## Musings on a Dice Progression Mechanic

For many years I have had interest in implementing an opposed-die roll dice progression mechanic in a game. Many years ago Cory Ring and I wrote a small set of rules for the HMGS MidSouth Dispatch (newsletter) that featured such a mechanic. The problem is that there isn’t enough variance between a d4 and a d12 and then there is the big gap between d12 and d20. The gap can be filled with two dice, but then you don’t get the same uniform distribution of results that a single die achieves.

Recently, I found a company (http://mathartfun.com/shopsite_sc/store/html/DiceShop.html) that sells d14, d16, d18, d22, and d24. I wrote to them, and they were able to sell me 10 of each such that each type of die was a unique color. Since these are uncommon shapes I wanted to be able to say, “roll the blue one and always mean the d14 — or whatever shape was blue. They arrived recently, and I have begun to think about how to employ them.

The basic notion is that abilities would have a base die as a part point. Modifiers would then change the die rolled. The attacker and defender would each roll a die, with the higher roll winning. I have also thought it might be interesting if the difference in the rolls somehow indicated the level of success. For instance if the attacker’s roll is three times the defender’s that might indicate some sort of critical hit.

On a recent flight for work, I began to wonder about the probabilities of winning under these types of rules. One of the reasons that this die progression approach appeals to me is that someone rolling a d4 ** COULD** defeat someone rolling a d24. But what is that probability? So out came Excel. The table below shows the chance of the attacker (rolling the dice along the left of the table) defeating a defender (rolling the dice across the top of the table).

So, if an attacker roll d4 and the defender rolled d24, the attacker would have just a 6% of winning. Note that the attacker must roll *higher* than the defender to get a hit, so ties go to the defender. On the other hand, if the attacker rolled d24 and the defender rolled d4, the attacker would have a 905 chance of winning. Again, ties go to the defender.

Looking at this chart, I was pretty happy with the way the probabilities laid out. Then I stated wondering why things weren’t summing to 100%. For instance, why was P(Victory, d4 vs. d24) + P(Victory, d24 vs. d4) not equal to 1? Then Duncan made a comment that helped me figure it out. It’s those ties. Since some rolls are losses for both d4 vs. d24 and d24 vs. d4 those were the missing percentages.

The table (above) shows the probabilities of ties that are always failures. For a d4 vs. anything, there are 4 rolls that are always ties: 1:1, 2:2, 3:3, and 4:4. For d4 vs. d4, this is 25% of the total rolls possible (16). To check my math, I then inverted the first table…

so the defender is down the left and the attacker is across the top. Then I added all three tables together, yielding this:

Except for one cell (it looks like two, but this table is symmetrical about its diagonal) at 99%, all the math adds up. I rechecked all the math and didn’t find an error, so I’m chalking it up to round-off errors.

So, if anyone has stayed with me this far, I think the math shows that from a probability standpoint, the die progression mechanic is viable.

I am planning to implement this with something melee heavy so that weapons get a base attack die and skill and circumstances modify the die. The defender’s armor gets a base defense die, with skill and circumstances modifying it. I may try this in a couple of weeks with some Robin Hood figures.

## Comments