There are many commercial roleplaying games out there, but often the best games are ones you design yourself.
In my previous posts I tried to outline the basics of RPGs and their component conflicts. In this post I will explore some of the ways different RPGs model chance.
Most RPG dice schemes can be generalized as rolling k different n-sided dice (commonly abbreviated kdn) and comparing the results to some pre-determined pattern. Those patterns are typically one of the following.
In a few cases, special dice are used that don’t have the normal 1-though-n numbering; for example, FUDGE uses dice with sides +1, 0, and –1 (written dF); doubling dice (sides 1, 2, 4, 8, 16, and 32) and dice with pictures instead of numbers are also sometimes used.
It is common for RPGs to include some type of special-casing in order to handle very low-probability events. This typically takes the form of either “criticals”, where a 1 is always a failure and a n is always a success, or “exploding”, where each n adds another die to the pool.
It is also not unusual for damage to use different dice than do contests. This is done because damage is about how much is done rather than if an attempt succeeds.
Some examples from successful RPGs:
Always roll 1d20, and try to meet or exceed a cut-off value. Usually expressed as 1d20 + skill ≥ difficulty instead of 1d20 ≥ difficulty – skill, but the two are mathematically equivalent.
Sometimes these games use “opposed rolls”, which mathematically means 2d20 is compared to a cut-off – 1dn = 1dn – (n + 1) (because “1d20 + a ≤ 1d20 + b” is equivalent to “2d20 ≤ b – a + 21”).
Always roll 4dF and try to meet or exceed a cut-off value. This is the same as 4d3, since 4dF ≤ x is the same as 4d3 ≤ x + 8. Getting significantly more or less than the cut-off value has meaning.
Always roll 3d6, and try to keep the total under a cut-off value.
I’ve never actually played, but I hear that L5R rolls (skill + attribute)d10k(attribute) to meet or exceed a cut-off value. Explosions are kept if their originating dice are kept.
Roll 1d(skill) to meet or exceed a cut-off value, usually 4. Dice explode.
Roll (skill + modifier)d(6 ≥ 5) and try to meet or exceed a cut-off value. Every roll of kd(6 ≥ 5) ≥ x is also treated as a roll of kd(6 ≥ 2) ≤ (k + 1) ÷ 2. Some dice explode, but most do not. The exact amount by which the 6 ≥ 5 check succeeds or fails is significant, but the 6 ≥ 2 check is a simple pass/fail.
I include this example to point out that dice can get complicated…
Roll (skill)d(attribute) to meet or exceed a cut-off value. Dice explode. I cannot seem to recall which game this was… probably an older version of something.
Whatever the method used, rolling dice always comes down to a probability. There are also other issues to ask: what is the resolution (i.e., how much “barely better” can I be?) and how easily do probabilities map to numerical character attributes?
The graphs in this section use SVG; at the time of writing, they do not work in Internet Explorer. In this section I present sever graphs. Each shows the odds of getting at least a particular value (in black) as well as the odds of getting a specific value (in red). In all cases the first black bar’s height is 1 (100%).
Rolling one die gives equal chance of any outcome. Rolling two heaps probability in the middle region; three or more starts to become bell-shaped.
A single die’s histogram:
2d20:
3d6:
4dF:
8dF:
The overall impact of having k > c in these rolls is to make low rolls unlikely. What follows is various kd6kc; the left column is xd6k1 while the diagonal is xd6kx (or simply xd6).
This approach gives a left-leaning distribution that peaks at ((n – c + 1) ÷ n)k.
Histograms from 1, 2, 5, 9, and 20 dice:
Dice are not the only way to resolve action. Some games use a deck of cards or other source of randomness. Some resolve every interaction purely according to the GM’s pleasure. Some have some sort of trade-off mechanic by which players can choose which attempt will fail and which will succeed.
As with all elements of RPGs, the only limits are your imagination and the tolerances of the other participants in the game.
Looking for comments…