Captain Anorak's
Guide to Gaming
Dice Rolling Systems
Task resolution means that a character tries to do something
(a task) and there is a chance of him succeeding or failing dependent upon
his stats. In most game systems, the task is resolved (ie. the success or failure
of the task is determined) by rolling dice. This is also called a test.
Some task resolution systems determine not only whether the task is a success,
but the degree of success of the task.
LINEAR SYSTEMS
A linear system for dice-rolling is one in which a set of dice is rolled,
and the result is compared to a number derived from the character’s stats,
and possibly other factors like difficulty.
There are basically two common linear systems:
The roll-under system: A character has a stat value. A modifier may
be applied to this to represent factors such as difficulty. Some set of dice
is rolled. If the roll is less than or equal to the stat value (plus any modifier),
the test is a success. If the roll is greater than the stat value, the test fails.
A degree of success can be determined from the number of points by which a
test succeeds or fails.
The matching-totals system: A character has a stat value. The GM
chooses a target number. Some set of dice is rolled and added to the stat value
to give the test’s total. If the total is greater than or equal to the target
number, the test is a success. If the total is less than the target number, the
test is a failure. A degree of success can be determined by the number of points
between the total and the target number.
These two systems add up to the same thing mathematically, but they
get there by different routes. To illustrate this, observe these two systems.
In both systems, the stats of normal human characters fall in the range 3 to 7,
with an average of 5. Anyone with a stat below 3 would be disabled, or with
a stat above 7 would be supernormal.
Roll-under: The following difficulty modifiers are added to the stat:
| Very hard |
Hard |
Medium |
Easy |
Very easy |
| -4 |
-2 |
+0 |
+2 |
+4 |
To make a test, the difficulty modifier is added to the character’s stat, and
then for success a D10 roll must be less than or equal to this value.
Matching totals: Target numbers are determined based on
the difficulty of the task:
| Very hard |
Hard |
Medium |
Easy |
Very easy |
| 15 |
13 |
11 |
9 |
7 |
To make a test, D10 is rolled and added to the value of the stat to give
the total. The test is a success if this total is equal to or greater than
the target number.
Both these systems give the following chances of success:
|
Stat: |
3 |
5 |
7 |
| Diff |
|
|
|
|
| Easy |
|
50% |
70% |
90% |
| Medium |
|
30% |
50% |
70% |
| Hard |
|
10% |
30% |
50% |
So, roll-under and matching-total systems ultimately boil down
to exactly the same probabilities. The difference between them
is in how the GM and players think of them. It’s all about the level
of difficulty. Using roll-under, the GM may choose a
difficulty modifier. Using matching-totals, the GM must
choose a difficulty level. The psychological difference between
these two things is crucial.
If the GM is not forced to think about the level of difficulty,
then the temptation is great simply to forget about difficulty
altogether. Thus with a roll-under system, the GM is tempted to
say ‘Make a roll,’ without ever considering the difficulty. We then
have a situation where if a character has Climb skill 54% then he
succeeds at climbing tasks 54% of the time, regardless of how
difficulty they are. Whether climbing over a garden wall or scaling
a sheer ice-face he still has a 54% chance of success. This is
plainly absurd.
You may well object that a good GM will think about the
difficulty for any roll, and so will always apply the appropriate
modifier. But my experience as a GM is that even when I
intend to do this, often in the breach I don’t think about it, but
just say ‘Make a roll,’ without considering difficulty. Also, if you’re
writing a game for other people to run, you can not assume that
the GM will always be the sort of person who will think about this
as a matter of course.
THE SPREAD QUESTION
Spread is the range of possible values which an entity
in the game system can possess. There are three spreads in
linear systems: the stat spread, the dice spread, and the difficulty
spread. In the example systems described above, the stats are
rated from 3 to 7, so they have a spread of 5 possible values. The
set of dice used is 1D10, which has a spread of 10. The range
of difficulties has a spread of 9 points (from +4 to -4, or from 7 to
15).
It is important to think about the relative spread of stats, dice
and difficulties when designing a game system. The ratios of these
spreads influences the outcomes of tests. Here are some
examples of the different ratios of spreads in different games:
GURPS: It’s hard to define the exact spread of
stats which a character can have, because no upper limits are
applied, but almost all normal humans have primary stats in the
range 6-14. A few special characters may go outside this
range, but it’s pretty rare. So we can say that the stat spread is
about 9. The dice set used is 3D6, which has a spread of 16.
The roll-under system is used, and as far as I recall at the time
or writing, difficulty modifiers run to about +2 to -2 for most tasks,
a spread of 5.
Stat spread: about 9
Dice spread: 16
Difficulty spread: about 5 (though extreme tasks may be more)
MegaTraveller: Primary stats (in the form in which they
are used in tests) have values 0, 1 or 2. Skill values of 0, 1 or 2
are fairly common,
but anything higher is rare. A stat or skill may be used on its own, or
two but no more may be added together. Thus overall, the stat added
to the roll will generally be 0-4.
The matching-total system is used, with the dice set 2D6.
The dificulty levels range from 3 (Simple) to 15 (Formidable).
Stat spread: 4
Dice spread: 11
Difficulty spread: 13
All Flesh Must Be Eaten: Primary stats range from
1 to 5, 1 and 5 being very rare. The matched-total system is used,
with the dice set 1D10. I’m not sure about the difficulty levels.
Stat spread: 5
Dice spread: 10
Difficulty spread: ?
CHANGES IN THE SPREAD
Call of Cthulhu: Rolls on primary stats are a bit funny
in CoC, because they require a percentile roll on a stat
multiplied by some factor. For instance, PCs have Intelligence
2D6+6 (max 18, med 13, min 8, spread 11). A percentile roll on
INTx5% would be easy-ish (the average character has a 65%
chance of success), while INTx3% is hard-ish (average 39%).
The complication is that the spread changes with difficulty.
In an INTx5% roll, characters of INT 18 and INT 8 (the
maximum and minimum possible values for PCs) have
respectively 90% and 40% chances of success, a spread of
51. In an INTx3% roll, the same characters have chances
54% and 24%, a spread of 31. In the very hard INTx1% roll,
the the chances are 18% and 8%, so the spread has dropped to 11.
So as a side-effect of making the rolls more difficult, the game
system has also, incidentally and unintentionally, made the spread
smaller as well.
NON-LINEAR SYSTEMS
THE DICEX SYSTEM
Dicex
is roughly the system used for the old Star Wars game
(actually a stat was listed as a dice value not a number, but it works
out the same way). When a stat roll has to be made, dice are rolled
dependent on the stat. The roll is 1D6 for each full three points of
stat, plus one for each point of stat left over. Thus a stat of 7 would
roll 2D6+1, a stat of 9 3D6. A task has a difficulty level, and if the
roll of the dice equals or beats this level, it's a success.
This is a very nice system to play. Mathematically it's rather
imperfect: look at the numbers.
| Stat |
Dice |
Minimum roll |
Mean average roll |
Maximum roll |
| 3 |
1D6 |
1 |
3.5 |
6 |
| 4 |
1D6+1 |
2 |
4.5 |
7 |
| 5 |
1D6+2 |
3 |
5.5 |
8 |
| 6 |
2D6 |
2 |
7 |
12 |
| 7 |
2D6+1 |
3 |
8 |
13 |
| 8 |
2D6+2 |
4 |
9 |
14 |
| 9 |
3D6 |
3 |
10.5 |
18 |
| 10 |
3D6+1 |
4 |
11.5 |
19 |
| 11 |
3D6+2 |
5 |
12.5 |
20 |
The mean roll is always slightly above the stat value (by about
10-20%). But the maximum and minimum values jump around pretty
erratically. Worst of all, a higher stat can have a lower minimum value -
for instance, a stat of 5 has a mininmum possible roll of 3, but for a stat of
6 the minimum is only 2. This means that a character with a stat of 5 can not
fail at a difficulty 3 task, but a character with a stat of 6 can.
However, these irregularities are minor inconveniences and I still think
it's a good, playable system. It is really easy to use, convenient, and it
converts a fixed number into a random number of about the same value, which
can be very useful.
THE ICON SYSTEM
The 'Icon System' is used by Last Unicorn games for their RPGs like the
Star Trek series. It works like this: a character has a stat (human
average 2). The player rolls a number of D6s equal to this stat. One is called
the 'drama die'. If the drama die is a 6, then the result is the value of the
drama die plus the next highest die. If not, then it is simply the highest
value of any die. If a roll is being made on a skill, the stat is rolled as
above and then the skill is added to the result.
For a simple stat roll, the result spread is as follows:
| Result: |
1 | 2 | 3 | 4 |
5 | 6 | 7 | 8 |
9 | 10 | 11 | 12 |
| Stat value 1 |
16.67% | 16.67% | 16.67% | 16.67% |
16.67% | 16.67% | 0.00% | 0.00% |
0.00% | 0.00% | 0.00% | 0.00% |
| Stat value 2 |
2.78% | 8.33% | 13.89% | 19.44% |
25.00% | 13.89% | 2.78% | 2.78% |
2.78% | 2.78% | 2.78% | 2.78% |
| Stat value 3 |
0.46% | 3.24% | 8.80% | 17.13% |
28.24% | 25.46% | 0.46% | 1.39% |
2.31% | 3.24% | 4.17% | 5.09% |
| Stat value 4 |
0.08% | 1.16% | 5.02% | 13.50% |
28.47% | 35.11% | 0.08% | 0.54% |
1.47% | 2.85% | 4.71% | 7.02% |
| Stat value 5 |
0.01% | 0.40% | 2.71% | 10.04% |
27.02% | 43.15% | 0.01% | 0.19% |
0.84% | 2.25% | 4.75% | 8.63% |
I think this is a really really stupid system. With two or more
dice, 5 times in 6 it generates a number from 1 to 6, and 1 in 6 times
it generates a number from 7 to 12. Whether your stat is 2 or 5, you
have a 1 in 6 chance of rolling a number above 6. So what it basically
comes down to is this: regardless of your stat, you have a 1 in 6 chance
of rolling high.
A quick look at the probability distributions shows that they are
very uneven. For a stat of 3 or more, the chance of rolling 6 is high
(above 25%), but the chance of rolling 7 is very low (below 0.5%) and
then the probability increases for higher results. This makes a
probability distribution with a big dip in the middle. I don't see why
anyone would want to make stat tests with a distribution like that.