[Shangri-la] Dice Mechanics and Related Foo

[Lxndr]

Shangri-La is meant to be the 2nd major project of Twisted Confessions.  It was originally a hacked-out FUDGE variant that I used in a pbem/chat combination game, but that game collapsed about a year ago.  The setting and situation, however, are still interesting to me, and when someone pointed out that it'd make a good game, I was all over the idea.

Then Fastlane came along, smacking my muse around and saying "damnit, work on me!"  So Shangri-la sort of floated off into limbo.  (That happens a lot, which is why there are more "works in progress" on the TC site than there are finished projects, but that's neither here nor there).

In any case, limbo isn't the same as the graveyard, and I've spent some time, on and off, screwing with the setting.  A rather useful milestone in the backburner relations was when a dice system pretty much came to me in a dream (and since Shangri-la is supposed to represent the dreamlands, it seemed appropriate to use a dream system).  It was first used in my re-imagining of Creeks and Crawdads (an actual play post can be found here, but is now ready to move on to bigger and better things.

And that's what I want to talk about, the basic die mechanic and the other things that have grown out of it.  Here's a brief outline of the process, slightly different from the C&C iteration of the Impact System, as it was called in my dream.  Weird stuff, that.

* Players roll a variable number of six-sided dice, looking for fives and sixes.  These are "successes."
* If at least one six is rolled, players may choose to reroll.  All the dice are dropped again, and the successes on the 2nd roll are added to the first.  If no successes are rolled on the 2nd roll, all accumulated successes are forfeited, and it's treated as a zero.
* Players can choose to reroll until they either forfeit (lose everything) or stall (fives and no sixes).

My half-baked calculations have given me a rough idea of the average number of successes a particular die pool will give, including the possibility of rerolls, but I would love a more complete analysis, if anyone in here could help.  Here are results for the first ten numbers:

Code Select Expand

# of  average
dice  successes
1       0.40
2       0.96
3       1.73
4       2.76
5       4.15
6       5.97
7       8.36
8      11.47
9      15.48
10     20.64
The curve just gets steeper from there.  It's really kind of nice, I think, since each step displays a significant improvement over the previous step, especially after you pass the six-die point.  Once again, though, this is a half-baked attempt, a mix of brute force, guesswork, and flashes of inductive reasoning.

Here's what I'm considering, at this juncture:

* Characters have attributes that determine how many dice they roll.
* Characters have a something similar to tRoS weapon proficiencies, which add dice to attributes.  Each of these is a "regional" skill, related to a particular dreamland in the dreamworld.
* Character skills are also inspired by tRoS, being the target number (in this case, number of successes) needed to succeed at a skill.  Thus the lower a target number is, the better.
* Both # of dice rolled and the target number can be modified by environment in particular ways (which are outside the scope of this discussion).

So we have:  Roll [Attribute + Region + Modifier], and get at least [Skill + Adjustment]

I'm currently considering the "human average" for an attribute to be either 2 or 3, with a reasonable human maximum being 6.  It just feels appropriate that the best mankind can do is "almost as many successes as dice, on average."

Anyway, how does that look, as a framework?  Does anything not make sense?  Do my calculations for average-successes seem right?  Is there anything else you need to know?  Is this a good start, a bad one?

[Walt Freitag]

Alexander,

Here are probability distributions for various exact numbers of successes (5s and 6s rolled) for various numbers of dice.

Code Select Expand

# of         probability (%) of rolling exactly the indicated number of successes
dice    0       1       2       3       4       5       6       7       8       9      10      11      12

1     66.67%  33.33%
2     44.44%  44.44%  11.11%
3     29.63%  44.44%  22.22%   3.70%
4     19.75%  39.51%  29.63%   9.88%   1.23%
5     13.17%  32.92%  32.92%  16.46%   4.12%   0.41%
6      8.78%  26.34%  32.92%  21.95%   8.23%   1.65%   0.14%
7      5.85%  20.48%  30.73%  25.61%  12.80%   3.84%   0.64%   0.05%
8      3.90%  15.61%  27.31%  27.31%  17.07%   6.83%   1.71%   0.24%   0.02%
9      2.60%  11.71%  23.41%  27.31%  20.48%  10.24%   3.41%   0.73%   0.09%   0.01%
10     1.73%   8.67%  19.51%  26.01%  22.76%  13.66%   5.69%   1.63%   0.30%   0.03%   0.00%
11     1.16%   6.36%  15.90%  23.84%  23.84%  16.69%   8.35%   2.98%   0.75%   0.12%   0.01%   0.00%
12     0.77%   4.62%  12.72%  21.20%  23.84%  19.08%  11.13%   4.77%   1.49%   0.33%   0.05%   0.00%   0.00%

The zero-successes column is also the probability of forfeiting when attempting a reroll. The chance of being eligible for a reroll (not forfeiting or stalling) after a given roll is:

Code Select Expand

# of
dice    chance

1       16.67%
2       30.56%
3       42.13%
4       51.77%
5       59.81%
6       66.51%
7       72.09%
8       76.74%
9       80.62%
10      83.85%
11      86.54%
12      88.78%

Now, what you really need to know about this system's behavior is this: given a number of dice to roll, and a number of successes that the player is determined to make (or stall or forfeit trying), what are the odds of each outcome (making the target, stalling, or forfeiting)? This is a somewhat tricky calculation. Here are the results:

Code Select Expand

Probability of eventually reaching or exceeding target number of successes

# of     target number of successes
dice    1      2      3      4      5      6      7      8      9     10

1     .3333  .0556  .0093  .0015  .0003   low    low    low    low    low
2     .5556  .2346  .0984  .0414  .0174  .0073  .0031  .0013  .0005  .0002
3     .7037  .4156  .2467  .1469  .0872  .0519  .0308  .0183  .0109  .0065
4     .8025  .5659  .4012  .2867  .2040  .1452  .1034  .0737  .0524  .0373
5     .8683  .6820  .5365  .4271  .3386  .2682  .2127  .1686  .1337  .1060
6     .9122  .7690  .6461  .5503  .4679  .3968  .3368  .2860  .2428  .2061
7     .9415  .8331  .7316  .6512  .5802  .5153  .4576  .4067  .3614  .3211
8     .9610  .8799  .7973  .7308  .6724  .6168  .5650  .5181  .4752  .4358
9     .9740  .9139  .8473  .7925  .7455  .6999  .6557  .6144  .5762  .5404
10    .9827  .9386  .8853  .8401  .8024  .7661  .7296  .6946  .6617  .6305
11    .9884  .9563  .9141  .8767  .8463  .8178  .7886  .7596  .7319  .7056
12    .9923  .9690  .9359  .9050  .8802  .8580  .8351  .8115  .7885  .7666

# of     target number of successes
dice   11     12     13     14     15     16     17     18     19     20

1      low    low    low    low    low    low    low   0.000  0.000  0.000
2     .0001   low    low    low    low    low    low    low    low    low
3     .0038  .0023  .0014  .0008  .0005  .0003  .0002  .0001  .0001   low
4     .0266  .0189  .0135  .0096  .0068  .0049  .0035  .0025  .0018  .0013
5     .0840  .0666  .0528  .0419  .0332  .0263  .0209  .0165  .0131  .0104
6     .1750  .1485  .1261  .1070  .0909  .0771  .0655  .0556  .0472  .0401
7     .2853  .2536  .2253  .2002  .1779  .1581  .1405  .1248  .1109  .0986
8     .3996  .3664  .3360  .3081  .2825  .2591  .2376  .2179  .1998  .1832
9     .5066  .4750  .4453  .4176  .3915  .3671  .3442  .3227  .3026  .2837
10    .6008  .5724  .5453  .5195  .4949  .4716  .4493  .4280  .4078  .3885
11    .6803  .6557  .6320  .6092  .5873  .5661  .5457  .5260  .5070  .4887
12    .7455  .7250  .7049  .6853  .6663  .6479  .6300  .6125  .5955  .5790


Probability of forfeit result before reaching target number of successes
(including rolling no successes on first roll)

# of     target number of successes
dice    1      2      3      4      5      6      7      8      9     10

1     .6667  .7778  .7963  .7994  .7999  .8000  .8000  .8000  .8000  .8000
2     .4444  .5432  .6022  .6235  .6332  .6371  .6388  .6395  .6398  .6399
3     .2963  .3621  .4262  .4610  .4815  .4939  .5012  .5056  .5082  .5097
4     .1975  .2365  .2882  .3241  .3483  .3660  .3786  .3875  .3939  .3984
5     .1317  .1534  .1894  .2197  .2418  .2595  .2737  .2849  .2937  .3008
6     .0878  .0994  .1226  .1453  .1630  .1777  .1905  .2014  .2106  .2183
7     .0585  .0645  .0786  .0946  .1078  .1189  .1289  .1380  .1460  .1531
8     .0390  .0421  .0503  .0609  .0704  .0783  .0856  .0924  .0987  .1044
9     .0260  .0275  .0322  .0389  .0455  .0511  .0561  .0609  .0655  .0698
10    .0173  .0181  .0207  .0248  .0293  .0331  .0365  .0398  .0430  .0460
11    .0116  .0119  .0133  .0158  .0187  .0214  .0236  .0258  .0279  .0301
12    .0077  .0079  .0086  .0101  .0119  .0137  .0153  .0167  .0181  .0195

# of     target number of successes
dice   11     12     13     14     15     16     17     18     19     20

1     .8000  .8000  .8000  .8000  .8000  .8000  .8000  .8000  .8000  .8000
2     .6400  .6400  .6400  .6400  .6400  .6400  .6400  .6400  .6400  .6400
3     .5107  .5112  .5115  .5117  .5118  .5119  .5119  .5120  .5120  .5120
4     .4016  .4039  .4056  .4067  .4075  .4081  .4086  .4089  .4091  .4092
5     .3063  .3108  .3143  .3170  .3193  .3210  .3224  .3235  .3244  .3250
6     .2250  .2306  .2353  .2394  .2428  .2458  .2482  .2503  .2521  .2536
7     .1594  .1650  .1700  .1744  .1783  .1818  .1849  .1877  .1902  .1923
8     .1097  .1145  .1189  .1230  .1267  .1301  .1332  .1361  .1387  .1411
9     .0739  .0776  .0811  .0845  .0876  .0905  .0932  .0958  .0982  .1004
10    .0489  .0517  .0543  .0568  .0592  .0615  .0637  .0657  .0677  .0696
11    .0321  .0340  .0359  .0377  .0394  .0411  .0427  .0443  .0458  .0472
12    .0209  .0222  .0235  .0247  .0259  .0271  .0283  .0294  .0305  .0315


Probability of stalling before reaching target number of successes

# of     target number of successes
dice    1      2      3      4      5      6      7      8      9     10

1      N/A   .1667  .1944  .1991  .1998  .2000  .2000  .2000  .2000  .2000
2      N/A   .2222  .2994  .3350  .3494  .3556  .3581  .3592  .3597  .3599
3      N/A   .2222  .3272  .3921  .4313  .4542  .4679  .4761  .4809  .4838
4      N/A   .1975  .3106  .3892  .4477  .4888  .5180  .5389  .5537  .5643
5      N/A   .1646  .2740  .3532  .4196  .4723  .5136  .5465  .5726  .5933
6      N/A   .1317  .2313  .3044  .3691  .4254  .4727  .5126  .5467  .5756
7      N/A   .1024  .1897  .2543  .3120  .3658  .4135  .4553  .4926  .5258
8      N/A   .0780  .1524  .2083  .2573  .3049  .3494  .3895  .4261  .4598
9      N/A   .0585  .1205  .1685  .2090  .2491  .2882  .3246  .3583  .3898
10     N/A   .0434  .0940  .1351  .1683  .2008  .2339  .2657  .2954  .3234
11     N/A   .0318  .0725  .1074  .1349  .1608  .1877  .2146  .2402  .2644
12     N/A   .0231  .0554  .0849  .1079  .1283  .1497  .1718  .1935  .2140

# of     target number of successes
dice   11     12     13     14     15     16     17     18     19     20

1     .2000  .2000  .2000  .2000  .2000  .2000  .2000  .2000  .2000  .2000
2     .3599  .3600  .3600  .3600  .3600  .3600  .3600  .3600  .3600  .3600
3     .4855  .4865  .4871  .4875  .4877  .4878  .4879  .4879  .4880  .4880
4     .5718  .5771  .5810  .5837  .5856  .5870  .5880  .5887  .5892  .5895
5     .6096  .6226  .6329  .6411  .6476  .6527  .6568  .6600  .6625  .6646
6     .6001  .6209  .6386  .6536  .6663  .6771  .6863  .6941  .7007  .7063
7     .5553  .5814  .6047  .6254  .6438  .6601  .6746  .6875  .6989  .7091
8     .4907  .5191  .5451  .5689  .5908  .6108  .6292  .6460  .6615  .6757
9     .4195  .4474  .4735  .4980  .5209  .5424  .5626  .5815  .5993  .6159
10    .3503  .3759  .4004  .4237  .4458  .4669  .4871  .5062  .5245  .5419
11    .2877  .3102  .3321  .3531  .3733  .3928  .4116  .4297  .4472  .4641
12    .2336  .2528  .2716  .2900  .3077  .3250  .3418  .3581  .3740  .3894

"low" means less than .00005
All probabilities are in conventional form, multiply by 100 for percentages.

- Walt

[Lxndr]

Dude.  Thank you very much.  The probabilities are a much more useful bit than the "average successes" which is all I was able to get (and I'm not sure how accurate my calculations are anyway).  My number looked a little high compared to yours, but in both cases it's obvious that there's a devastating curve.

So if I arbitrarily set "completely unskilled" at 10, then a human attribute range of 1-5 ("low" to 11%) with an average of 2 (at 0.02%) or 3 (0.65%) seems fitting.  If the regional "proficiencies" are also 1-5, then a paradigm human who's learned nearly everything there is to know about a region, can perform even unskilled activities at 63.05%.

That at first sounds outrageous, but 5 dice in a proficiency should suggest mastery over that part of the dreamworlds, even the portions in which your character is otherwise untrained, especially if your character is a "paradigm human" (i.e. maxxed out in the appropriate attribute).  So I'm not too worried about it, at this juncture, though playtesting might give me reason to worry later.

Once again, thanks Walt!  That's a rather large heaping of help.