## Modifier Math

Last night I was toying around with an idea for a mechanic for a possible Cinemechanix Elevator Pitch. Initially, the plan was to make it work like a Trope: each point would either raise or lower the character’s Hero Die depending on if it was positive or negative. That’s all well and good, but it seemed like it might be nice to make the mechanic work slightly differently for gameplay reasons. Let’s say the new mechanic was measuring cybernetics. Making the mechanics for cyber-whatever work a little differently than normal-but-similar-whatever makes the cyber-whatever feel a little more special and central.

Disclaimer 1: This post is about math. You have been warned.

Disclaimer 2: I suck at math. I let the Troll Dice Roller do the math for this post, but there may be flaws in my assumptions here. In the past, though, I’ve found that looking at the kinds of numbers I’m looking at here can be useful for getting a general idea of where dice mechanics break down. I thought it might be interesting or useful for aspiring game designers.

I came up with a roll modifier (a modifier that you add to the die roll before determining success or failure) because it’s a simple mechanic not used for any of the core “better at something” rules. In fact, it’s so simple that it got me wondering why I didn’t use roll modifiers instead of Boosts (die increases) in the first place. I knew there was a reason, and eventually remembered that there’s a point where roll modifiers can make the die rolls a moot point, mainly because they raise the minimum roll as well as the average and maximum. If my roll modifier is 10 higher than yours and we’re rolling a d10, I win no matter what we roll. Since Bumps increase the die type, they increase the average roll and the maximum without increasing the minimum that much (it just goes up by 1 every time you roll over a d12). That makes die type increases a better standard bonus than just adding to the roll, at least if you want the die rolls to remain relevant.

Still, I felt like roll modifiers might be ok if they stay small enough that they don’t invalidate the dice, so I ran the math on that using the same character with different types of “I’m better at that” modifiers to see how the math worked out. We’ll say he’s swinging a sword.

First is our standard character without any modifiers. He’s an accountant with heroic potential (Hero Factor 4), but no real combat training or experience. When he attacks with a sword, he rolls a Default Die of d12 and a Hero Die of d4, which means:

- Roll Range: 2-16
- Average Roll: 9
- Typical roll range (average -/+ standard deviation): 6-12

Our next character is a freshly-trained member of the city watch. He’s been through basic training, so he gets a Concept Bump on his Default Die, but doesn’t have any Trademarks in swordsmanship and his Hero Factor is low enough that his Hero Die is still a d4. That means he rolls d20 + d4, resulting in:

- Roll Range: 2-24
- Average Roll: 13
- Typical roll range (average -/+ standard deviation): 8-18

That’s a pretty big improvement, but since the Concept Bump is supposed to represent “the thing you are,” it makes sense. Since I want to see how the rolls change with different kinds of modifiers, the best way I can think of to make them equivalent is to make sure they all have a maximum roll of 24, which brings us to the character with a higher Hero Die. To get a 24, he needs to be rolling a Hero Die of d12. It doesn’t really matter whether the d12 comes from being very powerful (Hero Factor 12), extremely well-trained in sword swinging (HF 4 with 4 Boosts in Sword Fighting), or some combination of the two (HF 8 and 2 Boosts in Swording). He’s not a professional fighter, so he rolls a Default Die of d12 and a Hero Die of d12, yielding:

- Roll Range: 2-24
- Average Roll: 13
- Typical roll range (average -/+ standard deviation): 8-18

While the statistical calculations are slightly different (2d12 rolls tend to clump toward the middle, while d20 + d4 results in a flatter distribution, with a 5% chance for all but the top and bottom 3 rolls), the range, average, and typical rolls are the same as for the previous character once you round everything to a whole number. A Hero Factor 4 Warrior who put his Trademarks in Archery is about evenly matched (in a sword fight) with a Hero Factor 4 Cheese Maker who dumped all his Trademarks into Sword Fighting. Gandalf is also evenly matched with them because of his Hero Factor 12 even if he doesn’t get a Concept Bump for fighting and doesn’t have any Sword Trademarks (for purposes of this example, we’re going to assume Gandalf is using classic D&D rules). So far, so good.

[Side note: Just in case you’re worried about that 1st level fighter, here’s what his numbers look like when he gets his Hero Die for swords up to d12 through either training (Trademarks) or becoming more badass (Hero Factor increases) and rolls d20 +d12:

- Roll Range: 2-32
- Average Roll: 17
- Typical roll range (average -/+ standard deviation): 12-23]

The other standard Cinemechanix modifier, Effect Bonus, is trickier to deal with because it only applies to successful rolls. For our unmodified character (the one who rolls d12 + d4) to get an Effect Bonus that will give him a 24 maximum Effect, he needs an Effect Bonus of +8 (since his normal maximum is 16). If we assume the other modified characters are rolling average (13) on every roll, the non-combatant but very strong giant will only hit (and get his +8 modifier) a little over 20% of the time. He’ll cause at least 21 points of damage when he does, though.

That brings us to the non-core modifier that I’m testing, the roll bonus. Like Effect bonus, it’s going to need to be +8 to give our pacifist human with a super-cool magic sword (we’ll say) a maximum of 24. Unlike Effect bonus, this modifier is applied before success or failure is determined, which means his rolls look like this:

- Roll Range: 10-24
- Average Roll: 17
- Typical roll range (average -/+ standard deviation): 14-20

So yeah, a character with a +8 roll modifier is much more powerful than a character with a more-or-less equivalent number for any of the other kind of bonuses. His typical roll range is closer to our experienced/trained professional, and his average is the same 17, but there’s one big difference: while the character with a d12 Hero Die and a Concept Bump will roll 9 or less about 15% of the time, the Hero Factor 4 character with a +8 roll bonus never rolls less than a 10. The two “equivalent” characters fare even worse: the one with just a Concept Bump rolls 9 or less 32.5% of the time, while the character with a Default Die of d12 and a Hero Die of d12 rolls 9 or less 25% of the time. The smaller standard deviation range also means that two characters with similar roll bonuses will pound on one another forever without either gaining an advantage (the original dice pool mechanic for Cinemechanix had similar problems).

So if I’m going to use roll bonuses, I need to make sure that there’s a limit to how high I can go. This actually works fine for what I was thinking about using them for, since there’s a fairly low upper limit, but I probably need to run some lower numbers just to be sure. Since I suspect the roll modifier I’m thinking of will top out at about +4, let’s see how the d12 + d4 character with the special thing at +1 to +5 (just in case it goes higher than I expect) does against our two standard characters (with either a Concept Bump or a d12 Hero Die; Range 2-24, average roll 13, typical roll 8-18). For this one, I’m adding a couple of extra entries for the % of time that the standard characters would roll less than the minimum roll for a character with the roll bonus.

### +1 Roll Modifier:

- Roll Range: 3-17
- Average Roll: 10
- Typical roll range (average -/+ standard deviation): 7-13
- % Concept Bumped Character rolls less than 3: 1.25
- % d12 Hero Die Character rolls less than 3: 0.694

### +2 Roll Modifier:

- Roll Range: 4-18
- Average Roll: 11
- Typical roll range (average -/+ standard deviation): 8-14
- % Concept Bumped Character rolls less than 4: 3.75
- % d12 Hero Die Character rolls less than 4: 2.083

### +3 Roll Modifier:

- Roll Range: 5-18
- Average Roll: 12
- Typical roll range (average -/+ standard deviation): 9-15
- % Concept Bumped Character rolls less than 5: 7.5
- % d12 Hero Die Character rolls less than 5: 4.167

### +4 Roll Modifier:

- Roll Range: 6-19
- Average Roll: 13
- Typical roll range (average -/+ standard deviation): 10-16
- % Concept Bumped Character rolls less than 6: 12.5
- % d12 Hero Die Character rolls less than 6: 6.944

### +5 Roll Modifier:

- Roll Range: 7-20
- Average Roll: 14
- Typical roll range (average -/+ standard deviation): 11-17
- % Concept Bumped Character rolls less than 7: 17.5
- % d12 Hero Die Character rolls less than 7: 10.417

The minimums start to get a little out of whack at +5, which makes sense given that it’s the point where the average roll exceeds the average roll for our characters using other types of modifiers. Since +5 is only there in case someone figures out a way to squeeze out a higher-than-expected roll bonus, we should still be ok. The character with the roll bonus has the advantage at the lower end of the spectrum, but this is balanced out by the lower minimum roll. On an average roll, the character with a roll bonus has a typical minimum roll that’s two higher than a standard character, but his maximum is two lower. Overall, I think it’s safe to say that a +4 roll bonus is roughly equivalent to either a Concept Bump or 4 Hero Die Boosts.

After running the numbers, it looks like roll bonuses will work if you need a slightly different kind of modifier for Cinemechanix, but you have to make sure that you don’t let the bonus get too big or things might get out of hand. I’m guessing that if I test some characters at other power levels, I’d find that the roll bonus starts getting too powerful when the average roll with a roll bonus exceeds the average roll for a similar character with more standard modifiers. I also suspect that the die roll becomes irrelevant when the average roll for a character with a roll bonus exceeds the typical maximum (average + standard deviation) for a character with normal modifiers, especially for die combinations where the roll distribution is more curved.

I’ll probably go with it for the thing that inspired all this math, but I’ll make sure the modifier maxes out at +4.