[7W] Dice mechanics, need sound math advice

[Justin Marx]

Hi there,

I am having some problems with figuring out the probability curves when trying to work this dice mechanic for my game Seven Worlds. This is the basic dice mechanic for the system, and I recently changed it from something far more cumbersome, which however cumbersome, I could work the probability curves. I am toying with this new idea as an alternate, but my statistical probability skills have reached their limit and some sound advice from the math inclined would be really helpful.

Basically, all traits have two ratings - a primary and secondary rating. The primary rank is between 1 and 12, and sets the TN (equal to 24 minus primary rank, so a Primary Rank of 8 gives a TN of 16). The primary rank represents general competency in a field or skill, etc.

The secondary rank runs from 0 to 6. It represents expertise in individual tasks within the general skill - specialities in other words. The secondary rank determines the amount of dice to be used - in this case d12s.

The two ranks are related, although indirectly. In general, the higher one's Secondary Ranks, and the more skills taken within the trait group, the higher the primary rank, but it is not a linear progression. In general however, high Primary means high Secondary ranks.

OK, now all die rolls are made on 2d12, summed and then compared to the TN. However, depending on the secondary rank, a number of dice are rolled as follows:

Rank 0 = roll 4 dice, take the two lowest
Rank 1 = roll 3 dice take the two lowest
Rank 2 = roll 2 dice (normal)
Rank 3 = roll 3 dice take the two highest
Rank 4 = roll 4 dice take the two highest
Rank 5 = roll 5 dice take the two highest
Rank 6 = roll 6 dice take the two highest

I am not sure how to compute the odds on this. I suspect that with a rank of 6, you are getting extremely high results almost all the time. The rationale behind this system is that I want a fate mechanic which is still heavily based upon trait ranks - as in low ranks rarely score high results, while high ranks rarely make mistakes. In play, ranks 5 and 6 are restricted often anyway, for practical purposes 4 is the highest most people will attain (5 and 6 come into play when dealing with divinity), but I still would like to know how to work out a rough probability for this system.

This could have been posted in RPG theory, as a general case when dealing with mechanics of this kind (I have only ever seen it in MechWarrior 2nd edition, where else is a similar mechanic used?), but as I am working on this for my specific game, I posted it here.

So, the questions to avoid ambiguity:
a) How do you figure out the probabilities for this sort of dice mechanic, and
b) Does this dice mechanic work in fulfilling the design goals I listed above?
c) Is a d12 too big a dice for this sort of mechanic? Would a d6 work better? I am using d12's cause I like 'em.

Thanks for any advice in advance,
Justin

[Christoph Boeckle]

Hello

I calculated the probabilities with John Hope's Simple Dicer program. Can't remember where I found it, but you might be able to google it.

The probabilities aren't analytical, but averages after a lot of tries.
I don't know the difference between Mean and Median, and have absolutely no clue what Mode is, but maybe somebody can help us out.

You read the charts by looking up the result in the first column, then the second column is for obtaining the result exactly, next one for rolling under and equal and last one is for rolling over and equal.

Hope that helps!

I can't help you with the other questions for the while being, sorry.

Code Select Expand


Rank 0

4D12, Keep Lowest 2, Summed
Mean: 8.2
Median: 8
Mode: 6 at 10.0%

#  % =    % <=  % >= 
2  3.7%  3.7%  100.0%
3  6.4%  10.1% 96.3%
4  8.3%  18.4% 89.9%
5  9.4%  27.8% 81.6%
6  10.0% 37.8% 72.2%
7  9.8%  47.6% 62.2%
8  9.5%  57.1% 52.4%
9  8.6%  65.8% 42.9%
10 7.7%  73.4% 34.2%
11 6.5%  79.9% 26.6%
12 5.3%  85.2% 20.1%
13 4.2%  89.4% 14.8%
14 3.2%  92.6% 10.6%
15 2.4%  95.0% 7.4% 
16 1.8%  96.8% 5.0% 
17 1.2%  98.1% 3.2% 
18 0.9%  98.9% 1.9% 
19 0.5%  99.4% 1.1% 
20 0.3%  99.7% 0.6% 
21 0.2%  99.9% 0.3% 
22 0.1%  100.0% 0.1% 
23 0.0%  100.0% 0.0% 
24 0.0%  100.0% 0.0% 


Rank 1

3D12, Keep Lowest 2, Summed
Mean: 10.0
Median: 10
Mode: 9 at 8.4%

#  % =    % <=  % >= 
2  2.0%  2.0%  100.0%
3  3.6%  5.6%  98.0%
4  5.1%  10.7% 94.4%
5  6.2%  16.9% 89.3%
6  7.2%  24.1% 83.1%
7  7.8%  32.0% 75.9%
8  8.2%  40.1% 68.0%
9  8.4%  48.5% 59.9%
10 8.2%  56.7% 51.5%
11 7.8%  64.6% 43.3%
12 7.2%  71.8% 35.4%
13 6.2%  78.0% 28.2%
14 5.3%  83.3% 22.0%
15 4.3%  87.6% 16.7%
16 3.5%  91.1% 12.4%
17 2.8%  93.9% 8.9% 
18 2.1%  96.0% 6.1% 
19 1.6%  97.6% 4.0% 
20 1.1%  98.7% 2.4% 
21 0.7%  99.4% 1.3% 
22 0.4%  99.8% 0.6% 
23 0.2%  99.9% 0.2% 
24 0.1%  100.0% 0.1%


Rank 2

2D12, Summed
Mean: 13.0
Median: 13
Mode: 13 at 8.3%

#  % =    % <=  % >= 
2  0.7%  0.7%  100.0%
3  1.4%  2.1%  99.3%
4  2.1%  4.2%  97.9%
5  2.8%  7.0%  95.8%
6  3.4%  10.4% 93.0%
7  4.2%  14.6% 89.6%
8  4.8%  19.4% 85.4%
9  5.5%  25.0% 80.6%
10 6.3%  31.2% 75.0%
11 7.0%  38.2% 68.8%
12 7.7%  45.8% 61.8%
13 8.3%  54.2% 54.2%
14 7.7%  61.8% 45.8%
15 6.9%  68.7% 38.2%
16 6.2%  75.0% 31.3%
17 5.6%  80.5% 25.0%
18 4.9%  85.4% 19.5%
19 4.2%  89.6% 14.6%
20 3.5%  93.0% 10.4%
21 2.8%  95.8% 7.0% 
22 2.1%  97.9% 4.2% 
23 1.4%  99.3% 2.1% 
24 0.7%  100.0% 0.7% 


Rank 3

3D12, Keep Highest 2, Summed
Mean: 16.0
Median: 16
Mode: 17 at 8.3%

#  % =    % <=  % >= 
2  0.1%  0.1%  100.0%
3  0.2%  0.2%  99.9%
4  0.4%  0.6%  99.8%
5  0.7%  1.3%  99.4%
6  1.1%  2.4%  98.7%
7  1.6%  4.0%  97.6%
8  2.1%  6.1%  96.0%
9  2.8%  8.9%  93.9%
10 3.5%  12.4% 91.1%
11 4.3%  16.8% 87.6%
12 5.2%  22.0% 83.2%
13 6.3%  28.3% 78.0%
14 7.1%  35.4% 71.7%
15 7.9%  43.3% 64.6%
16 8.2%  51.5% 56.7%
17 8.3%  59.9% 48.5%
18 8.2%  68.0% 40.1%
19 7.8%  75.8% 32.0%
20 7.2%  83.0% 24.2%
21 6.2%  89.3% 17.0%
22 5.1%  94.4% 10.7%
23 3.6%  98.0% 5.6% 
24 2.0%  100.0% 2.0% 


Rank 4

4D12, Keep Highest 2, Summed
Mean: 17.8
Median: 18
Mode: 20 at 9.9%

#  % =    % <=  % >= 
2  0.0%  0.0%  100.0%
3  0.0%  0.0%  100.0%
4  0.1%  0.1%  100.0%
5  0.2%  0.3%  99.9%
6  0.3%  0.6%  99.8%
7  0.5%  1.1%  99.4%
8  0.9%  1.9%  98.9%
9  1.2%  3.2%  98.1%
10 1.8%  4.9%  96.8%
11 2.4%  7.3%  95.1%
12 3.2%  10.6% 92.7%
13 4.2%  14.8% 89.4%
14 5.3%  20.1% 85.2%
15 6.5%  26.5% 79.9%
16 7.7%  34.2% 73.5%
17 8.7%  42.9% 65.8%
18 9.5%  52.4% 57.1%
19 9.9%  62.3% 47.6%
20 9.9%  72.2% 37.7%
21 9.4%  81.6% 27.8%
22 8.3%  89.9% 18.4%
23 6.4%  96.3% 10.1%
24 3.7%  100.0% 3.7% 


Rank 5

5D12, Keep Highest 2, Summed
Mean: 19.0
Median: 20
Mode: 21 at 11.9%

#  % =    % <=  % >= 
2  0.0%  0.0%  100.0%
3  0.0%  0.0%  100.0%
4  0.0%  0.0%  100.0%
5  0.0%  0.0%  100.0%
6  0.1%  0.1%  100.0%
7  0.2%  0.3%  99.9%
8  0.3%  0.6%  99.7%
9  0.5%  1.1%  99.4%
10 0.8%  2.0%  98.9%
11 1.3%  3.2%  98.0%
12 1.9%  5.1%  96.8%
13 2.6%  7.7%  94.9%
14 3.7%  11.3% 92.3%
15 4.8%  16.1% 88.7%
16 6.3%  22.4% 83.9%
17 7.7%  30.1% 77.6%
18 9.3%  39.4% 69.9%
19 10.6% 50.0% 60.6%
20 11.6% 61.5% 50.0%
21 11.9% 73.4% 38.5%
22 11.3% 84.8% 26.6%
23 9.4%  94.1% 15.2%
24 5.9%  100.0% 5.9% 


Rank 6

6D12, Keep Highest 2, Summed
Mean: 19.8
Median: 20
Mode: 22 at 14.0%

#  % =    % <=  % >= 
3  0.0%  0.0%  100.0%
4  0.0%  0.0%  100.0%
5  0.0%  0.0%  100.0%
6  0.0%  0.0%  100.0%
7  0.1%  0.1%  100.0%
8  0.1%  0.2%  99.9%
9  0.2%  0.4%  99.8%
10 0.4%  0.8%  99.6%
11 0.6%  1.4%  99.2%
12 1.0%  2.5%  98.6%
13 1.6%  4.0%  97.5%
14 2.4%  6.4%  96.0%
15 3.4%  9.8%  93.6%
16 4.8%  14.5% 90.2%
17 6.3%  20.9% 85.5%
18 8.5%  29.4% 79.1%
19 10.3% 39.6% 70.6%
20 12.3% 52.0% 60.4%
21 13.5% 65.4% 48.0%
22 14.0% 79.4% 34.6%
23 12.3% 91.7% 20.6%
24 8.3%  100.0% 8.3% 





[nsruf]

I don't know the difference between Mean and Median, and have absolutely no clue what Mode is, but maybe somebody can help us out.

The mean is the weighted average of all outcomes.

The median is a number such that the probability to get a result that is greater or equal is at least 50%, and the probability to be less or equal is also at least 50%.

The Mode is the number with the largest probability to appear.

For assessing how competent a PC is, the median is probably the best measure, as it gives you the largest target number where his chance to succeed is still at least 50%.

[nsruf]

So, the questions to avoid ambiguity:
a) How do you figure out the probabilities for this sort of dice mechanic, and
b) Does this dice mechanic work in fulfilling the design goals I listed above?
c) Is a d12 too big a dice for this sort of mechanic? Would a d6 work better? I am using d12's cause I like 'em.

Re b): If I  understand you correctly, a skill will look something like this

Swordfighting 8 (target number = 24 - 8 = 16)
- Slash 2 (roll 2 dice)
- Stab 3
- Parry 1
- Disarm 2

Where the 8 is determined by performing some calculations with the 2, 3, 1, and 2. Correct?

I think this does meet your design goals in principle, but the chance of success for an average person will be very low. Example: If I have 2 dice in all specialization, I need a target number of 13 (the median) to have an above 50% of success for any maneuver. But TN 13 means a primary rank of 11, which is one short of the maximum. So unless there are some hefty situational modifiers to TN, PCs willbe rather incompetent at most tasks.

A second problem I have with your mechanic is more of a pet peeve of mine: your numbers are not in very meaningful! E.g., If my skill is listed as 7/1, I need to transform this into "target number = 17, 1 penalty die" before making a check. Listing the skill as "17/-1" would be much easier., IMO.

Re c): No sane person can doubt that we need more games using the d12. That's all I have to say about that. ;)

[Rob Carriere]

Justin,
I would recommend sticking with a program like Artanis suggests. You can do these calculations analytically, but you're talking several pages of algebra and every time you think, "Hm...what would this do?" you can go do the algebra again. Not really fun.

I'll also be the party poop and suggest that if you can't do the math, you might not want the rules. There's two parts to that. The first is that you run the risk of gaffes like the original White Wolf botch rule, or their first edition probability example, which won't help make you look like you know what you're doing.

The second and more important part is that if you can't do the math, neither can most of your target audience. So, everybody out there will be playing blind ("I think this score means I'm kinda good, but I have no idea whether that means doing X is a good bet.").

Say I'm writing an adventure for your game. I have a guy with sword fighting/8, I'm estimating no PC will have sword fighting over 7. Now, is this guy a wall they can't get through, tough opposition, a walkover, or what? Would you want me to be writing the adventure if I can't answer that question?

I have to agree on the d12, though. :-)
(and, to answer your question: d12 being a big die can be simply compensated by setting the TN high enough and making the steps in the TN big enough. You're much more likely to run into trouble with the small dice where the number of different outcomes can easily get too small to accomodate a decent range of capability.)

SR
--

[Kynn]

Wandering Monsters High School (my game) uses pretty much the same mechanic.  Roll Nd6, keep only the highest 2 scores, and N ranges from 2 to 4 -- depending on your "grade" (C = 2d6, B = 3d6, A = 4d6).  For a D, roll 2d6 and keep only the highest die, and for an F roll 2d6 and keep the lowest.

[Josh Roby]

So, the questions to avoid ambiguity:
a) How do you figure out the probabilities for this sort of dice mechanic, and
b) Does this dice mechanic work in fulfilling the design goals I listed above?
c) Is a d12 too big a dice for this sort of mechanic? Would a d6 work better? I am using d12's cause I like 'em.

a) While I can do most probability math, I generally find it easier to make a spreadsheet or simple program to just "roll dice" a couple thousand times and see what usually happens.  It's lots faster.  The charts that Artanis provided should do well.

b) You want a relatively deterministic dice mechanic, and anything that has the player roll more dice than actually get used will flatten your probability curve (which you want for determinism), so you're on the right page, there.

c) On the other hand, the higher your die size, the less predictable your results are.  If you set up the same system using d6s instead of d12s, your individual dice would have a smaller range and the results would be more preditable.  To put it another way, on a d12, your highest roll is 12 times your lowest roll, meaning that a 'fluke' roll may make the character twelve times more effective than they 'should' be.  On a d6, however, that fluke roll is only half as curve-throwing.

Additionally, I'm curious why a player's competencies determine two inputs to the system -- why does a expert swordsman have more resources to reach a lower bar than a beginner?  Do you have any mechanics to take situation (darkness, wounds, tickling) into account?  Do you have any resources available to the player to throw the normal probability when they want to (hero points, drama points, willpower)?

[Justin Marx]

Artanis, a billion thanks for the spreadsheet of the probabilities, will try to find the website ASAP.

I understand the point with the relative meaninglessness of arbitrary ranks, especially if the maths if difficult to compute. The first point is that the Primary and Secondary Ranks are not linearly correlated.... there are outside modifiers that influence the Primary. Having competencies of 2 (the average score of competence) in everything will probably give you a Primary in the range of 6 to 9, depending on stats and other stuff (its easy to compute, but is trait specific). In any case, the problem is the same, as far as chance of success - if 2 is competent, but has a less than probably chance of getting a success, then the system is obviously broken.

The main reason for this trait system of 12 and 6 is because traits are usually quite large in breadth, and the primary rank is not as important as the secondary. Instead of swordsmanship, a skill would more likely be Marskmanship, with Secondary Skills of Lasers, Rail Guns, Bows etc. General competency (a steady hand to aim with) helps a lot, but specialising should pay off more. I like this because it is deterministic and is what I want in the game.

With this in mind, there are TN modifiers, as well as meta-stats to pump into skill rolls, and there is a relatively constant TN modifier (advantage - used in most, if not all skill rolls) that modifies the roll between -6 and +6. Reducing the Original TN down from 24 to 18 changes things at the low end, but it makes massive success very high at the other end. Meta-mechanics are not used all the time, and most NPCs have very little of this stuff, even the powerful ones, and if your NPC mechanic can't even fix a broken cart-wheel, then no amount of meta-stats will change that. Advantage modifiers are constantly applied however. But this other mechanism is, and should be, seperate from the basic mechanics.

Hmm...... going to have to work on this one some more. The margin of success and failure is also important, extremely important, the original system was that only the Primary Rank determines chance of success, while the the Secondary Rank determined the rate of success or failure depending on dice rolls. The problem was that it was too ungainly, and required division arithmetic, which was simple enough, but was far too crunch heavy for sustained play.

Thanks for all the advice, I suspect that this mechanism is broken. The rationale is to have a relatively straightforward dice computation, using d12, with ranges of success and failure, and a two tiered trait system to reflect specialised and general ability. The Primary influences chance of success, while the Secondary influences the range of that success or failure. If anyone knows of any systems that do this, or any similar mechanisms, I'd love to have a look. I haven't given up on the dice pools like this yet, but I am working it like a ticking clock in my head presently. I'll be posting tomorrow once I have worked it again.

Justin

[nsruf]

The rationale is to have a relatively straightforward dice computation, using d12, with ranges of success and failure, and a two tiered trait system to reflect specialised and general ability. The Primary influences chance of success, while the Secondary influences the range of that success or failure. If anyone knows of any systems that do this, or any similar mechanisms, I'd love to have a look.

If you keep the distinction primary = target number and secondary = number of dice, the following system might work:

Roll a number of d12 equal to your secondary trait and take the maximum. If the result is greater or equal to your primary trait, you succeed with a margin of success equal to the difference. If the result is less, you fail with a margin of failure again equal to the difference.

This system has the following interesting properties:

1. With two or more dice, the probability to roll a given number is strictly increasing from 1 to 12. So a specialist can expect pretty consistent results, with most of his rolls in the 10-12 range. And his occasional failures are likely to be mild, as his chance to roll really low is small.

2. You don't need low target numbers to succeed more than half of the time with 2 dice. The probability to roll a 9 or higher on 2 dice is 5/9 = 56%.

3. Probabilities are easy to work out analytically. Your chance to get a T or higher on D dice is 1 - ((T - 1) / 12) ^ D, which is easily evaluated by Excel.