very strong attribute, they will need to have a weakness in another attribute. Now while that seems all nice and balanced, there are people in the world who have been born with a silver spoon in theirs mouths, so to speak, and who have no attribute problems at all, and there are people who are truly unfortunate in there base stats (both make for good characters). But the point-buy system seems to exclude these possibilities. I don't use a point-buy system with my players because I think my players would feel limited by it (not numerically limited), but conceptually limited. What do you all think?
I hope this won't be a thread hijack and detract from the statistical analysis.
What you see as an inherent weakness of the system is what I see as the advantage. Yes, people are born totally above or below average in real life. However, D&D is not real life. D&D is a game, and it's generally good for those playing a game to be on relatively equal footing. The point buy system solves the "problem" of one player being much more powerful than the others because of lucky rolls. Of course, if you don't see this as a problem, then the point buy system doesn't give you much. I think most arguments come down to whether you see a potential unbalance through ability scores as a problem. The point buy system takes out the randomness of ability rolling to avoid any flukes.
The one other aspect I really like about the point buy system is how it centers the ability scores. I've often found the mentality among players that 16 isn't really that high an ability score, and no character can compete without an 18 in their primary stat. I disagree with this. With the elite array (the point buy system), you get two scores at +2 (a 15 and a 14). These scores are well above average, and should be seen as such. I like that the 25pt buy makes a 16 really quite high, and an 18 almost impossible to have.
So your fighter with an 18 STR in all likelihood really is the strongest guy in the kingdom, not simply on par with other adventurers. If you want to play charismatic wizards, intelligent fighters, or strong rogues, you just put the 15 in your primary stat and the 14 in your secondary stat. If the bard has a 15 CHA and the wizard has a 14 CHA, then you've got a charismatic wizard who can hold his own with the party's face. Sure, he might not be AS charismatic (and the difference may increase as you gain levels), but he's close. And even if you put a 13 or a 12 in that other stat, you're still above average in that area, and not too far below where the other guy may be (imagine the wizard with a 13 CHA. Not too shabby). I like this distribution a lot, and it's my primary reason for using the point buy system.
I have some questions:
1: Has anyone looked into this scenario: Roll 4 dice, then have the option to reroll 1 of them, then drop the lowest. I'd be interested to see how to calculate this...
2: Given the choice between a 25 point buy and the official 4d6 method, why does it appear that you are misunderstanding the spirit of the 25 point buy: to make sure that you don't get a character that is below-average (by hero standards). By rolling, there is about a 35% chance of rolling stats that would be below 25 points...
3: What's kurtosis?
3d6 Flat 3d6 Accumulated
03 00.46% 03- 00.46%
04 01.39% 04- 01.85%
05 02.78% 05- 04.63%
06 04.63% 06- 09.26%
07 06.94% 07- 16.20%
08 09.73% 08- 25.93%
09 11.57% 09- 37.50%
10 12.50% 10- 50.00%
11 12.50% 11- 62.50%
12 11.57% 12- 74.07%
13 09.73% 13- 83.80%
14 06.94% 14- 90.74%
15 04.63% 15- 95.37%
16 02.78% 16- 98.15%
17 01.39% 17- 99.54%
18 00.46% 18- 100.00%
I had a DM who insisted that rolling 4, rerolling the lowest and dropping the new lowest was waaaaay different that rolling 5, dropping the 2 lowest...
Roll 4
subtract one
Roll 1
subtract one
Roll 5
subtract two...
The same !! >_<
Weird control freak he was.
Anyways, useful thread.
I use a point buy system which is above the average, so my players don't have to ditch stats to be playable or fit their absolute need of a high charisma for an almost non speaking character (usually dressed in dark, with a hood, and, y'know, looking mysterious...), and also are all equal.
I've come to dislike randomness in character generation, because I hate to see someone come with a pleasing character concept they can't play because of low rolls (for those multiple ability dependant characters like a paladin or a skillful rogue).
3 = -2
6 = -1
8 = 0
9 = 1
10 = 2
11 = 3
12 = 4
13 = 5
14 = 6
15 = 8
16 = 10
17 = 13
18 = 16
This has worked well: rather than paying more for heavily reducing a stat, you get less in return (to discourage min-maxing).
When I added these into the equation (multiplying these values by the number of times that result comes up on 4d6-drop the lowest), I achieved this result for an average point-buy:
28.84
So, rounding down to 28 seems fine, and I'm perfectly content with using that as my new standard.
I'm unsure what discarding bad arrays do as the PHB orders, but I have a sneaking suspicion that it is close to 32 points.
I then sorted these values in ascending and in descending order, on the basis that if you take a score "near the top", it ought to be balanced with a score "equally near the bottom". This gave me a set of 1269 pairs of values. I used a cross-tabulation to count the number of occurrences of each pair.