D&D General Deck-based Ability Score Generation

[Yaarel]

Assuming six ability scores, consider the following point-buy numbers.



Score (Cost)

6 (−4 points)
7 (−3 points)
8 (−2 points)
9 (−1 point)

10 (0 points)
11 (1 point)

12 (2 points)
13 (3 points)

14 (6 points)
15 (9 points)

16 (18 points)
17 (27 points)

18 (54 points)
19 (81 points)

20 (162 points)

 

[clearstream]

It is a matter of taste. Each table will probably evolve its own preference, or multiple preferences depending on the setting concepts.

For me, I like numbers that are low but non-negative. So, +0 to +3 is ideal. But a +4 or even a −1 can add flavor.

I like something in the range perhaps −2 to +3 at 1st level, leaving scope for ASIs as characters advance.

It looks like the new format will have all player characters add a +2 and +1, or else three +1s, on top of whatever scores get generated.

But this deck method could tailor to numbers to already be equivalent to this improvement step, and not do it.

I like giving players that bit of control, on top of the deck. That's why I tend to tune my decks slightly low - to make space for those three points.

 

[Stalker0]

Hehe I was trying to calculate the various number probabilities with just some brute force tactics (aka just generate a giant list of random number combinations and see what shakes out).

In my head I was like "Combination of 20,18 isn't that large"..... but then I realized because the timing of the numbers matters for what gets added into what number....its more like "Permutation of 20,18". And that is a VERY LARGE number!

Since cards are deterministic unlike dice, and the timing of the cards is important, calculating the probabilities is much more difficult. It actually would make for a very solid advanced probability question.

Best I could do was determine the chance of getting a 6 with this array:

[2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6]

was .5%

 

[clearstream]

Hehe I was trying to calculate the various number probabilities with just some brute force tactics (aka just generate a giant list of random number combinations and see what shakes out).

In my head I was like "Combination of 20,18 isn't that large"..... but then I realized because the timing of the numbers matters for what gets added into what number....its more like "Permutation of 20,18". And that is a VERY LARGE number!

Since cards are deterministic unlike dice, and the timing of the cards is important, calculating the probabilities is much more difficult. It actually would make for a very solid advanced probability question.

Best I could do was determine the chance of getting a 6 with this array:

[2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6]

was .5%

The distributions are indeed hard to analyse! One way I use is just to Monte Carlo them. A friend wrote a deck shuffling plugin for me, for Excel. So I can create a deck, clone it many times, and apply some basic statistical analysis to get a sense of how it performs.

Another approach is to focus on combinations rather than permutations, to get a sense of how many ways there will be to get each value. Say we are drawing pairs - we don't care about the order of the cards in the pair - so we don't need to worry about all possible permutations because (for example) 5+6 will be identical to 6+5 for our purposes. If you allow assign-as-desired, the ordering of the pairs in the deck is also irrelevant, further simplifying, because in that case (for example) 12, 17, 7 is the same as 7, 12, 17 etc.

I've found I like 12-card arrays with relatively low average values. Like this one - 8, 7, 7, 7, 6, 6, 5, 5, 4, 4, 4, 3 - which gives swingy scores from 7-15 summing to 66. Average 11.5 after TCoE ASIs. Final modifiers generally sum to +3 to +4. In my campaign, scores are allocated as drawn (and I am not concerned about 'surprise' - or rather I think of that as a feature of the whole array, and not of a given score or of total point value variance).

 

[Stalker0]

The distributions are indeed hard to analyse! One way I use is just to Monte Carlo them. A friend wrote a deck shuffling plugin for me, for Excel. So I can create a deck, clone it many times, and apply some basic statistical analysis to get a sense of how it performs.

Another approach is to focus on combinations rather than permutations, to get a sense of how many ways there will be to get each value. Say we are drawing pairs - we don't care about the order of the cards in the pair - so we don't need to worry about all possible permutations because (for example) 5+6 will be identical to 6+5 for our purposes. If you allow assign-as-desired, the ordering of the pairs in the deck is also irrelevant, further simplifying, because in that case (for example) 12, 17, 7 is the same as 7, 12, 17 etc.

I've found I like 12-card arrays with relatively low average values. Like this one - 8, 7, 7, 7, 6, 6, 5, 5, 4, 4, 4, 3 - which gives swingy scores from 7-15 summing to 66. Average 11.5 after TCoE ASIs. Final modifiers generally sum to +3 to +4. In my campaign, scores are allocated as drawn (and I am not concerned about 'surprise' - or rather I think of that as a feature of the whole array, and not of a given score or of total point value variance).

Except the pair model isn’t quite sufficient. For example, a classic problem is “draw 6 cards, what is the possibility of a pair?”

that doesn’t work for this example because the order of the cards is particularly important. A 2,2,3 in the first 3 cards generates a 7, but a 2,2,X,3 (two 2s, a different card, then the 4th card a 3) generates different numbers. Cards are spacially connected to each other, so I’m not sure how to calculate that, gnat probability math seems way over my head

 

[jgsugden]

I get the idea, but in general it seems like a loss of control over a point buy system. You get roughly the same range but give people less control.

In my experience, whether a PC has straight 18s or no score greater than 14, the PC works fine. The relative difference in power is noticeable, but not game breaking, and a DM that accounts for it with strategic introduction of magic items or storyline benefits can balance it out without difficulty. To that end, I settled on my system and have not looked back:

Each player rolls 3d6 6 times. For each trio of rolls, they replace the lowest number with a 4. This gives a range of 6 to 16 (with only one in 36 PCs getting that 16). If they accept them 'in order' I give them a minor boon. If they change the order, no boon. A minor boon is a weak feat, a minor magic item, an ally, etc... If they're unhappy, they can use point buy.

I've been very happy with it since implementing it.

 

[clearstream]

I get the idea, but in general it seems like a loss of control over a point buy system. You get roughly the same range but give people less control.

That's right. It's not intended to replace point-buy, which is a good method for when a group wants 100% control. Rather it is intended to be an alternative to random methods like rolling dice.

In my experience, whether a PC has straight 18s or no score greater than 14, the PC works fine. The relative difference in power is noticeable, but not game breaking, and a DM that accounts for it with strategic introduction of magic items or storyline benefits can balance it out without difficulty. To that end, I settled on my system and have not looked back:

Each player rolls 3d6 6 times. For each trio of rolls, they replace the lowest number with a 4. This gives a range of 6 to 16 (with only one in 36 PCs getting that 16). If they accept them 'in order' I give them a minor boon. If they change the order, no boon. A minor boon is a weak feat, a minor magic item, an ally, etc... If they're unhappy, they can use point buy.

I've been very happy with it since implementing it.

Part of the problem-definition for the deck-generation solution, is overshadowing. The method is designed to put characters on a mechanically fair footing. It's also simple in application, so that it levels the playing field for system mastery.

 

[payn]

I like, but ultimately, it just seems like extra work to get to a stat array. This is where all my stat gen travels take me. I may try this next time I'm doing a Traveller game though.

 

[clearstream]

Except the pair model isn’t quite sufficient. For example, a classic problem is “draw 6 cards, what is the possibility of a pair?”

that doesn’t work for this example because the order of the cards is particularly important. A 2,2,3 in the first 3 cards generates a 7, but a 2,2,X,3 (two 2s, a different card, then the 4th card a 3) generates different numbers. Cards are spacially connected to each other, so I’m not sure how to calculate that, gnat probability math seems way over my head

I mean more this situation. Picture a simplified deck of six cards - three 3s and three 4s - from which we'll draw and sum two cards for each score. (I referred to those two summed cards as a 'pair'.)

Say the deck look like this after shuffling - 3, 3, 4, 3, 4, 4 - so that our scores would be 6, 7, 8. For our purposes, that is identical to a deck that looks like this after shuffling - 3, 3, 3, 4, 4, 4 - because that will also make our scores 6, 7, 8. It doesn't matter what order the middle pair are drawn in. It also doesn't matter which 3 and which 4 because for our purposes 3s are all the same, and 4s are all the same.

So long as we're agnostic as to where each score is allocated - so 8, 6, 7 is as good as 6, 7, 8 - say because we are allocating-as-desired (or are happy with anything we get), then - 4, 3, 4, 4, 3, 3 - is also identical to the above shuffled decks (for our purposes).

If precise card order mattered, we'd have to worry about 720 permutations. Given that the ordering within pairs doesn't matter, we can worry about just 30 permutations. And given that the ordering of pairs doesn't matter, we can worry about just 15 combinations. At least, that is my understanding.

Does that sound right? Or have I gone wrong somewhere?

 

[Stalker0]

I mean more this situation. Picture a simplified deck of six cards - three 3s and three 4s - from which we'll draw and sum two cards for each score. (I referred to those two summed cards as a 'pair'.)

Say the deck look like this after shuffling - 3, 3, 4, 3, 4, 4 - so that our scores would be 6, 7, 8. For our purposes, that is identical to a deck that looks like this after shuffling - 3, 3, 3, 4, 4, 4 - because that will also make our scores 6, 7, 8. It doesn't matter what order the middle pair are drawn in. It also doesn't matter which 3 and which 4 because for our purposes 3s are all the same, and 4s are all the same.

So long as we're agnostic as to where each score is allocated - so 8, 6, 7 is as good as 6, 7, 8 - say because we are allocating-as-desired (or are happy with anything we get), then - 4, 3, 4, 4, 3, 3 - is also identical to the above shuffled decks (for our purposes).

If precise card order mattered, we'd have to worry about 720 permutations. Given that the ordering within pairs doesn't matter, we can worry about just 30 permutations. And given that the ordering of pairs doesn't matter, we can worry about just 15 combinations. At least, that is my understanding.

Does that sound right? Or have I gone wrong somewhere?

Unfortunately no that's not correct.

Here is another way we could arrange those same cards:

33 43 44 (6 7 8)
34 34 34 (7 7 7)

That's a permutation not a combination, as order does matter. Now you are correct that within our blocks order doesn't matter, so the theoretical number of permutation is smaller, but it still grows very very quickly. And we never really get back down to "combination level"

As another example, just take 3 and 4s but now its 12 draws (2 draws per stat x 6 stats)

33 33 33 44 44 44 (6 6 6 8 8 8)
34 34 33 33 44 44 (7 7 6 6 8 8)

Same exact cards just drawn in a slightly different order produces a different stat block. So yes unfortunately the position of the card draws is quite important to what numbers you get. And this problem gets progressively worse when you look at 3 draws per stat and more than 2 types of cards. But even just a 20 card deck where I draw 12 cards from (aka 2 cards x 6 stats), is a 10 to the power of 13 number in terms of permutations (aka 10 trillion)..... far more than most programs can easily calculate (for reference excel only has 17 billion cells, you would need to utilize 588 worksheets all completely filled to the brim to display all of the possibilities). Even if I can cut that by 4 or 6 or heck even 10 by noting certain permutations are "the same".....I still have a very complex problem to solve.