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?
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).
Over the last few years my group and I have played a number of systems that had mechanics you could game to get XP or other advantages, or had some of their mechanics boil down to a solvable math problem. Both of those situations ended up detracting from the experience, either drawing more attention to themselves rather to the game being played, or just being bland mechanics.
RPG mechanics as math problems
The first category of mechanics are essentially math problems - mechanics that for any given situation have a correct solution on what to do to maximise your outcome.
This attack basically gives you a lot of extra dice to a roll, but makes the roll Difficulty higher. It's basically a move you want to use either when you need a hail mary, or the Difficulty is already so low it doesn't matter much. For other scenarios, whether or not to use this move would require running the numbers, but there is still a definitive yes or no answer to whether using it is a good move or not. Figuring it out for certain however requires some complicated math of using your AnyDice-fu.
DOGS Growth
DOGS presented a similar math problem to its players when it came to Growth.
DOGS is a system in which you have stats that take form of multiple dice of a given size - "3D4", "4D6", etc. When you undergo Growth, you get to either increase the number of dice for a given stat, or increase the size of those dice - so from "3D4" you can go into either "4D4" or "3D6". Turns out there is an optimal way of progressing through those dice to get the best result on average:
So for example, 3D6 gives you on average 0.5 higher roll than 4D4, while 5D8 is better than 7D4 by 5 whole points on average. The game doesn't explain those concepts to the players and it's simple and abstract enough that these things shouldn't matter, but for a problem-solving player it's a solvable mechanic.
Mouse Guard and optimal combat
Mouse Guard is a system with its own little combat / conflict engine that relies on picking actions (Attack, Defend, Feint and Manoeuvre) and seeing how they interact with one another. Attack lowers opponent's Disposition (HP essentially), Defend heals your Disposition, Manoeuvre is a way to get an advantage on next rolls, while a Feint is like an Attack with caveats - if played against an Attack, Feint does nothing, but if played against a Defend, Defend does nothing.
During our first game of Mouse Guard we soon learned that this setup creates a simple First Order Optimal Strategy - just always Attack. Attack vs Attack or Defend gets you closer to resolving the conflict, Attack trumps Feint, and Manoeuvre often isn't useful enough to trump dealing damage to an opponent. Attack, Attack, Attack!
FOO (First Order Optimal) Strategy
Towards the end of the game this has ended up being such a simple and optimal strategy that for our next game of Mouse Guard we had to switch the rules to give Attack a hard counter not to devolve every conflict of any type into "press A to win if you press it faster than your enemy".
Cortex and marginally useful SFXs
Recently our group picked up the modular system Cortex. We only played a few sessions of it so far, but one thing that stood out to me was how "starchy" (boring) some of its special powers were.
First of all, Cortex is another mixed-die system that puts a big emphasis on making pool of dice and manipulating your dice. So if you're Iron Man, you can have say, Eccentric Billionaire at D10, Ganius Scientist at D12 and Mk1 Iron Man Suit at D6 and roll those all together to do something.
On top of that, one module you can use in Cortex are Power SFX (special effects). Those are some extra powers your character can use that are tied to a Power Set that can alter the game a bit. So for example you can be Tony Stark with Iron Man Power Set, and one of your SFX could be "Immunity" where you spend a Power Point to negate a specific attack, simple enough.
However, a lot of those SFX are boring dice manipulators. For example - Focus lets you take two dice and turn them into one bigger die. Boost lets you shut down one power to increase the die on another die. Dangerous gives you an extra small die to roll, but changes the size of other dice. Multipower lets you use more than one dice from a given pool but they all are decreased a step or more. Versatile lets you split one die into two or more smaller dice. The list goes on.
I've ran some numbers and a lot of these powers are marginally useful. Say, turning 3D8 into 2D8+2D6 with Versatile gives you an average roll that's 0.42 higher, but gives you 0.45 smaller Effect and 0.2 more Hitches. Without going into what those are, those numbers are marginally useful. Sometimes the numbers increase marginally, sometimes they decrease, but from what I've seen it's not a big effect overall.
Unfortunately to get those numbers I had to spend a few hours programming and debugging a Cortex dice simulator. It's hard to make an informed decision as to whether a power is useful or not without a chart, and trying to play a game well that's filled with unknowable probabilities would just be the case of blind luck.
One way or another - the stat-focused SFX and similar mechanics can be one of two things - either boring because they don't change much about the roll, or having an optimal way to play it, in which case you're not engaging with the mechanics, you're solving a math problem. Either way the mechanics become irrelevant because they're either "use them always", "don't bother with them ever", or "use them under specific circumstances". Since Cortex is based on complex math with no glaringly obvious answer, I honestly can't be bothered to use these SFX.
Honourable mention - Exalted, Paranoia Combat
Honourable mention in this category should also go to Exalted. I won't elaborate much on it since this section is already getting long, but there are two things that are worth mentioning that make this epic game of sword and sandal capital H Heroes boring: Paranoia Combat and Minuscule Incremental Charms.
Paranoia Combat was a strategy from 2nd edition Exalted where the optimal way of winning the Rocket Tag combat was to turtle up and play in the most boring way possible.
Minuscule Incremental Charms were special powers you could buy with XP that would give you just small bonuses to rolls or change tiny things that were rather boring in themselves. Things like Triumph-Forged God-Body that gave you double-9s on Athletic rolls instead of double-10s, or Wyld-Forging Focus that started wyld-shaping at a higher phase. All of those were such small tweaks that they might not be worth the mental load, and weighing their effect vs XP cost would be a small math problem in itself.
Gaming mechanics for profit
Most RPG mechanics that you can game for profit I've come across were focused on being able to farm XP, or at least streamline the way you earn XP. While not a problem in itself (who cares if the party got more XP if they're having fun doing it - you're not competing with anyone), it can start to become a problem when it draws too much attention away from doing things in the game and having fun and onto "brrrr the number goes up"
Chronicles of Darkness - punch me in the face for XP
In a similar vein, the systems also let you earn XP by a number of other ways, like turning fails into botches. This can create some animosity between players when someone is invested in some scene going well, while other players are there to mess things up just to farm up some extra XP - "I failed to impress this character, I opt to botch it instead and make them hate us. Too bad they knew something about your lost sister, guess we'll never find what they knew!".
DOGS - Growth vs Consequences optimisation
Another entry for DOGS, this time about maximising the rate at which your character growths, as opposed to optimising how they grow.
DOGS is a system where you Grow when you suffer Consequences as a result of a conflict you had. To become stronger, you have to get into conflicts, get beaten up a bit, etc. However, Consequences can also have lasting effects if they are bad enough - if you roll too high on them, you may even have to step down your stats, essentially netting you zero, or potentially giving you some net negative sessions. Once again, there is a mathematically optimal way of playing:
Which is basically to get a 3D4 Consequence - it has the best chances of being a net gain. You get such small Consequences by essentially keeping non-violent in conflicts, which to an extent is a "mechanic as a metaphor" for the system.
Mouse Guard and farming Checks
Mouse Guard is a system where you grow your character by practice - aka the more you use a skill, the better you get essentially. As with any such system, the first way of farming it is by doing things all the time, which can encourage you to hog the spotlight. This can be a bit of a problem, but then there is more.
The game is broken up into two parts - the GM turn and the players' turn. During the GM's turn (which lasts about half of the session, not "a turn"...) you can earn "Checks", which you spend during the players' turn to do things and make rolls. You earn those Checks by using your Traits against yourself ("I am Small, therefore I have problems lifting this large log!"). You can use a Trait against yourself once per roll, which means the more you act and roll on GM's turn, the more Checks you can earn to act more during the players' turn.
Moreover, during a conflict you can easily earn a lot of Checks if you play in a very boring way. Essentially, during a conflict you pick actions to take - Attack, Defend, Feint and Manoeuvre. When you Defend, you essentially try to recover your HP. Since the conflict only ends when one of the party's HP goes down to zero, if you turtle up you will be rolling for a long time, letting you earn Checks for every roll. In a lot of cases you can also earn a lot more Checks during a fight under specific circumstances - breaking a tie in enemy's favour or giving an enemy more dice in a vs conflict. So if you play like a turtling asshole and have enough dice, you can in theory earn a lot of Checks.
This strategy has one counter though, Feinting makes you unable to roll Defend. You can try anticipating it though by throwing an Attack that trumps Feint into the mix to make your opponent have to Defend and recover. It's not perfect, but it can work...
While that turtle Defence is an extreme example, I have played in some sessions where a less extreme form of Check farming was involved, which later resulted in pretty neat things being accomplished during the players' turn.
Conclusions
There are a number of games out there that rely on math obscurity to give a sense of depth or agency. However, solving the game mechanic from a mathematical sense is only so fun, and once solved the complexity is replaced with an optimal way to play the game, which isn't fun. Making the math behind it harder is not making the choices more meaningful, just the decisions harder to make informed. Try pruning such mechanics from your game if possible.
Similarly, there are games that can be exploited by players to gain some disproportionate amount of XP and what have you that detract from the game by rewarding boring play.
Or in other words - if you are designing a new game system, try asking a math nerd or a game developer to break it. They might do the math and show you how balanced your system can be, and you can guide your players to playing the game well with that math as well.
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.
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.
