Method to roll ability scores to statistically match various Point Buy levels

[Mr. Grogg]

I rolled several million characters up tonight thanks to Excel, and learned something useful along the way that I know others may be looking for. :)

I have players who really like to roll for ability scores instead of point-buying. I have no problem with that per se, as long as I see them do it. (I actually prefer rolling dice myself - point buy creates min/maxed arrays. As long as the DM witnesses the rolls, I think it's a good thing to introduce more randomness.)

The problem is that Paizo's AP's are all balanced for a 15-point buy, and there was no obvious way to statistically mirror that with dice rolls. Google failed me.

So I thought for a bit.

Rolling 4d6 and dropping the lowest six times is known to be pretty close to a 20-point buy.

Rolling 3d6 six times is known to give very low power characters which will be well-below the 15-point buy on average.

So then I thought, what if I blend it by having them roll 4d6 dropping the lowest 5 times, and rolling 3d6 once? What number of points would that be equal to on average? (Spoiler alert: About 18.6. And we'll get it closer to 15 in a moment.)

So I made a spreadsheet that rolled 100,000 stat blocks at a time using 4d6 drop the lowest six times, discarded any block with any stat of less than 7 (since Pathfinder's point-buy doesn't allow it, I couldn't compare those characters directly -- and I'm a nice enough guy I'd let them re-roll a six or worse anyway), calculated the points it would take to buy each character, and then spit out an average point cost for the 100,000 stat blocks.

This revealed that the average for 4d6 drop the lowest six times is 20.7 points. That's mostly just a confirmation of what was already known.

Then I zeroed one of the four dice on one ability score on all 100,000 blocks. This made it the same as rolling 4d6 dropping the lowest 5 times, and rolling 3d6 once. The average for that turned out to be 18.6 points. Still a little higher than my target 15.

Then I zeroed a die on another ability score and recorded the average. And again. And again.

Then I went the other way: One of the methods described in the core book is rolling 2d6+6 six times for truly heroic characters. So I did that and found the average to be a whopping 41.9 point value!

Then I replaced one 2d6+6 stat at a time with 4d6 drop the lowest.

So, several million stat blocks later, I give you the equivalencies for blended rolling methods. Anytime you see 4d6, that is shorthand for 4d6 drop the lowest. Also remember that it discards any stat block containing one or more ability scores of 6 or less.

2d6+6 six times = 41.9 point value
2d6+6 five times, 4d6 once = 38.4 point value
2d6+6 four times, 4d6 twice = 34.9 point value
2d6+6 three times, 4d6 3 times = 31.3 point value
2d6+6 two times, 4d6 4 times = 27.8 point value
2d6+6 one time, 4d6 5 times = 24.2 point value
4d6 6 times = 20.7 point value
4d6 5 times, 3d6 once = 18.6 point value
4d6 4 times, 3d6 twice = 16.2 point value
4d6 3 times, 3d6 3 times = 14.6 point value
4d6 twice, 3d6 4 times = 12.4 point value
4d6 once, 3d6 5 times = 10.4 point value
3d6 six times = 8.4 point value

So, if you want to run an AP with a 15-point buy but your players are addicted to rolling their scores, you can be just slightly strict and have them roll 4d6 drop the lowest three times and 3d6 three times and, on average, it will be the same as spending 14.6 point. Or you can be slightly more generous and let them roll 4d6 drop the lowest four times and 3d6 twice and it will be statistically the same as spending 16.2 points.

I tried fine tuning towards 15 points in a number of ways, but they got so kludgy that it wasn't worth it.

I think the sleekest and most natural-sounding way to go if you want 15 point equivalency is to tell your players to roll half their scores with 3d6 and half with 4d6 drop the lowest.

Hopefully somebody will find the above equivalency list useful.

Game on!
Mr.Grogg

[Coridan]

I don't think this will work in practice as well as it does over millions of rolls. We use 2d6+6 and generally I would prefer 20 point buy over it.

[Mathius]

It is nice to know where these equivalences lie but I think I would prefer to find a system that will on average give point buy x and 90-95 percent of the rolls will be with in 3 points of point by x. With random dice comes major differences in player power and that can spoil fun.

[cnetarian]

Unfortunately the game system doesn't really work well with this conversion. Having an odd ability score is not much better than having an even score one point less (a 12 instead of a 13), there are some feats which use odd ability score breakpoints but almost everything else is based on even ability score breakpoints. Spells, magic items, class bonuses etc have ability scores increase in increments of two points to reflect this reality - wish type and leveling permanent ability increases excepted. Compare a 42 point buy character with all 15s versus a 30 point character with all 14s and they are functionally almost identical. Point buy lets you allocate ability scores to meet the breakpoints without waste. (my last rolled character had 6 odd attribute scores, at level 16 through use of two wish spells I managed to get them all to be even.)

---edit---
to clarify most of the functionality of ability scores comes from the equation:

(score - 10) / 2

this strongly favors even ability scores.

[Knight Magenta]

I think a simple way of making odd scores useful is to allow odd value enhancment bonus items

[Orfamay Quest]

Also, much of the utility of point buy is not in the mean but the variance; as a stupid example, flip a coin for each attribute. If it's heads, the attribute is 10, if it's tails, the attribute is 16. On average, you get five points per attribute per flip, for a 30 point system.

I don't think I know anyone who would prefer this system to 30 point buy.

Even worse, flip a single coin and you're either all 10s or all 16s. Still on average 30 point buy.

[EWHM]

Rolled attributes are normally effectively lower than their point buy equivalent because they aren't usually very optimized. There's the odd attribute issue mentioned before (although most prereqs for feats are at odd attributes, especially 13), but also the issue that rolled characters are typically more well rounded than point buy characters.

For instance, 16,16,14,12,12,10
That's 29 points. How many players given the choice of that template would take it over a 25 point buy? Hell, a lot of players making SAD characters who are highly optimized would take a 20 point buy over it.

[cnetarian]

Knight Magenta wrote:

I think a simple way of making odd scores useful is to allow odd value enhancment bonus items

Could be, but what I intended was that the calculation of equivalent rolls to point buy needed to take this in to account. Instead of using the book chart for point buy, use something like this:

• 7 ..... -4
  
• 8 ..... -1.2
  
• 9 ..... -1
  
• 10 ..... 0
  
• 11 ..... 0.2
  
• 12 ..... 2
  
• 13 ..... 2.3
  
• 14 ..... 5
  
• 15 ..... 6
  
• 16 ..... 10
  
• 17 ..... 12
  
• 18 ..... 17

This is only a rough guideline, but if applied to a rolled array should produce a point-buy equivalence value closer to that of actual utility of the attributes.

[Kydeem de'Morcaine]

This is interesting information and I appreciate you providing it for those times when people insist on rolling scores.

I personally don’t like rolling scores. The averages are not a problem. The outliers that are still within the range of possibilities are the problem.

Almost every time I have been in a group that rolled abilities, I see one or both of the following situations.
One PC has abilities amazingly higher than everyone else. Now there is a tendency to for everyone else to feel like the sidekick to the true hero.
One PC has abilities amazingly lower than everyone else. Now that player has a tendency to feel like the younger brother tagging along with the older kids who is only tolerated because mom made you take him.

For a short couple sessions, a one shot module, or a campaign designed around a hero (or younger brother) it can still work out. And sometimes it is fun to see if I can keep up with the big boys even with my poor stats. But not everyone enjoys that. At least not all the time. With our group’s scheduling, an AP can easily last a couple years. It can get very old when I have to spend the next 2 years struggling to survive that which the rest of the group finds easy. And every time the group find something challenging, I need raised from the dead. Again.

As an example. A new group I joined rolled using the 2d6+6 method. One of the PC’s has close to 45 point buy equivalent. My PC has only a 7 point buy equivalent! Now I am still going to try and make a go of it and I picked as close as you can get to a NAD (No Attribute Dependant) class as you can get. The summoner. From some of what has been said, I think my system mastery is a bit higher than several of the other players and the GM let me beg the CR+1 race, fetchling. But it will still likely be tough to survive.
And what if it had randomed out the other way? What if the guy with low system mastery had gotten the 7 and I had the 45? Would he have had fun playing in the same group as my PC in that case? I don’t know for sure, but I suspect not.

That is why I prefer point buy, so everyone is at the same ground zero starting point.

[Majuba]

Mr. Grogg - nice job! I did something similar a while back, though slightly less robust than yours.

I like how you accounted for the effective 'rerolls' on sets or scores of 6 or less - since that effectively raises the point-buy of the rolls.

Did you make any adjustments for other stat sets that you would allow a reroll on? (I'm thinking of the old 3.5 method which allowed rerolls when the total ability score modifiers were less than +2)

[Malith]

Interesting information. Im going to have to come back to this information later.

[mamaursula]

Mr. Grogg wrote:

I rolled several million characters up tonight thanks to Excel, and learned something useful along the way that I know others may be looking for. :)

I have players who really like to roll for ability scores instead of point-buying. I have no problem with that per se, as long as I see them do it. (I actually prefer rolling dice myself - point buy creates min/maxed arrays. As long as the DM witnesses the rolls, I think it's a good thing to introduce more randomness.)

The problem is that Paizo's AP's are all balanced for a 15-point buy, and there was no obvious way to statistically mirror that with dice rolls. Google failed me.

So I thought for a bit.

Rolling 4d6 and dropping the lowest six times is known to be pretty close to a 20-point buy.

Rolling 3d6 six times is known to give very low power characters which will be well-below the 15-point buy on average.

....
Then I zeroed a die on another ability score and...

I am a huge fan of the Point Buy system, primarily because all of my dice think they're sooooo clever rolling low and making me cry. We have used PB at our table for years as a result. PB can lead to Min-Maxing, IF the DM let's it happen. If you don't want it, don't allow it, it's pretty straight forward.

Thank you for running the numbers though, I love the stats. It's nice to know that you can approximate level starting stats for your players and continue to use dice. You know, if your dice like you.

[xevious573]

Honestly if I wanted stat variance with die rolling I would stick to a point buy system like this:

Standard Fantasy: 8 pts + 2d6 additional pts (10 - 20 pts, Avg 15)
High Fantasy: 13pts + 2d6 additional pts (15 - 25 pts, Avg 20)
Epic Fantasy: 18pts + 2d6 additional pts (20 - 30 pts, Avg 25)

On average each player will get the appropriate number of build points for the level of the game but it does allow for some variance.

[Knight Magenta]

Variance in ability scores is ok if the campaign has a high mortality rate. You just make a new guy and it will all average out.