Equivalent Dice Theorems of RPGs

My group and I have played a good amount of PbtA games (Fellowship, Legacy, Dungeon World, etc.). After getting used to them we did a one shot game of iHunt, which used the FATE system. During the session my GM remarked how FATE is making him roll again to set our difficulty and how he got used to not having to do that in PbtA. This got me thinking - "was that roll even necessary?", which lead me down a math rabbit hole...

Lets back up and start from beginning.

FATE dice rolls

The FATE system uses FATE dice, an alternative set of D6s that can roll +1, 0 and -1:

FATE Dice

To figure out how much you rolled, you take four FATE Dice, roll them, add their results together and add whatever skill modifier your character has. Then that is either compared to a static number determined by the GM for a "passive opposition", or another roll with modifiers for an "active opposition".

The second situation was what my GM remarked about, and when you think about it - you really don't need to have more than one side rolling dice in this system.

FATE Dice are a bit different from the standard dice - their average roll is a "0", and you have both positive and negative 1s on it. The dice is symmetrical - it doesn't matter if you roll a FATE dice or its opposite, the result is the same.

So if you wanted to avoid the GM having to ever roll dice, you would just make the player roll 8 FATE Dice and give them a passive opposition instead and it would be exactly the same roll (4 GM dice turn into the player rolling 4 opposite dice, which in this system is the same as normal dice, therefore 4+4=8 dice total roll).

This got me thinking - could something similar be done in other systems?

Equivalent dice and rolls

After thinking about it, turns out you can do something similar. Here is a more formal explanation of what that entails if you like math, but to summarise it based on D6s:

Rolling a D6 and rolling "7-D6" is the same - you get the same results. Based on this you can turn any versus roll into a single roll by one side that uses all the dice vs a static number.

If you subtract the average of 3.5 from every side of the D6, you get a symmetrical die D6Sym with sides {-2.5, -1.5, -0.5, 0.5, 1.5, 2.5}. Based on that, rolling a D6 and rolling "3.5+D6Sym" is the same. While this doesn't help much by itself, it allows you to easily make a statistical analysis of rolls involving multiple dice (since the average will always be 0, so you can easily compare these binomial distributions).

Based on the last one, I did some programming to figure out the statistics of rolling various amounts of dice...

Dice roll statistics

This part is probably the hardest to understand. Basically, it boils down to this:

The goal was to figure out rolling how many dice is "good enough" - when you don't need to roll more dice to get "random enough" results.

The more dice you roll, the closer the results is to a binomial distribution, but there are some diminishing returns. After you roll about 3-4 dice the results don't get much better.

Size of the dice rolled doesn't change things that much beyond making the results more granular. Rolling 5D4 is comparable to rolling 5D12.

So where does this all lead us?

Conclusions

When designing a system, you don't really need to roll a lot of dice - rolling more than 3-4 gets a bit excessive and doesn't improve the probabilities of the roll too much.

When you have a versus roll, you only need to have one side of the conflict roll, while the other would provide a static difficulty. The exact math of a roll can be a little complicated, but it's mostly fixed for any given amount of dice.

If you don't want to roll a lot of dice, you can instead roll fewer but bigger dice to get a granular enough result (again providing you're rolling those 3-4 dice).

So after all that, I can say that the GM never needs to roll dice in FATE - the 4 FATE dice the player rolls should be good enough of randomness in most situations. The rest would be taken care of by a static difficulty for them to beat based on how challenging the enemy is.

The same principles could be applied to a lot of systems. Maybe not something that involves a lot of dice manipulation and tricks like CORTEX, but others - maybe. There is definitely room for some systems designed from the ground-up to minimise the amount of rolls you make (similarly to how Chronicles of Darkness limited the amount of chain rolls).

Related articles:

• Make every roll impactful

Related links:

• Central Limit Theorem
• Reddit threads 1 & 2