### Dice mechanics

Apr. 26th, 2017 04:56 pm**luckykaa**

In Tabletop RPGs skills are modelled using dice and a "skill" parameter. You state your intention to attempt a task. The Game Master (GM) asks you to roll. The mechanics of the system ensure that people who are good at a skill are more likely to succeed than those who are not.

Dungeons and dragons mostly relies on a roll of a 20 sided die. Some games use percentile dice, or sets of polyhedrals. But how do we acurately model skill?

Let's compare two archers

John Klutz (skill level: "incompetent") has barely a clue. He holds the bow incorrectly, takes a bad stance and tries pulling the arrow rather than drawing the string. The arrow will still go in a direction approximating forwards. But he'll rarely get a bullseye.

At the other end of the scale, we have the legendary archer (Robin Hood, or Katniss Everdeen). They never miss!

A fluke for John Klutz is hitting the centre ring. A calamity is hitting someone behind him.

For the legendary archer though, we expect that. Even a mere Olympic archer will hit yellow more often than not. A fluke result for a legend is Robin Hood hitting the arrow that's already in the target. A calamity is Katniss not hitting the centre when it mattered most.

Flukes and calamities are what makes things exciting. They add more than a simple pass/fail. Robin Hood doesn't just win! He passes into legend! Katniss needs to do something incredible to make up for the failure. These are million to one to one shots, and in fiction, according to Terry Pratchett, "million to one chances crop up nine times out of ten"! And they should, but in games, it should be nine

We can simply roll a die. A standard six sided die (d6 in gaming parlance). This makes the extremes come up a little too often. 1 in 6 attempts are flukes. 1 in 6 are calamities. That's far too frequent. You can make them less common with 20 sided dice (d20), or even percentile dice.

It lacks something though. All results are as likely as each other. In a skill based challenge they're not. In my archery example, the target has a yellow circle, a red circle, a blue one and a black one each of the same thickness. But because the diameter is larger, the area is larger. The red circle has 3 times the area of the yellow centre circle. John Klutz is more or less hitting random points on the target so he'll get 3 times as many reds as yellows. The expert is aiming for the centre, so should get substantially more yellows.

Mathematically, it makes more sense to model this as a normal distribution. Experts get a tall, thin distribution curve peaking at a higher value. Unskilled get a wide, flat curve. This is more fully discussed (with diagrams) here, so that covers the basics and the details. Something I do think is particularly useful to illustrate though is the ideal probability curve.

The "result" on the x-axis is the difficulty. So, for example, the difficulty of hitting the centre ring might be 4 or 5. The difficulty of the "Robin Hood" shot is 9 or 10.

The following uses Anydice.com for modelling the results.

Personally I'm quite fond of "d6" based systems. They're very accessible. Most people have a few dice kicking about.

For some reason, the above treatise doesn't consider simply rolling three 6-sided dice. This can be applied simply by adding skill level to the result. This gives us a nice set of ranges. The average is 10.5. The Standard Deviation is 2.96. So most of the time values fall between 8 and 15. These are the mundane results. You perform the task as well as can be expected. 2-sigma values (5-15) are the range including good days and bad days. This means 4 and 17 are the outliers (we get each of them 3/216 of the time, or on average once every 72 rolls). We get our 3's and 18's on average every 216 rolls. This can be a major life threatening calamity of legend making success. Gamers roll a lot of dice. This will happen every few sessions, but not every session.

We don't see the flattening but that's less important than the basic shape. I like this one because it's really simple! It's something that can be explained in 10 seconds to someone with no experience.

Another option is to go backwards. Higher skill involves fewer dice. Subtract from 24 to get a value. This does mean we can above average is positive and below average is negative. One thing I don't like is that it feels backwards. Its not intuitive. Higher numbers are worse. More dice are worse.

Even better, there's a "roll and keep" mechanic. Roll a bonus die per level. Keep the best 3. The more bonus dice you add, the taller and thinner the curve gets. This works really well. The only downside is that a lot of dice are quite unweildy for the highly skilled characters.

There are other mechanics that I might have a look at later. "Exploding Dice" is an interesting one. If you roll a 6, then you roll another die and add. This does mean there isn't an upper limit though, and the curves are the wrong way round. The less skilled are more consistent.

My conclusion here: The roll and keep mechanic is probably the best, both for elegant modelling and for general nice results. The next best is probably roll a bunch of six sided dice and add the modifiers. Noit sure which oif these I go for in my homebrew game but they're certainly my favourites.

Dungeons and dragons mostly relies on a roll of a 20 sided die. Some games use percentile dice, or sets of polyhedrals. But how do we acurately model skill?

Let's compare two archers

John Klutz (skill level: "incompetent") has barely a clue. He holds the bow incorrectly, takes a bad stance and tries pulling the arrow rather than drawing the string. The arrow will still go in a direction approximating forwards. But he'll rarely get a bullseye.

At the other end of the scale, we have the legendary archer (Robin Hood, or Katniss Everdeen). They never miss!

A fluke for John Klutz is hitting the centre ring. A calamity is hitting someone behind him.

For the legendary archer though, we expect that. Even a mere Olympic archer will hit yellow more often than not. A fluke result for a legend is Robin Hood hitting the arrow that's already in the target. A calamity is Katniss not hitting the centre when it mattered most.

Flukes and calamities are what makes things exciting. They add more than a simple pass/fail. Robin Hood doesn't just win! He passes into legend! Katniss needs to do something incredible to make up for the failure. These are million to one to one shots, and in fiction, according to Terry Pratchett, "million to one chances crop up nine times out of ten"! And they should, but in games, it should be nine

*games*out of ten. Not nine attempts! We do want it to be possible for this to happen.We can simply roll a die. A standard six sided die (d6 in gaming parlance). This makes the extremes come up a little too often. 1 in 6 attempts are flukes. 1 in 6 are calamities. That's far too frequent. You can make them less common with 20 sided dice (d20), or even percentile dice.

It lacks something though. All results are as likely as each other. In a skill based challenge they're not. In my archery example, the target has a yellow circle, a red circle, a blue one and a black one each of the same thickness. But because the diameter is larger, the area is larger. The red circle has 3 times the area of the yellow centre circle. John Klutz is more or less hitting random points on the target so he'll get 3 times as many reds as yellows. The expert is aiming for the centre, so should get substantially more yellows.

Mathematically, it makes more sense to model this as a normal distribution. Experts get a tall, thin distribution curve peaking at a higher value. Unskilled get a wide, flat curve. This is more fully discussed (with diagrams) here, so that covers the basics and the details. Something I do think is particularly useful to illustrate though is the ideal probability curve.

The "result" on the x-axis is the difficulty. So, for example, the difficulty of hitting the centre ring might be 4 or 5. The difficulty of the "Robin Hood" shot is 9 or 10.

The following uses Anydice.com for modelling the results.

Personally I'm quite fond of "d6" based systems. They're very accessible. Most people have a few dice kicking about.

For some reason, the above treatise doesn't consider simply rolling three 6-sided dice. This can be applied simply by adding skill level to the result. This gives us a nice set of ranges. The average is 10.5. The Standard Deviation is 2.96. So most of the time values fall between 8 and 15. These are the mundane results. You perform the task as well as can be expected. 2-sigma values (5-15) are the range including good days and bad days. This means 4 and 17 are the outliers (we get each of them 3/216 of the time, or on average once every 72 rolls). We get our 3's and 18's on average every 216 rolls. This can be a major life threatening calamity of legend making success. Gamers roll a lot of dice. This will happen every few sessions, but not every session.

We don't see the flattening but that's less important than the basic shape. I like this one because it's really simple! It's something that can be explained in 10 seconds to someone with no experience.

Another option is to go backwards. Higher skill involves fewer dice. Subtract from 24 to get a value. This does mean we can above average is positive and below average is negative. One thing I don't like is that it feels backwards. Its not intuitive. Higher numbers are worse. More dice are worse.

Even better, there's a "roll and keep" mechanic. Roll a bonus die per level. Keep the best 3. The more bonus dice you add, the taller and thinner the curve gets. This works really well. The only downside is that a lot of dice are quite unweildy for the highly skilled characters.

There are other mechanics that I might have a look at later. "Exploding Dice" is an interesting one. If you roll a 6, then you roll another die and add. This does mean there isn't an upper limit though, and the curves are the wrong way round. The less skilled are more consistent.

My conclusion here: The roll and keep mechanic is probably the best, both for elegant modelling and for general nice results. The next best is probably roll a bunch of six sided dice and add the modifiers. Noit sure which oif these I go for in my homebrew game but they're certainly my favourites.