Task Resolution Using Dice Pools: Heavy Lifting

Last night I wrote about switching Echelon to using dice pools instead of a single roll with modifiers.

I had an intuitive sense it would do some good things for me, but I thought I’d do some math and see just how it behaves.

Overall, I think it’ll do good things.

Dice Pools in Echelon

The draft rules at this point are:

  • One eight-sided die plus one die per tier, for all abilities.
  • Training or other improvement lets you improve dice ‘up to that tier’ by one size.
  • Double training or further improvement lets you improve dice ‘up to that tier’ by one size, and stacks.
  • Roll dice, each roll equal to or greater than ‘5’ counts as a success, each roll of ‘1’ counts as a ‘complication’.
  • You may choose to spend successes to buy off complications, but are not required to.

Superficial Examination

It’s worth doing a quick examination of the mean number of expected successes.

For each die size, the number of expected successes is equal to (chance of success – chance of complication) * (number of dice).

Tier Dice d6 (1/6) d8 (3/8) d10 (5/10) d12 (7/12)
Pre-Basic 1 0.17 0.375 0.5 0.583
Basic 2 0.33 0.750 1.0 1.167
Expert 3 0.50 1.125 1.5 1.750
Heroic 4 0.67 1.500 2.0 2.333
Master 5 0.83 1.875 2.5 2.917
Champion 6 1.00 2.250 3.0 3.500
Legendary 7 1.17 2.625 3.5 4.083
Epic 8 1.33 3.000 4.0 4.667

It seems pretty clear that where d8 is the baseline, having to roll d6s is bad, and rolling d12s is good. On average, though, the mean number of successes rarely differ by that much. Going to a larger die certainly can be expected to provide more successes, but not a huge number more, and the full range of successes has the same limit. Nobody in the Master tier can get more than five successes, and even reinforced ability (two bumps up from d8 to d12) means you can expect only one more success.

This suggests that while there are certainly benefits to specialization of ability an reinforcement, it is likely to not be overwhelming.

A more detailed examination for each tier can be found below, for each die type from d6 to d12. Using d8s is the baseline and a target number of 5, d6 indicates some kind of restriction, while d10 and d12 are improvements. For simplicity I show only the net results (successes – complications) rather than all the permutations, and the percentages to two decimal places. Anything less than 1/10,000 is of no interest to me, except that it is functionally not going to be seen in practice.

The ‘d# Chance’ column shows the percentage chance of scoring that many successes or complications when rolling the indicated dice. The ‘d# At Least’ column shows the percentage chance of scoring at least that many successes or complications when rolling the indicated dice. That is, the ‘d8 At Least’ value for the ‘1 row’ shows the percentage chance of scoring a net of one success or more when rolling d8s, while the ‘d8 At Least’ value for the ‘-1 row’ shows the percentage chance of scoring a net of one complication or more when rolling d8s.

I am using my standard tier definitions below, where they exist. “Pre-Basic” is introduced here and does not have a standard definition.

Pre-Basic (Level 0, One Die)

I have difficulty imaging any creatures actually being found in this tier. They have no talents, no levels, nothing. However, There might be very minor tests that are below even the weakest statted creature.

Net d6 Chance d6 At Least d8 Chance d8 At Least d10 Chance d10 At Least d12 Chance d12 At Least
1 33.33 33.33 50.00 50.00 60.00 60.00 66.67 66.67
0 50.00 N/A 37.50 N/A 30.00 N/A 25.00 N/A
-1 16.67 16.67 12.50 12.50 10.00 10.00 8.33 8.33

Basic Tier (Levels 1-4, Two Dice)

This tier replaces the ‘0-level’ characters from AD&D and largely replaces the ‘NPC classes’ from D&D 3e and 3.5. The bulk of the non-adventuring characters in a setting should likely be in this tier.

Introductory adventures and ‘apprentice-level’ adventures might take place in this tier, but these are expected to be uncommon.

Most skilled craftsmen and general soldiers are in this category. They are competent at what they do but usually lead fairly predictable lives.

Net d6 Chance d6 At Least d8 Chance d8 At Least d10 Chance d10 At Least d12 Chance d12 At Least
2 11.11 11.11 25.00 25.00 36.00 36.00 44.44 44.44
1 33.33 44.44 37.50 62.50 36.00 72.00 33.33 77.77
0 36.11 N/A 26.56 N/A 21.00 N/A 17.36 N/A
-1 16.67 19.44 9.38 10.94 6.00 7.00 4.17 4.86
-2 2.78 2.78 1.56 1.56 1.00 1.00 0.69 0.69

Expert Tier (Levels 5-8, Three Dice)

This is the start of the ‘adventuring tiers’, where characters get involved in situations beyond the norm. They are quite within normal human abilities of our world (ignoring such things as magic), if more impressive than normal.

Adventures in this tier are often fairly straightforward, be they dungeon crawls, urban adventures, or simple wilderness adventures.  Most have fairly limited and easily managed scope.

Elite troops (Navy SEALS, Army Rangers, etc.) and Olympic athletes may be good examples of people you might find in this tier. Better-trained, generally mentally and physically tougher than the norm.

Net d6 Chance d6 At Least d8 Chance d8 At Least d10 Chance d10 At Least d12 Chance d12 At Least
3 3.70 3.70 12.50 12.50 21.60 21.60 29.63 29.63
2 16.67 20.37 28.13 40.63 32.40 54.00 33.33 62.96
1 30.56 50.93 30.47 71.09 27.00 81.00 23.61 86.57
0 29.17 N/A 19.34 N/A 13.50 N/A 9.90 N/A
-1 15.28 19.91 7.62 9.57 4.50 5.50 2.95 3.53
-2 4.17 4.63 1.76 1.95 0.90 1.00 0.52 0.58
-3 0.46 0.46 0.20 0.20 0.10 0.10 0.06 0.06

Heroic Tier (Levels 9-12, Four Dice)

At this point characters are reaching or exceeding the limits of real-world credibility (which isn’t to say that they are impossible, just incredible).  They can do things that are somewhat beyond the capabilities of all but the very best in the world.  Where many people fit into the previous tiers and can be described as groups, in this tier we find specific people who made their mark on history.

There are mundane animals from the real world that might be found in this tier — they aren’t necessarily incredible, but they are beyond human capability (most of the major predators are stronger, faster, and tougher than a human can be, for example).

Adventures in this tier are more wide-ranging.  If Expert adventures take place around a single town or small region (such as in a ‘points of light’ setting) and have limited local consequences, a Heroic adventure might involve traveling between regions or have consequences affecting a region.

Julius Caesar.  Genghis Khan.  Isaac Newton.  Sun Tzu.  Joan of Arc.  Albert Einstein.  All could be considered of the heroic tier — though as people who actually existed, probably toward the lower end.

Net d6 Chance d6 At Least d8 Chance d8 At Least d10 Chance d10 At Least d12 Chance d12 At Least
4 1.23 1.23 6.25 6.25 12.96 12.96 19.75 19.75
3 7.41 8.64 18.75 25.00 25.92 38.88 29.63 49.38
2 19.14 27.78 27.34 52.34 28.08 66.96 26.54 75.93
1 27.78 55.56 24.61 76.95 19.44 86.40 15.28 91.20
0 24.77 N/A 14.87 N/A 9.45 N/A 6.41 N/A
-1 13.89 19.68 6.15 8.18 3.24 4.15 1.91 2.39
-2 4.78 5.79 1.71 2.03 0.78 0.91 0.41 0.48
-3 0.93 1.00 0.29 0.32 0.12 0.13 0.06 0.07
-4 0.08 0.08 0.02 0.2 0.01 0.01 0.01 0.1

Master Tier (Levels 13-16, Five Dice)

At this point we are beyond human capability in the real world and have to lean on literary and mythical figures.  For reference, the equivalent levels in D&D have wizards who can teleport and clerics who can raise the dead.

Adventures in this tier might have consequences that affect an entire kingdom.  This might be leading a rebellion against an unjust ruler, dealing with an invasion from the Icy North, recovering a lost artifact that causes dissension and strife in the Grand Temple and a schism among the faithful.

Using those guidelines, the following are possible examples.

  • Vlad Taltos in the earlier Dragaera books.  He can teleport (though he doesn’t like being teleported in the first place, and really doesn’t like doing it himself, he’ll get someone else to teleport him when he can — in D&D it might be dimension door except for the range).  He gets more powerful (probably Champion; I don’t think he’d be higher tier than Morollan or Aliera) later in the series.
  • Belgarath the Sorcerer, the Eternal Man, would probably be at this tier.  He does teleport, but it’s probably closer to dimension door (very limited range) and polymorph.  He might be Champion tier (he and his peers have some ability to affect weather, and he does summon some pretty big demons), but he doesn’t feel quite like it to me.  I’d be inclined to park him here.
Net d6 Chance d6 At Least d8 Chance d8 At Least d10 Chance d10 At Least d12 Chance d12 At Least
5 0.41 0.41 3.13 3.13 7.78 7.78 13.17 13.17
4 3.09 3.50 11.72 14.84 19.44 27.22 24.69 37.86
3 10.29 13.79 21.48 36.33 25.92 53.14 26.75 64.61
2 20.06 33.85 24.90 61.23 22.68 75.82 19.29 83.90
1 25.33 59.18 20.08 81.31 14.31 90.13 10.30 94.20
0 7.21 N/A 11.73 N/A 6.72 N/A 4.15 N/A
-1 12.67 19.17 5.02 6.96 2.39 3.15 1.29 1.65
-2 5.02 6.51 1.56 1.94 0.63 0.77 0.30 0.36
-3 1.29 1.49 0.34 0.38 0.12 0.14 0.05 0.06
-4 0.19 0.21 0.05 0.05 0.02 0.02 0.01 0.01
-5 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00

Champion Tier (Levels 17-20, Six Dice)

At his point we’ve got resurrection and teleport without error.

Adventures in this tier can be expected to have far-reaching effects, often affecting multiple kingdoms or an entire continent.

  • Morrolan and Aliera from the Dragaera books.  They are certainly more powerful than Vlad for much of the series, and certainly less powerful than Sethra Lavode (whom much of the Empire holds in awe).
Net d6 Chance d6 At Least d8 Chance d8 At Least d10 Chance d10 At Least d12 Chance d12 At Least
6 0.14 0.14 1.56 1.56 4.67 4.67 8.78 8.78
5 1.23 1.37 7.03 8.59 14.00 18.66 19.75 28.53
4 5.04 6.41 15.53 24.12 22.16 40.82 25.10 53.64
3 12.35 18.76 21.97 46.09 23.33 64.15 21.60 75.24
2 20.19 38.95 22.06 68.16 17.98 82.13 13.92 89.16
1 23.23 62.17 16.51 84.66 10.59 92.73 6.95 96.11
0 19.27 N/A 9.42 N/A 4.88 N/A 2.75 N/A
-1 11.61 18.56 4.13 5.92 1.77 2.39 0.87 1.14
-2 5.05 6.95 1.38 1.79 0.50 0.63 0.22 0.27
-3 1.54 1.90 0.34 0.41 0.11 0.13 0.04 0.05
-4 0.32 0.36 0.06 0.07 0.02 0.02 0.01 0.01
-5 0.04 0.04 0.01 0.01 0.00 0.00 0.00 0.00
-6 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Legendary Tier (Levels 21-24, 7 Dice)

This is the pinnacle of mortal ability.  Characters in this tier may well be pursuing immortality.  There is true resurrection and wish; the bounds of reality are fairly elastic at this point.

Adventures in this tier might have consequences that span the entire world.

  • Sethra Lavode from the Dragaera books is almost certainly in this tier, and probably near the top end.
Net d6 Chance d6 At Least d8 Chance d8 At Least d10 Chance d10 At Least d12 Chance d12 At Least
7 0.05 0.05 0.78 0.78 2.80 2.80 5.85 5.85
6 0.48 0.53 4.10 4.88 9.80 12.60 15.36 21.22
5 2.32 2.85 10.60 15.48 17.96 30.56 22.41 43.62
4 6.84 9.69 17.69 33.17 22.04 52.60 22.33 65.95
3 13.74 23.43 21.21 54.38 20.00 72.61 16.77 82.72
2 19.89 43.33 19.27 73.65 14.08 86.69 9.91 92.63
1 21.40 64.73 13.66 87.31 7.90 94.60 4.73 97.37
0 17.38 N/A 7.66 N/A 3.58 N/A 1.85 N/A
-1 10.70 17.90 3.41 5.03 1.32 1.82 0.59 0.79
-2 4.97 7.20 1.20 1.62 0.39 0.50 0.15 0.19
-3 1.72 2.23 0.33 0.41 0.09 0.11 0.03 0.04
-4 0.43 0.51 0.07 0.08 0.02  0.02 0.01 0.01
-5 0.07 0.08 0.01 0.01 0.00 0.00 0.00 0.00
-6 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00
-7 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Epic Tier (Level 25+, Eight Dice)

I mention this tier only to make it clear that I expect there to be something after Legendary.  I don’t yet know exactly what this tier will look like, nor anything after that.

Many characters in this tier are effectively immortal… unless something more powerful comes along.  They are likely in direct service to (or in contention with) gods.  They may be learning to manipulate the fundamental elements of reality.

While planar travel may be present in lower tiers, adventures in this tier can have consequences that affect entire planes.

  • The Jenoine from the Dragaera books are probably at this level.  Sethra was exhausted facing off against a single one while in her seat of power.  Defeating several of them took a team consisting of Champion and Legendary characters, at least three Great Weapons (and one ‘Very Good Weapon’), and a few gods… and even then there were casualties.
Net d6 Chance d6 At Least d8 Chance d8 At Least d10 Chance d10 At Least d12 Chance d12 At Least
8 0.02 0.02 0.39 0.39 1.68 1.68 3.90 3.90
7 0.18 0.20 2.34 2.73 6.72 8.40 11.71 15.61
6 1.02 1.22 6.93 9.67 14.00 22.39 19.27 34.87
5 3.52 4.74 13.33 23.00 19.60 41.99 21.76 56.64
4 8.39 13.13 18.56 41.56 20.41 62.40 18.63 75.27
3 14.64 27.77 19.80 61.36 16.66 79.06 12.66 87.93
2 19.37 47.14 16.71 78.07 10.97 90.03 7.03 94.96
1 19.81 66.95 11.36 89.43 5.93 95.96 3.24 98.20
0 15.82 N/A 6.29 N/A 2.66 N/A 1.25 N/A
-1 9.90 17.23 2.84 4.28 0.99 1.39 0.41 0.54
-2 4.84 7.32 1.04 1.44 0.30 0.40 0.11 0.14
-3 1.83 2.48 0.31 0.40 0.08 0.10 0.02 0.03
-4 0.52 0.65 0.07 0.09 0.02 0.02 0.00 0.01
-5 0.11 0.13 0.01 0.01 0.00 0.00 0.00 0.00
-6 0.02 0.02 0.00 0.00 0.00 0.00 0.00 0.00
-7 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
-8 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Closing Comments

Well, that was tedious.

Overall I think the frequencies shown above look workable. By and large going to more dice is better despite the greater chance of having complications, because the complications are usually easily negated by spending successes. The tables show only net successes or complications. If I wished I could go to greater detail in the examination and differentiate between (four successes and one complication) and (three successes and no complications), but I think that is not necessary. Cursory examination of the raw data shows me that the cases where successes and complications cancel each other out are generally the less common cases, and I am quite happy for a PC to accept negative effects for greater positive effect. This is still bound by the total number of successes possible for the tier — a Master-tier character (five dice) cannot get more than five successes, and if there are four successes and one complication the player chooses between three successes and four successes with one complication, still less than the five successes that could be possible. I don’t expect the ability to choose between negating complications and not will prove overpowering.

In the end, more dice are better than fewer dice of the same size, and bigger dice are better than smaller dice of the same number. There is some possibility of more complications in total, largely mitigated by negation by trading successes, and more net complications, mitigated by the small chance that that will happen and the greater capacity of the character to deal with increasing complications.

And really, at the Legendary tier do you really want to see minor complications? Or have complications, as infrequent as they are, commensurate with your own stature? If it’s going to go wrong, Let It Go Wrong.

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2 Comments

  1. Pingback: Task Resolution Using Dice Pools | Echelon d20 - An RPG framework based on the d20 system.

  2. Pingback: Task Resolution Using Dice Pools: Heavy Lifting Part 2 | Echelon d20 - An RPG framework based on the d20 system.

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