approachable theory logo. (By Brie Sheldon) |

*Hi all! I have a post today from Michael "Karrius" Mazur (email) about tabletop RPG dice math. Michael is a tabletop RPG player, more often a GM than not, and in his own words, he's "always had an interest in tinkering with and designing game systems." I asked him his favorite part about roleplaying games and he said it was that roleplaying games are a creative hobby with a low barrier to entry - and I like that too!*

*This post is definitely a lot of information, but I think Michael explains it simply and approachably. Please enjoy!*

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Mathematical balancing can be an intimidating subject for RPG designers, but crucial for making a game that works like the designer intends is the solid foundation of having the correct dice rolling method. While familiarity with some methods of rolling are usually understood by designers, what questions to ask when deciding on a dice rolling system, the differences between said systems, and how to pick what’s most suited to the game can be a tough subject. The first question to tackle is often one of how many dice are appropriate.

The most familiar method of rolling dice in RPGs is the simple “roll a 1d20, add a number to it, and compare to a target number,” due to its use in Dungeons and Dragons and related spin-offs. One common modifier to that is instead of rolling 1d20, use the sum of 3d6. These two methods make for useful comparisons of the difference between rolling a single die vs rolling multiple dice and adding them. Both average out to the same result (10.5), and have similar maximums (20 vs 18) and minimums (1 vs 3). The following graph shows the probability curve of rolling each result on a 1d20 and 3d6, comparatively.

This chart shows the probability of rolling a specific result, visual by Michael. Full details in alt text. |

Rolling the maximum or minimum are far less likely due to having fewer combinations of dice that can achieve them. While a 1d20 can be expected to roll within a half point of the average result (10-11) one time in ten, and roll the maximum result one time in twenty, a 3d6 rolls within a half point of the average an expected one time in four, and roll the maximum result one roll out of every 216. Rolling multiple dice causes the “typical” results to be far more likely, and the extremes to be far less so.

*click through for more!*

The next step on the analysis is to understand that when you’re playing the game, for most rolls you’re not looking to roll an exact result - you’re looking to roll that number or higher - to succeed on the difficulty of a task. The following graph shows what the probability is of succeeding on a task for at various results needed, comparing 1d20 to 3d6.

This chart displays the probability of rolling a specific result or greater, visual by Michael. Full details in the alt text. |

Situations where a skilled character repeatedly fails at a given task over and over become far less likely, which can ease frustration or keep a challenging monster fight from becoming trivial due to a few bad rolls from the monster. Systems that roll 1d20 allow a wider variety of task difficulties, which become possible for characters to attempt sooner, but continue to have a noticeable failure rate as a character grows. Rolling 1d20 gives a character less reliability, but also more leeway if they’re being forced into attempting more difficult tasks.

Deciding which tasks - reliable or risky - you want your dice system to encourage isn’t the only change in player behavior you’ll see from it either. An important consideration is how valuable a +1 bonus is to the success at any given task. This won’t be obvious to all players, but some will understand how this works, and others will pick up a sense of it as you play. The following graph shows the probability of a +1 bonus changing a failure to a success for each roll needed.

The probability of a +1 bonus having changed a failure to a success is the exact same as the probability of rolling the number one fewer than whatever you’re trying to roll. As such, the graph of the +1 bonus helping is the same as the graph of individual roll results, except shifted one number higher. If you’re rolling a 1d20, no matter what task you’re performing, a +1 bonus is giving a 5% greater chance of success. If you’re rolling 3d6, that instead varies.

Deciding which tasks - reliable or risky - you want your dice system to encourage isn’t the only change in player behavior you’ll see from it either. An important consideration is how valuable a +1 bonus is to the success at any given task. This won’t be obvious to all players, but some will understand how this works, and others will pick up a sense of it as you play. The following graph shows the probability of a +1 bonus changing a failure to a success for each roll needed.

This chart shows the probability of a +1 helping, visual done by Michael. Full details in the alt text. |

For tasks which are near impossible or very easy to perform, the +1 might not be worth sacrificing much for. But for tasks that you already have a near-even chance at, the value is very high - over twice as much as you would be getting on a d20! Players are more incentivized to spend resources or time in order to get any small bonus they can when they expect they’re close to the average of an attack or defense.

All of the above shows the difference between 1d20 and 3d6 - but what about other dice, or rolling even more? The following graph shows a comparison between six types of dice rolling methods, all with roughly even averages of 10-11.

This chart shows the probability of rolling a specific result, visual done by Michael. Full details in the alt text. |

When so few numbers are likely to be rolled, this means that characters gaining small bonuses or penalties can very quickly put them in the realm of instant success or failure. It only takes a +4 to go from the average value to an instant success on a 7d2, as opposed to a +10 on a d20, meaning there’s not a lot of room for individual bonuses to make a difference and still have a chance of failure. This however also allows a stratification in difficulty - it doesn’t require very large bonuses for an expert character to be able to complete tasks a non-trained character cannot, keeping numbers reasonable if you want multiple tiers of difficulty.

In summary, there are multiple considerations that go into choosing the dice rolling method for your game. A single large die, or fewer dice in general, is more appropriate if you want a game to have multiple conditional sources of bonuses such as gear, positioning, and teamwork, to keep tasks from ever seeming too reliable, and to encourage risky maneuvers. Multiple dice, or smaller dice, are more appropriate if you want to make small bonuses matter a great deal, allow characters to treat lower-difficulty tasks as trivial, or to have hard tiers in difficulty. Neither method is inherently superior; instead, both are proper tools for their tasks.

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**Thanks so much to Michael for the excellent article! Please share around and I hope you all gained something from reading this post on approachable theory!**

*If you would like to write an approachable theory post, send an email to Brie with your name, pronouns, and pitch. Responses may be delayed over the next two weeks as Brie is recovering from grad school, but they'll get back as they can.*

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*Updated 5/9/18 12:55pm Eastern to change "odds" to "probability." Failure on Brie's part to not catch that mathematical terminology difference. Sorry!*

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