This month, we will look at some simple dice-roll systems and discuss them in terms of probabilities and other properties discussed earlier.

One die to rule them all

The simplest dice-roll mechanism is to use a single die. The result can be modified by ability, difficulty and circumstance in various ways. Probability calculations are easy, as each result on a dN has probability ¹/_N.

There are many single-dice systems, but the best known is the d20 system that originated in D&D. Here, a d20 is rolled, ability is added and a threshold (determined by difficulty) must be exceeded. Some opposed actions are handled by a contest of who gets the highest modified roll, while other opposed actions are treated by using properties of the opponent (such as armour class) to determine a fixed (i.e., non-random) difficulty rating. If the unmodified die (i.e., before attribute bonuses and skills are added) is high enough (e.g., 18-20) and the modified roll is a success, there is a chance of critical success: If a second roll also indicates a successful action, the action is critically successful. As far as I recall, there is no mechanism for critical failures in the standard d20 mechanics.

Diminishing returns are in d20 handled by increasing costs of level increases.

Another single-die system is HârnMaster, where you roll a d100 and must roll under your skill (rounded to nearest 5) to succeed. If the die-roll divides evenly by 5, the success or failure is critical (so there is a total of four degrees of success/failure). The effective skill may be reduced by circumstance such as wounds and fatigue, but difficulty does not directly modify the roll. In opposed actions, both parties roll to determine degree of success and the highest degree wins. Ties in degree of success normally indicate a stalemate to be resolved in later rounds.

Diminishing returns are in HârnMaster handled by letting increases of skills be determined by dice rolls that are increasingly difficult to succeed (you must roll over the current ability on a d100 to increase it).

Talislanta (4th and 5th edition) uses a d20 to which you add ability and subtract difficulty. An ``Action Table'' is used to convert the value of the modified roll to one of five degrees of success/failure. Opposed rolls use the opponent's ability as a negative modifier to the active players roll.

Diminishing returns are in Talislanta (like in d20) handled by increasing cost of skill increases.

So, even with the same basic dice-roll mechanism (rolling a single die), these systems are quite different due to differences in how difficulty and circumstance modify the rolls, how opposed actions are handled, how degree of success is determined and how diminishing returns are achieved.

If we cut each of the above systems to the bone, we have one system where you roll a die, add your ability and compare to a fixed or difficulty-determined threshold. In the other, you roll a die and compare directly to your skill. Though these look different, they behave the same way if the threshold in the first method is fixed: Increased ability will linearly increase the probability of success (until success is certain). Modifiers applied to the roll or threshold will also linearly increase or decrease the success probability. Such linear modification of probability is the most basic property shared by (nearly) all single-dice systems.

I will pass no judgment about which of the above systems is best (and, indeed, this will depend on what you want to achieve), just make a few observations:

• If opposed actions have both players roll dice, but only one player rolls in an unopposed action, there is a larger spread on opposed actions than on unopposed actions.

• In the HârnMaster system, 20% of all failures and successes will be critical, regardless of ability. The d20 system and Talislanta both give higher proportion of critical successes to higher abilities, though d20 tops at 16.7% for critical success (for tasks where 18-20 give criticals).

• Even though HârnMaster uses a d100, the fact that skills are rounded to nearest multiple of 5 before adding them to the roll means that there are only 20 essentially different die-roll results (if we ignore criticals). Hence, HârnMaster is not really any more fine-grained than systems that use a d20. The steps between each multiple of 5 are, however, used to keep track of progress in experience towards the next ``real'' skill level.

• If both attribute and skill modify the roll, the typical ranges of each of these will determine if the game favours training over raw talent or vice-versa. d20 translates attributes to modifiers at the rate of two to one, which makes (unmodified) skill differences more significant than similar differences in attribute (though the latter helps in more situations). This signifies that skills are more important than attributes. A different way of achieving this is to make it cheaper to increase skills than to increase attributes, so skills will, generally, increase more quickly than attributes.

Adding a few dice

A variant of rolling one die is adding up a few dice, but otherwise use the result as above (i.e., adding it to the ability, require it to be less than the ability, etc.).

An example is Stefan O'Sullivan's Fudge system, that to the ability number of the character adds four ``Fudge dice'' that each have the possible values -1, 0 and 1 (so a single Fudge die is equivalent to d3-2). This gives values from -4 to 4 that are added to the ability, which is then compared to the difficulty. The roll has a bell-like distribution centered on 0. Centering rolls on 0 has the advantage that ability numbers and difficulty numbers can use the same scale, so you can use an opponent's ability directly as difficulty without adding or subtracting a base value.

Another way of getting zero-centered rolls is the dn - dn method: Two dice of different colours (or otherwise distinguishable) are rolled, and the die with the ``bad'' colour is subtracted from the die with the ``good'' colour. The distribution is triangular and equivalent to dn + dn shifted down n+1 places (i.e., to 2dn-n-1). Yet another way of getting the same distribution is, again, to roll a good die and a bad die, but instead of subtracting the bad from the good, you select the die with the smallest number showing and let it be negative if it is on the bad die and positive if it on the good die. Ties count as 0. For example, if the good die shows 6 and the bad die shows 4, the result is -4. Using this method, you replace a subtraction by a comparison, which many find faster. It takes a bit more effort yo explain, though.

Also equivalent to dn - dn is letting both sides in a conflict add a dn to their abilities and then compare the results. For unopposed actions, the GM acts as opponent, so he adds the dn to a predetermined difficulty number. This allows the GM to hide the exact difficulty of an action (or ability of an NPC) from the players by rolling the opposing die secretly. It also means that players get to roll whenever they are involved in an action (even if they are on the receiving end), which keeps them active.

In general, having one side roll dn and the other roll dm is equivalent to letting the first side roll dn-dm or dn+dm-(m+1). So there is no basic difference between having both sides roll and only one side roll (apart from constant offsets), so long as the the total number of dice rolled is the same.

The advantage of dn - dn (or equivalent) over Fudge dice is that you don't need special dice. But you do need players and GMs to be in agreement of which dice are good and bad before the dice are rolled. If you use dn+dn-(n+1), you don't need this agreement, but you need one more arithmetic operation.

Since zero-centered dice-rolls (by definition) always have average 0, you can fairly easily get different degrees of randomness. For example, with dn - dn, you can use different n for different tasks: If the task has a low degree of variability, use d4s, if it has average variability, use d8s and if it has high variability, use d12s or even d20s. With Fudge dice, you can use three, four or five Fudge dice in a roll depending on how variable you want the result to be.

All of the above have non-flat distributions (and if more than two dice are involved, the distribution will be a bell curve), so adding a constant modifier will not increase the probability of success by a fixed percentage (as it does in single-dice systems). Some people dislike this by saying that the same modifier benefits some people more than others, but you can argue that this is the case for single-die systems too (if you, for example, look at the relative increase in success chance). So, again, it boils down to what the designer wants to achieve.

Instead of adding a number of dice and comparing the sum to an ability rating, you can compare each die value individually to a (lower) ability rating and count how many are below the threshold. You can then directly translate this number into a degree of success. If three or more dice are rolled, the degree of success will have an asymmetric bell-like distribution which is skewed towards low or high results depending on whether the ability is lower or higher than the mean value of a die. This mechanism limits the effective range of abilities to the range of a single die, but for games that operate with low granularity of abilities, this won't be a problem. And a d20 should accommodate enough ability levels to satisfy most.

Coming up...

Next month, we will look at more complex systems, including systems where the number or type of dice may depend on character ability.