Revisiting Another Dice Mechanic

I was thinking at lunch today about Another Dice Mechanic, and thought I saw a few things that could be made better.

Recapping Yet Another Dice Mechanic

Each tier has a die associated. Pre-basic is d2, Basic is d4, Expert is d6, Heroic is d8, Master is d10, Champion is d12, and Legendary is d12. When you need to make a check, you roll a die based on your tier (a starting PC, assuming the Expert tier, rolls 1d6). If you have a relevant talent — you want to climb a cliff and you are a Mountaineer (who naturally knows how to climb cliffs), an Earth Mage (magical affinity with stone), or Spider-Man{tm} (no explanation needed) — you can add another die to your roll, based on the tier of the talent. Just as talents of the same type (cornerstone, common, and capstone) don’t stack, but different ones do, I’ll say you can add up to one die for each of a cornerstone, a common, and a capstone talent. Earth Mage Mountaineer Spider-Man doesn’t get three extra dice, assuming all three talents are common talents.

Target numbers for challenges of each tier are based on 50% success rate for an untrained character: Pre-basic target number is 2, Basic is 3, Expert is 4, Heroic is 5, Master is 6, Champion is 7, and Legendary is 11. This is a big jump and I’d like to see it not there, but that’s how it is, at least until d16s become more common.

This means a character who is trying to meet an equal-tier challenge succeeds half the time. A character with two dice can expect success 3/4 of the time, with three dice can expect success 7/8 of the time, and with four dice can expect success 15/16 of the time. I’m pretty satisfied with that… but what happens if a character tries for something harder?

Well, an Expert character attempting a Heroic check needs to make a target of 5. This happens 1/3 of the time. Not bad. With two dice it becomes 5/9 of the time, better than half! With three dice it’ll be 19/27 (better than 2/3 of the time), and with four dice it’ll be 65/81 slightly more than 4/5 of the time.

The same Expert attempting a Master check needs to make a target of 6. This happens 1/6 of the time… unlikely, but far from impossible. With two dice it becomes 11/36 (much better!), three dice 125/216 (about 42%, a bit better than 2/5), and with four dice 671/1296 (slightly better than half). I’m a little surprised it’s that good, but this is a character heavily tuned (all three talents align) to the task.

Obviously having four dice is very good. I like that.

Now, a Champion challenge. Target is 7, which happens… never. No matter how many dice the character has, the task is impossible. Which actually isn’t as horrible as it sounds; in D&D terms this is a level 1-4 character trying to do something suitable for a level 13-16 character.

Still… looking down, the Pre-basic creature can never, ever succeed at even a Basic task. While an Expert character being incapable of doing a Champion task, a Pre-basic creature not even being able to reach up one tier seems like it could use some adjustment.

PB B X H M C L
n d 2 % 3 % 4 % 5 % 6 % 7 % 8 % 9 % 10 % 11 % 12 %
1 2 50.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
1 4 75.00 50.00 25.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
1 6 83.33 66.67 50.00 33.33 16.67 0.00 0.00 0.00 0.00 0.00 0.00
1 8 87.50 75.00 62.50 50.00 37.50 25.00 12.50 0.00 0.00 0.00 0.00
1 10 90.00 80.00 70.00 60.00 50.00 40.00 30.00 20.00 10.00 0.00 0.00
1 12 91.67 83.33 75.00 66.67 58.33 50.00 41.67 33.33 25.00 16.67 8.33
1 20 95.00 90.00 85.00 80.00 75.00 70.00 65.00 60.00 55.00 50.00 45.00
2 2 75.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
2 4 93.75 75.00 43.75 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
2 6 97.22 88.89 75.00 55.56 30.56 0.00 0.00 0.00 0.00 0.00 0.00
2 8 98.44 93.75 85.94 75.00 60.94 43.75 23.44 0.00 0.00 0.00 0.00
2 10 99.00 96.00 91.00 84.00 75.00 64.00 51.00 36.00 19.00 0.00 0.00
2 12 99.31 97.22 93.75 88.89 82.64 75.00 65.97 55.56 43.75 30.56 15.97
2 20 99.75 99.00 97.75 96.00 93.75 91.00 87.75 84.00 79.75 75.00 69.75
3 2 87.50 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
3 4 98.44 87.50 57.81 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
3 6 99.54 96.30 87.50 70.37 42.13 0.00 0.00 0.00 0.00 0.00 0.00
3 8 99.80 98.44 94.73 87.50 75.59 57.81 33.01 0.00 0.00 0.00 0.00
3 10 99.90 99.20 97.30 93.60 87.50 78.40 65.70 48.80 27.10 0.00 0.00
3 12 99.94 99.54 98.44 96.30 92.77 87.50 80.15 70.37 57.81 42.13 22.97
3 20 99.99 99.90 99.66 99.20 98.44 97.30 95.71 93.60 90.89 87.50 83.36
4 2 93.75 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
4 4 99.61 93.75 68.36 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
4 6 99.92 98.77 93.75 80.25 51.77 0.00 0.00 0.00 0.00 0.00 0.00
4 8 99.98 99.61 98.02 93.75 84.74 68.36 41.38 0.00 0.00 0.00 0.00
4 10 99.99 99.84 99.19 97.44 93.75 87.04 75.99 59.04 34.39 0.00 0.00
4 12 100.00 99.92 99.61 98.77 96.99 93.75 88.42 80.25 68.36 51.77 29.39
4 20 100.00 99.99 99.95 99.84 99.61 99.19 98.50 97.44 95.90 93.75 90.85
  • ‘n’ is the number of dice rolled
  • ‘d’ is the size of the dice
  • ‘PB’, ‘B’, ‘X’, ‘H’, ‘M’, ‘C’, ‘L’: tier that has this target number (Pre-Basic, Basic, Expert, Heroic, Master, Champion, Legendary)
  • number %’ is the percent of successes rolled for that target

The success numbers and percentages shown in bold above are for when the creature is rolling dice for a task of the same tier (that is, “1,512/87.50” for row 3d12 is highlighted to show the results for a Champion creature with three dice rolling against a Champion task). Note that for all tiers they show the same chance of success at any particular number of dice (anyone rolling two dice against a challenge of the same tier can expect success 75% of the time).

It is evident that a Pre-basic creature can never achieve a Basic task (which is probably a problem), a Basic creature can never achieve a Heroic task (might be a problem), an Expert creature can never achieve a Champion task (probably not a problem), a Heroic creature can never achieve a Legendary task (probably not a problem), and a Master creature can never achieve a Legendary task (might be a problem again).

Can we widen the range when adding more dice? It seems reasonable that someone who is better trained might not only be more likely to punch above his weight (succeed at a higher-tier task), but be able to reach even farther (succeed at even higher-tier tasks) than someone of the same tier who is not as trained.

Expanding the Range

It turns out the solution is fairly straightforward.

Every double rolled adds one to the total.

That easy. Well, slightly complicated in that ‘each double’ isn’t just counting the number of dice that match each other, but the number of pairs between them. That is, two matching dice make one pair, but three matching dice (A = B = C) are three pairs: (A, B), (B, C), and (A, C). Four matching dice are six pairs. This works in my favor, I think, but it is a slight complication.

PB B X H M C L
n d 2 % 3 % 4 % 5 % 6 % 7 % 8 % 9 % 10 % 11 % 12 %
1 2 50.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
1 4 75.00 50.00 25.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
1 6 83.33 66.67 50.00 33.33 16.67 0.00 0.00 0.00 0.00 0.00 0.00
1 8 87.50 75.00 62.50 50.00 37.50 25.00 12.50 0.00 0.00 0.00 0.00
1 10 90.00 80.00 70.00 60.00 50.00 40.00 30.00 20.00 10.00 0.00 0.00
1 12 91.67 83.33 75.00 66.67 58.33 50.00 41.67 33.33 25.00 16.67 8.33
1 20 95.00 90.00 85.00 80.00 75.00 70.00 65.00 60.00 55.00 50.00 45.00
2 2 100.00 25.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
2 4 100.00 81.25 50.00 6.25 0.00 0.00 0.00 0.00 0.00 0.00 0.00
2 6 100.00 91.67 77.78 58.33 33.33 2.78 0.00 0.00 0.00 0.00 0.00
2 8 100.00 95.31 87.50 76.56 62.50 45.31 25.00 1.56 0.00 0.00 0.00
2 10 100.00 97.00 92.00 85.00 76.00 65.00 52.00 37.00 20.00 1.00 0.00
2 12 100.00 97.92 94.44 89.58 83.33 75.69 66.67 56.25 44.44 31.25 16.67
2 20 100.00 99.25 98.00 96.25 94.00 91.25 88.00 84.25 80.00 75.25 70.00
3 2 100.00 100.00 25.00 12.50 0.00 0.00 0.00 0.00 0.00 0.00 0.00
3 4 100.00 100.00 81.25 32.81 3.12 1.56 0.00 0.00 0.00 0.00 0.00
3 6 100.00 100.00 94.44 80.09 54.63 15.28 0.93 0.46 0.00 0.00 0.00
3 8 100.00 100.00 97.66 91.60 80.86 64.26 40.62 8.79 0.39 0.20 0.00
3 10 100.00 100.00 98.80 95.70 90.20 81.70 69.60 53.30 32.20 5.70 0.20
3 12 100.00 100.00 99.31 97.51 94.33 89.41 82.41 72.97 60.76 45.43 26.62
3 20 100.00 100.00 99.85 99.46 98.78 97.71 96.20 94.16 91.53 88.21 84.15
4 2 100.00 100.00 100.00 62.50 12.50 12.50 6.25 0.00 0.00 0.00 0.00
4 4 100.00 100.00 100.00 74.22 24.22 10.94 1.17 0.78 0.39 0.00 0.00
4 6 100.00 100.00 100.00 94.91 77.62 39.81 8.26 3.40 0.23 0.15 0.08
4 8 100.00 100.00 100.00 98.39 92.92 80.96 59.25 24.02 3.69 1.46 0.07
4 10 100.00 100.00 100.00 99.34 97.10 92.20 83.31 68.88 47.11 15.96 1.95
4 12 100.00 100.00 100.00 99.68 98.60 96.24 91.95 84.99 74.49 59.47 38.83
4 20 100.00 100.00 100.00 99.96 99.82 99.51 98.96 98.06 96.69 94.75 92.07

Interpretation of the column headings and numbers/percentages is as above.

The range of creatures of all tiers is expanded… perhaps too far now, for the pre-basic creatures, but I’ll come back to that. As the number of dice increases, the chance of lower tier creatures succeeding at tasks of the same tier is greater than the chance of higher-tier creatures succeeding at tasks of the same tier.

A Pre-Basic through Expert creature trying to make a roll against a challenge of the same tier literally cannot fail once four dice dice are rolled (and Pre-Basic and Basic can’t fail when three dice are rolled, and Pre-Basic can’t fail when two dice are rolled). The same cannot be said of a Legendary creature attempting a Legendary task (not quite 95% chance of success). This is entirely because the smaller dice reach a point where doubles are unavoidable, which increases the minimum possible roll. A Pre-Basic creature with four dice literally cannot roll lower than 4 on four dice (2 twice and 1 twice, for 2 base + 2 for two pairs of doubles), but cannot roll higher than 8 (2 four times, for 2 base + 6 for doubles). A Basic creature with four dice also cannot roll lower than 4 on four dice (one of each value from 1..4, the smallest way to get no doubles), but could roll as high as 10 (4 base + 6 for doubles).

In any case, as much as it seems odd that a ‘Pre-Basic’ creature with the right build cannot fail at tasks above its tier, to some degree, I’m actually pretty okay with it. Pre-Basic was looking awfully anemic, and this makes it so they actually can be relevant… if they’re lucky, or properly suited to the task.

A creature with no particular advantage, who rolls only a single die, has no chance of getting doubles. A Pre-Basic creature rolling one die cannot achieve a Basic task (a Pre-Basic creature rolling two dice has a 25% chance, and rolling three dice cannot fail), and a Basic creature rolling one die cannot achieve a Heroic (two tiers higher) task. Only with training or other advantage (i.e. one or more relevant talents) is it possible to get at least one double and get the final result higher.

Closing Comments

All in all, I think this is a good change. The ability to extend the range of success outside the creature’s tier is worth the slight increase in difficulty (counting doubles — which comes naturally to cribbage players like me anyway). You always want more dice, and you want the biggest dice you can get. There are situations where more, smaller dice are better than a fewer, bigger dice: 3d2 vs. 2d4 when trying to achieve a Basic task (100% vs. 81.25%)… but not an Expert task (25% vs 50%)… and again attempting a Heroic task (12.5% vs. 6.25%, bugger).

Okay, it’s not perfect. I’d be happier if 3d2 was better than 2d4 up to a point, then 2d4 overtook and stayed better than 3d2, but I can live with the curves crossing a couple times.

In any case, more dice of the same size are better then fewer dice of the same size, and the same number of bigger dice is better than the same number of smaller dice (almost always — there is one exception, 4d2 trying to make a Champion-tier check). The occasional odd interaction between different numbers of different-sized dice, I’ll just live with. They appear, on cursory examination, to mostly happen outside the range I’m interested in.

And besides, how else will a house cat kill a wizard if it can’t roll better sometimes?

Postscript: Count Number of Additional Dice

“We solved the problem. Now we fight the solution.”

I realized after initially posting that my solution-problem had itself a fairly simple solution.

Instead of counting all the doubles, I just count the number of times each doubled number comes up and subtract one. That is, when I roll doubles I add one, when I roll triples I add two (instead of the three I was adding before), and when all four values match I add three. If I roll two pairs of doubles I add two (one for each pair), same as I did before.

This works much, much better. Now more dice of the same size are always better than fewer dice of the same size, and for any particular number of dice bigger dice are better than smaller dice. It can sometimes be better to have a larger number of small dice than a small number of bigger dice (compare 3d4 to 2d6, up to the Expert target), but once the lines cross they stay crossed (2d6 is only slightly worse than 3d4 at making Expert checks, is much better at making Heroic checks, 20 times better at Master checks, and has at least a tiny chance of making Champion checks that 3d4 cannot even touch).

PB B X H M C       L
n d 2 % 3 % 4 % 5 % 6 % 7 % 8 % 9 % 10 % 11 % 12 %
1 2 50.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
1 4 75.00 50.00 25.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
1 6 83.33 66.67 50.00 33.33 16.67 0.00 0.00 0.00 0.00 0.00 0.00
1 8 87.50 75.00 62.50 50.00 37.50 25.00 12.50 0.00 0.00 0.00 0.00
1 10 90.00 80.00 70.00 60.00 50.00 40.00 30.00 20.00 10.00 0.00 0.00
1 12 91.67 83.33 75.00 66.67 58.33 50.00 41.67 33.33 25.00 16.67 8.33
1 20 95.00 90.00 85.00 80.00 75.00 70.00 65.00 60.00 55.00 50.00 45.00
2 2 100.00 25.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
2 4 100.00 81.25 50.00 6.25 0.00 0.00 0.00 0.00 0.00 0.00 0.00
2 6 100.00 91.67 77.78 58.33 33.33 2.78 0.00 0.00 0.00 0.00 0.00
2 8 100.00 95.31 87.50 76.56 62.50 45.31 25.00 1.56 0.00 0.00 0.00
2 10 100.00 97.00 92.00 85.00 76.00 65.00 52.00 37.00 20.00 1.00 0.00
2 12 100.00 97.92 94.44 89.58 83.33 75.69 66.67 56.25 44.44 31.25 16.67
2 20 100.00 99.25 98.00 96.25 94.00 91.25 88.00 84.25 80.00 75.25 70.00
3 2 100.00 100.00 12.50 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
3 4 100.00 100.00 79.69 31.25 1.56 0.00 0.00 0.00 0.00 0.00 0.00
3 6 100.00 100.00 93.98 79.63 54.17 14.81 0.46 0.00 0.00 0.00 0.00
3 8 100.00 100.00 97.46 91.41 80.66 64.06 40.43 8.59 0.20 0.00 0.00
3 10 100.00 100.00 98.70 95.60 90.10 81.60 69.50 53.20 32.10 5.60 0.10
3 12 100.00 100.00 99.25 97.45 94.27 89.35 82.35 72.92 60.71 45.37 26.56
3 20 100.00 100.00 99.84 99.45 98.76 97.70 96.19 94.15 91.51 88.20 84.14
4 2 100.00 100.00 100.00 6.25 0.00 0.00 0.00 0.00 0.00 0.00 0.00
4 4 100.00 100.00 100.00 70.70 17.19 0.39 0.00 0.00 0.00 0.00 0.00
4 6 100.00 100.00 100.00 94.21 76.23 37.73 5.56 0.08 0.00 0.00 0.00
4 8 100.00 100.00 100.00 98.17 92.48 80.30 58.40 22.97 2.44 0.02 0.00
4 10 100.00 100.00 100.00 99.25 96.92 91.93 82.96 68.45 46.60 15.37 1.28
4 12 100.00 100.00 100.00 99.64 98.51 96.11 91.78 84.78 74.25 59.19 38.50
4 20 100.00 100.00 100.00 99.95 99.81 99.50 98.94 98.03 96.66 94.71 92.03

To compare the three approaches, the table below shows the amount added to the base roll (i.e. highest die rolled) for doubles, triples, and quadruples (and ‘two doubles’, since it’s the only oddity after that).

Same Values Starting Model (Recap) Count All Doubles (Solution) Count Only Extras (PS)
1 0 0 0
2 0 1 1
3 0 3 2
4 0 6 3
2 x 2 0 2 2

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2 Comments

  1. “doesn’t three extra dice” < -- I think you accidentally a word.
    If you make Pre-Basic a d3 then it can still reach the Basic target. This spoils the die-size pattern a bit, but then again Legendary already started doing that.

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