Alternative Advantage and Disadvantage Rule

Dungeon Crawl Classic (DCC) has uses the dice chain to roll advantage and disadvantage ability checks. Advantage and disadvantage rolls use a higher (+1d) and lower die (-1d) in the dice chain chain, respectively. Ability checks with advantage may use a d24, and ability checks with disadvantage may use a d16. Advantageous and disadvantageous situations may also stack going twice or multiple times up and down in the dice chain (e.g., d20 +2d = d30). This is interesting comparing with the D&D 5e method, of rolling 2d20 and using either the highest or the lowest value. As compared before, the DCC method allow higher values to be rolled in advantage, or the maximum possible roll to be capped down in disadvantage. However, the DCC method requires the use of the Zochi dice (a.k.a. funky dice). 

I've heard of a couple of DMs using an alternative system using the normal, more available D&D dice. In this system, the d20 is summed or subtracted with another die from the standard D&D dice chain: [d4, d6, d8, d10, d12, d20]. It works exactly like the bardic aid from D&D 5e, 

\[ d20 \pm [d4, d6, ..., d20] + \textit{mod} \quad \textbf{vs} \quad \textit{DC} \]

Let's now look to the probabilities of this kind of dice roll to see if we like it.

Probabilities

I will use same kind of matrix representation to show the dice result as before. Because it is easy to scrip this in Python, I will show all matrices in a row.

\[d20 \pm d4\].

\[d20 \pm d6\].

\[d20 \pm d8\]

\[d20 \pm d10\]

\[d20 \pm d12\]

\[d20 \pm d20\]

Most values repeat on diagonals, but the values near the top and the bottom don't. In general terms, the probability distribution function is the same uniform 5% in the middle, and a wedge is cut from one side to be added to the other. The size of this wedge is the size of the advantage/disadvantage die. When both die are of the same type (d20 + d20) the roll probability distribution is a triangle.

Advantage

This is very clear in the advantage case. The average value rolled increases by the average of the advantage die (i.e., 2.5 for d4, and 3.5 for d6). In the case of probability of success (taking all values equal or higher the DC), this results in an increase of almost 5% per extra modifier. We can see below that d20 + d4 increases the success probability in about \( 2.5 \times 5\%  = 12.5\% \) as compared to rolling d20 alone. It also has significant chances of rolling 20, 21, 22 and 23, and no chance of rolling 1.

Alternative advantage rule summing an extra die. The reference is the d20.

Disadvantage

Same thing, but the wedge is not taken from the right side (high values) and passed to the left side (low values). 

Alternative disadvantage rule subtracting an extra die. The reference is the d20.

We may not care about negative DCs...

Alternative disadvantage rule subtracting an extra die. The reference is the d20. In fact, there is no negative DC in D&D...

Critical Success and Fumbles

Do we keep the 5% chance of critical success or fail? It doesn't make much sense, since those change with both the DCC and 5e rules. It is a bit more complicated to define a fair critical rule with this adv-dis rule. What is fair?

• In advantage, a fumble is a 1d20 + d = 2? That is about 1% chance.
• In advantage, a critical success is \( 1d20 + d \geq 20 \)? That is about 15% chance for d4, and 20% for d6 etc. In D&D5 it is about 10%.
• In disadvantage, a fumble is \( 1d20 - d \leq 1 \)? About 15%, 20% etc...
• In disadvantage, a critical success is \( 1d20 - d = 20 \)? About 1%...

Conclusions

• In terms of probability, this system is not too off from older versions of D&D (<5e), where +/- 1, 2, ..., 5 were given.
• The advantage-disadvantage system of D&D 5e changes the probability of DC 11 by +/-5 -- more than any other DC. For DC 1 and 20, the change is less than +/-1.
• The DCC system is always uniform.
• This system require less weird dice. * yey!
• This alternative system shown here is mostly uniform, but it works almost like D&D <5e. * yey!
• Rolling the extra die may be more fun than adding a +2. * yey!
• A fair critical success and failure rule can be used but it is a bit complicated to roll. Too much thinking. ! boo!
• This is the main drawback of the system.