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

Character competency, game feel and dice randomness

In the recent years I've explored a number of systems with my group. We've played some games with Godbound, Chronicles of Darkness, Powered by the Apocalypse and Star Trek Adventures systems. I'm a very system-focused player, so I enjoy exploring how the different systems play and what's their "game feel" like. Most importantly, I was really keen on exploring the most core mechanic of almost every role playing game - the dice mechanics.

Character competency

Whenever I would start playing a new system, I would inevitably ask myself - "does this roll as well as World of Darkness"? Vampire the Masquerade has been the first system I've played extensively and say what you will about how clunky it might be at times, the dice rolls in that game felt great. You always rolled a nice handful of dice (but not a crazy amount like Exalted or Shadowrun!), and you felt like your character was competent - it wasn't hard to roll a success, and you could rely on your character performing their specialities more often than not.

To contrast that, we've played about a year worth of Godbound between our Exalted game, Ancalia game, and doing one-shots. The system used a single d20 roll extensively, keeping with its OSR roots. Despite the system being geared towards feeling epic and grand, whenever a roll was required the characters didn't feel nearly as competent. It wasn't that uncommon to get a streak of low-value results and fail entirely, despite playing demigods.

I've been trying to figure out why those systems felt so different, and I think I figured something interesting out...

Binomial vs linear distribution

Two basic, simplified math concepts. Binomial distribution is a randomness distribution that looks like a bell-curve. It's similar to a normal distribution, but there are only a finite amount of values it can take. You can get extreme results from either end, but you're a lot more likely to land somewhere in the middle. 2D6 roll has a binomial distribution - you're most likely to end up with a 7, but 2 or 12 also happen on occasion. Linear distribution means you're as likely to get any one result as another one. 1D6 has a linear distribution - you're as likely to roll a 1, a 6 or a 3.

A useful spreadsheet to reference.

My guess is the difference in game feel between Chronicles of Darkness (the modern, updated version of World of Darkness, here is our game) and Godbound lies in the way dice rolling is handled. In CoD, the more skilled you are, the more dice you roll. Each dice landing on a 8, 9 or 10 is a success, 5 or more successes is an exceptional success. You often roll 5 or more dice. In Godbound, you always roll a single D20, add some modifiers and you have to roll above some threshold - 20 for example.

CoD feels good because you can rely on your dice rolls thanks to the binomial distribution - you know you're very likely to roll at least one success - having just two dice (a very paltry amount) you're already more likely to succeed than fail, 4 dice gives you 75%, 6 is over 88%, and 9 gives you 95%. Godbound feels worse due to the linear nature of its rolls - rolling a 1 is as likely as rolling a 10 or 20. You have to get +10 to reach 50% success rate, +15 gives you 75%, +18 gives you 90%, and you only get 95% at +19.

Why are those percentages important? Because that's basically halving the chance of failure - a half (0.5), a half again (0.25), and again (0.125) and again (0.0625).

Progression between 2, 4, 6 and 9 is a fairly linear one in CoD (the previous editions were a bit harder), while in Godbound you start somewhere in the middle of the progression and polishing up to the final few points is a challenge.

So subjectively at least, it seems having a binomial distribution in your game is the key to having a good game feel. Now, the question is, how many dice do you really need to pull this off?

Many dice, or just two?

I've played with two systems that tackle the binomial distribution in fundamentally different ways. First one is the already mentioned Chronicles of Darkness line of games. The other is Stars Without Number (it has a free edition, do check it out!, and here is our game), a game by the same author as Godbound.

In SWN, a skill roll is just a simple 2D6 roll, modified by your skill rating. So it's the simplest binomial distribution you can really get. The skills just shift the result. You usually have to get 7 or better to succeed at a basic task, 10 at something more complicated, or 12 at something very complicated.

How does this simple roll compare to rolling something like 10D10s in CoD? Well, looking at our handy spreadsheet, pretty favourably all things considered! Rolling 2D6 gives you a simple binomial distribution, and while rolling more D10s not only shifts the curve but also changes its shape a bit, it might not be enough to affect the game honestly (a binomial distribution that is thinner means results are very likely to end up near the centre, while a wider one as we see rolling a lot of D10s means there is a spread in the ranges - you're less likely to end up on the dead centre).

Both systems behave similarly - the higher your skill in SWN or the more dice you roll in CoD, your probability shifts upwards, meaning you're more likely to reliably land a success.

Honestly, as someone that was very much in love with CoD dice mechanics, it's surprising to see that rolling 2D6 is a fairly good substitute for having a handful of D10s.

Conclusions

It appears a binomial distribution from rolling two or more dice instead of one makes games feel a lot more satisfying and gives characters a degree of competency - the players can rely on their characters succeeding at a given task they're specialising in. It seems that you don't really need a lot of dice to achieve this either - rolling 2D6 and shifting the result accordingly might be enough to achieve this game feel. Rolling a single dice is generally the worst due to the linear nature of the probability distribution.

Other resources:

• Dice System Ease, Speed, and Probability Distribution in RPGs