Basic Math for RPG Design
I’ve been asked a couple times by people looking to make RPGs about what kind of math goes into designing an RPG, and if they need to study any particular mathematics in order to design a game. The short answers are – not that much, and no! I’m going to do a quick guide to the math you use when making RPGs in this blog post. This is intended for very fresh beginners, so if you’re an experienced game designer, you probably won’t find this tells you anything new. As a second disclaimer, I don’t have any education in math beyond high school, and I don’t consider myself mathematically skilled or talented. The language I use here is intended for explanation between laymen. As a third disclaimer, what I’m about to say is almost as much about psychology as actual math.
Keep It Simple
First of all, you usually want to keep the math in your game as simple as possible. This makes the game not just more accessible for players and GMs, but also more accessible for people who might write adventures for your game or make hacks of it. Think about using the smallest numbers you can that still cover the range of narrative outcomes you would like to happen. Avoid requiring players to do any math of their own except addition and maybe subtraction. If you do want to use a calculation that requires more advanced math – say, multiplying strength by toughness to yield carrying capacity – present the players with a table of outcomes instead of asking them to multiply the numbers.
There are exceptions to this rule – a small niche of games that revel in mathematical excess – but if you’re into making that kind of game, you probably don’t need to read this blog post.
As an anecdotal rule of thumb, if players are successful 75% of the time, they will believe that they were successful around half the time, and that the game they are playing is “fair”. You can calibrate around that number for the vibes you want your game to have. In a d20 game, a character with a +1 bonus has close to a 75% chance to roll a 10 or higher. In a 2d6 game, a character with a +1 bonus has close to a 75% chance to roll a 7 or higher. You can calculate these kinds of probabilities using a website like Anydice.
Percentile systems – also called d100 systems – are different, because players can always see exactly what the odds are. If they have a 50% in a skill, they will understand that they have a 50% chance of success. This lets most d100 games have lower chances of success and still feel satisfying to players. Many d100 games will give starting characters skills in the 30%-50% range. They’ll also include opportunities for characters to get situational, equipment, and teamwork bonuses that bring them closer to that 75% success chance, especially when it’s important.
The Peculiarities of Dice
Here are some basic things worth knowing about dice.
The average roll of a die is half it’s value plus 0.5. For example, the average roll on a d6 is 3.5, the average roll of a d10 is 5.5, and the average of a d20 is 10.5. I’m using average here to refer to the arithmetic mean.
Some games will have a player roll a dice twice and keep the higher number or keep the lower number. As a rule of thumb, rerolling and picking highest is equivalent to around a +4 on a d20. If you’d like more detail, you can use AnyDice to calculate the effect of rerolling, or read this article for a much more detailed look. (Warning: less basic mathematics await on the other end of that link.)
D100s are a combination of two ten-sided dice, one for the tens digit and one for the ones digit, so they have some neat tricks. They roll doubles one in ten times, so if you want something to have a 10% chance of happening – like a critical roll – you can have it happen on doubles. You can also allow players to reverse the digits; for example, a roll of 72 becomes a 27. The ability to reverse digits provides a bonus that scales with the original chance of success.
If you roll two of the same dice and add their values together, the outcomes will fit a bell curve instead of a linear distribution. You will be more likely to roll the median outcome than any other numbers, and the chance of each possible outcome gets lower the further you get to the extremes. In other words, when rolling a d20 you have as much chance of rolling a 7 as rolling a 20, but if you roll 2d6, you will roll a total of 7 more often than any other number; 16.66% of the time, specifically. Your odds of rolling a 12 are only 2.77%. This is easy to understand if you see it graphed: here’s a blog post with some good illustrations.
Using the sum of two dice will give less “swingy” outcomes, where players can expect most rolls to fall within the middle of possible values and will be surprised by the occasional roll at the highest or lowest values. It also means that small modifiers will have a big impact on the outcome, but there will be a diminishing effect as modifiers stack. For example, 2d10 have a 64% chance of rolling 10 or higher. 2d10+1 has a a 72% chance of rolling 10 or higher – a big difference! On the other hand, 2d10+4 has a 90% chance of rolling 10 or higher, while 2d10+5 has a 94% chance. That’s only half the actual value of going from +0 to +1.
Be careful about the statistical consequences of calling for multiple repeated rolls. Let’s say a character has a 70% chance of succeeding at a stealth roll. They’re probably a very sneaky character. Let’s say they have to make three stealth rolls to succeed at a task – sneaking in, getting past some guards, and sneaking out. Their chance of success goes down to around 34%. Now let’s say you have four party members each making a single roll to sneak into a building, with chances of success at 50%, 60%, 70%, and 80% - unusually high for a party of diverse protagonists in most RPGs. The odds they will succeed are around 17%. An adventure you write requiring this had better have a plan for what happens if they fail!
Cards Are Dice That Remember
I have a lot less experience working with cards, but if you draw a card from a standard deck, then return it to the deck and shuffle, then the deck is no different than a 52-sided die (or a 13-sided die if you only use the value of the card, or a 4 sided die if you only use the suit).
Cards become different than dice when drawn cards are not immediately returned to the deck. This makes the deck a set of dice with a memory. Now the value of each draw changes the possible outcomes left in the deck. If players can see drawn cards, this gives them an increasing amount of information about the likely outcome of the next draw. You can also use card draws as a way to choose from a table without the possibility of a result repeating.
Alternatives to Random Oracles
Not every RPG needs to include a source of random chance. Resolutions can be determined through fixed rules, point spends, or other mechanics. Amber Diceless was the first RPG to popularize this kind of design. Puppetland is one I’m particularly fond of. Belonging Without Belonging is an easy to hack diceless game engine that uses token spends.
The Big Picture
When your game includes a random element like dice or cards, you should make sure it works at the maximums, and that it works the way you intend in the most likely situations.
It’s easiest to start with the maximums. Figure out the highest and lowest a player can roll (or draw, etc.) in each situation and make sure the game is still playable. For example, if you’re including a damage table, and the maximum possible damage a character can do with all possible modifiers and the highest roll of the dice is 12, then the table needs to go up to 12.
For the common situations, you need to do a little more prediction. Take into account the way characters are built, the kinds of modifiers that are likely to come up, and the kinds of other characters they’re likely to encounter. In a trad game, you might look at the kinds of characters that players are likely to make, and compare them to the kinds of challenges and enemies they’re likely to face. You can make three sample characters for each type of challenge: one character that a player with good understanding of the system made specifically to excel at that kind of challenge, one character made to be decent at the challenge, and one character who is optimized for handling different kinds of challenges and should have difficulty with the challenge you’re testing. Figure out what the odds are for each of them to overcome the challenge, and make sure those are the odds you want them to have.
And that's it! There's really not much math that goes into most RPGs. I hope this blog post will help give some people the confidence they need to start making games.