## Need Mechanics/Math Help

### Need Mechanics/Math Help

Hey Twentysiders, wondering if anyone can help me improve a game mechanic!

So, my tabletop RPG, Tegwyn Saga, has a mechanic where your chance of success at an action is determined by comparing a number on your character sheet with an opposing number. This might be your accuracy vs. your target's evade to determine whether you hit, your magic vs. your target's aura to determine whether they resist your spell, your agility vs. the difficulty of a jump, your strength vs. the weight of something you want to lift, etc.

The catch is, I don't want to use a D&D mechanic where you roll a number and add a flat bonus, because this starts to go bad when your number and the opposing number are too far apart. It doesn't scale well. Ideally, 5 vs 5, 10 vs 10, and 100 vs 100 should all have the same odds of success. Also, 5 vs 10, 10 vs 20, 100 vs 200 should all have the same odds of success. This makes it tough to express mechanically without requiring the use of a chart every time you make a roll.

Currently, the math is as follows:

if a <= d, p = a/d * 1/2 = a/2d

if a > d, p = 1 - (d/a * 1/2) = 1 - d/2a

...where:

a is the attribute of the attacker (the one attempting the action)

d is the attribute of the defender (the thing opposing the action)

p is the probability of success for the attacker

Examples: To hit a target, it's my accuracy vs the target's evade. If we are even, my probability to hit is 1/2. If I have 20 accuracy and it has 40 evade, my probability to hit is 1/4. If I have 40 and it has 20, it is 3/4.

This scales nicely and works for any order of magnitude. You don't have to be within 20 of each other just because you roll a d20. I don't have to keep every number that uses this mechanic in a narrow range that makes sense to add to a d20. It's more about comparing relative quantities than absolute quantities. It doesn't matter if you have "ten more" than the opponent; it matters if you have "twice as much" as the opponent. I think this is more consistent with how we compare quantities on a day-to-day basis. It also compresses things the further you get from your opponent so that there is a diminishing returns on how much better you are than your opponent or vice versa. This makes it a little easier to design encounters, because it's harder to get situations where it's nearly impossible to win or lose because you've passed a threshold where the bad guys become so much better/worse that they're invincible/useless.

The problem is, the math is complicated enough to require a chart. The charts linked below let you compare your number and your roll to see what number you can hit. So, if you rolled a 7 on your d20, you'd go to the 7 column, and if you have 23 in the relevant attribute, you go to row 23. The result is the number on the opponent that you can beat. If you're rolling agility vs agility to try and chase down a thief, for example, your 7 and 23 would let you catch the thief if their agility is 14 or less. (The current chart just repeats the rows for each column so that you don't have to scan horizontally.)

EDIT: New chart: https://drive.google.com/open?id=0B7rlo ... authuser=0

Current chart: https://drive.google.com/file/d/0B7rloH ... sp=sharing

Old chart: https://drive.google.com/file/d/0B7rloH ... sp=sharing

(Wow, this is a long post. Hope it's an interesting problem for somebody?)

Can anyone think of another way to do this scalable opposed roll thing that doesn't require a chart? I'm open to changing the math up a little. The key things are that I'd like this to work even if your number and your opponent's aren't really close, and there should be a diminishing returns the further apart your and your opponent's numbers are.

So, my tabletop RPG, Tegwyn Saga, has a mechanic where your chance of success at an action is determined by comparing a number on your character sheet with an opposing number. This might be your accuracy vs. your target's evade to determine whether you hit, your magic vs. your target's aura to determine whether they resist your spell, your agility vs. the difficulty of a jump, your strength vs. the weight of something you want to lift, etc.

The catch is, I don't want to use a D&D mechanic where you roll a number and add a flat bonus, because this starts to go bad when your number and the opposing number are too far apart. It doesn't scale well. Ideally, 5 vs 5, 10 vs 10, and 100 vs 100 should all have the same odds of success. Also, 5 vs 10, 10 vs 20, 100 vs 200 should all have the same odds of success. This makes it tough to express mechanically without requiring the use of a chart every time you make a roll.

Currently, the math is as follows:

if a <= d, p = a/d * 1/2 = a/2d

if a > d, p = 1 - (d/a * 1/2) = 1 - d/2a

...where:

a is the attribute of the attacker (the one attempting the action)

d is the attribute of the defender (the thing opposing the action)

p is the probability of success for the attacker

Examples: To hit a target, it's my accuracy vs the target's evade. If we are even, my probability to hit is 1/2. If I have 20 accuracy and it has 40 evade, my probability to hit is 1/4. If I have 40 and it has 20, it is 3/4.

This scales nicely and works for any order of magnitude. You don't have to be within 20 of each other just because you roll a d20. I don't have to keep every number that uses this mechanic in a narrow range that makes sense to add to a d20. It's more about comparing relative quantities than absolute quantities. It doesn't matter if you have "ten more" than the opponent; it matters if you have "twice as much" as the opponent. I think this is more consistent with how we compare quantities on a day-to-day basis. It also compresses things the further you get from your opponent so that there is a diminishing returns on how much better you are than your opponent or vice versa. This makes it a little easier to design encounters, because it's harder to get situations where it's nearly impossible to win or lose because you've passed a threshold where the bad guys become so much better/worse that they're invincible/useless.

The problem is, the math is complicated enough to require a chart. The charts linked below let you compare your number and your roll to see what number you can hit. So, if you rolled a 7 on your d20, you'd go to the 7 column, and if you have 23 in the relevant attribute, you go to row 23. The result is the number on the opponent that you can beat. If you're rolling agility vs agility to try and chase down a thief, for example, your 7 and 23 would let you catch the thief if their agility is 14 or less. (The current chart just repeats the rows for each column so that you don't have to scan horizontally.)

EDIT: New chart: https://drive.google.com/open?id=0B7rlo ... authuser=0

Current chart: https://drive.google.com/file/d/0B7rloH ... sp=sharing

Old chart: https://drive.google.com/file/d/0B7rloH ... sp=sharing

(Wow, this is a long post. Hope it's an interesting problem for somebody?)

Can anyone think of another way to do this scalable opposed roll thing that doesn't require a chart? I'm open to changing the math up a little. The key things are that I'd like this to work even if your number and your opponent's aren't really close, and there should be a diminishing returns the further apart your and your opponent's numbers are.

Last edited by Cuthalion on Sat Jun 20, 2015 12:20 am, edited 1 time in total.

### Re: Need Mechanics/Math Help

Bless you for trying to keep this subforum alive.

As for the topic... I dunno. I'm not a mathematician by nature. But your formulas are simple enough, to me. If I played this game, I'd just do the math on my calculator each time I needed to, rather than look it up in the chart. Seems faster.

Maybe you or somebody could right a simple little program where you just entered "a" and "d" and the computer figured it out for you? Even I might even be able to do that, and I only have one Comp Sci class worth of experience.

Edit: I would say that if you just do the math each time, a percentile system (d100) works much better than using a d20, which is what your chart seems based off of. Eliminates a step.

Edit edit: I misread your chart. Makes much more sense now.

Yeah, I don't really see a way of simplifying this system. Yours is not the only system I've come across that uses a chart though; I believe Torg also used a chart for its rolls.

As for the topic... I dunno. I'm not a mathematician by nature. But your formulas are simple enough, to me. If I played this game, I'd just do the math on my calculator each time I needed to, rather than look it up in the chart. Seems faster.

Maybe you or somebody could right a simple little program where you just entered "a" and "d" and the computer figured it out for you? Even I might even be able to do that, and I only have one Comp Sci class worth of experience.

Edit: I would say that if you just do the math each time, a percentile system (d100) works much better than using a d20, which is what your chart seems based off of. Eliminates a step.

Edit edit: I misread your chart. Makes much more sense now.

Yeah, I don't really see a way of simplifying this system. Yours is not the only system I've come across that uses a chart though; I believe Torg also used a chart for its rolls.

Last edited by Supahewok on Sat Jun 13, 2015 7:35 pm, edited 1 time in total.

### Re: Need Mechanics/Math Help

Most obvious would be to use a direct ratio which means you only have to do one sum and it makes the whole thing a lot less fiddly. In this system the roll of the defender would be taken at its face value on the die while the attacker's value would be taken from the equation below:

(a/d)*r

a is the attribute of the attacker (the one attempting the action)

d is the attribute of the defender (the thing opposing the action)

r is the roll of the attacker

This is essentially taking your system and applying to die rolls not the raw probability which cleans it all up a bit. The problem with this is the division which makes the mental maths a bunch harder without a calculator (which I presume you're trying to avoid) or resorting to graphs (however one produced by this implementation would be cleaner).

The problem with your request is that mathematically the functions + and - are easy to use but create an additive scale (not proportional) while * and / create a logarithmic scale (proportional) but are difficult do carry out in your head. This means you have to three choices:

(a/d)*r

a is the attribute of the attacker (the one attempting the action)

d is the attribute of the defender (the thing opposing the action)

r is the roll of the attacker

This is essentially taking your system and applying to die rolls not the raw probability which cleans it all up a bit. The problem with this is the division which makes the mental maths a bunch harder without a calculator (which I presume you're trying to avoid) or resorting to graphs (however one produced by this implementation would be cleaner).

The problem with your request is that mathematically the functions + and - are easy to use but create an additive scale (not proportional) while * and / create a logarithmic scale (proportional) but are difficult do carry out in your head. This means you have to three choices:

- Use an additive scale and abandon proportionality (probably not what you want to do)
- Use a logarithmic scale but require a calculator or table
- Try and make the maths as simple as possible by rounding levels (e.g. to the nearest 5)

### Re: Need Mechanics/Math Help

Hmm, thanks for the feedback.

What do you guys think? Is it worth keeping the proportionality if it requires a chart for players in in-person games or online ones that don't have scripty buttons?

@Supahewok lol you're welcome, I try to keep the twentysided about the twentysided at least a little. I would love to ditch the chart, since players generally don't seem to like it. They usually seem to prefer just letting the GM do that, even though it means they're waiting for a black box to announce whether they hit rather than seeing on the chart what their result is (even if the GM still compares it to a hidden number and tells them whether they hit). But it's good that I'm not the only system that uses a chart.

Certainly this sort of thing would be trivial in a video game or using a tool that allows for scripting. (Roll20 does I think, but I don't have a paid account right now to be able to create scripts, and at any rate I don't want to invalidate traditional analog play.) I generated the charts, for example, by a spreadsheet with formulas that shifted the math around so that it takes a die roll and the attacker's number and solves for the defender's, rather than solving for the probability or die roll.

@JJR Your suggestion with the ratio is interesting. Are you saying you'd basically divide attacker by defender, then multiply it by the roll, and any high result (11-20 if using a d20) hits? Ex: 50/40 * 7 = 1.25 * 7 = 8.75 -> miss; 40/50 * 15 = 0.8 * 15 = 12 -> hit. That would be cool, but yeah, as you mention, it comes with the same caveats: it's still tough to do in the player's head, so you'd need a calculator or graphs. Plus, with this mode, only the GM can do the rolls unless you reveal monster stats to players.

I've toyed with ideas involving just rolling more dice as your attribute increases, maybe with an opposed roll from the opponent rather than just comparing vs their number directly. But nothing comes to mind that would keep the proportionality. And this method would be a lot more interesting if you could increase your die size or die quantity separately, but most players will not have 6d8 on them.

So, a 6 power, 2-skill monster attacking a 5 power, 3-skill creature would roll 6d6 and get a hit for each roll of 5 or 6. The other creature would roll 5d6 to attack and get a hit for each roll of 3, 4, 5, or 6.

It's an interesting mechanic, but I can't see how it really applies. Maybe my intuition sensor is miscalibrated. :P

What do you guys think? Is it worth keeping the proportionality if it requires a chart for players in in-person games or online ones that don't have scripty buttons?

@Supahewok lol you're welcome, I try to keep the twentysided about the twentysided at least a little. I would love to ditch the chart, since players generally don't seem to like it. They usually seem to prefer just letting the GM do that, even though it means they're waiting for a black box to announce whether they hit rather than seeing on the chart what their result is (even if the GM still compares it to a hidden number and tells them whether they hit). But it's good that I'm not the only system that uses a chart.

Certainly this sort of thing would be trivial in a video game or using a tool that allows for scripting. (Roll20 does I think, but I don't have a paid account right now to be able to create scripts, and at any rate I don't want to invalidate traditional analog play.) I generated the charts, for example, by a spreadsheet with formulas that shifted the math around so that it takes a die roll and the attacker's number and solves for the defender's, rather than solving for the probability or die roll.

@JJR Your suggestion with the ratio is interesting. Are you saying you'd basically divide attacker by defender, then multiply it by the roll, and any high result (11-20 if using a d20) hits? Ex: 50/40 * 7 = 1.25 * 7 = 8.75 -> miss; 40/50 * 15 = 0.8 * 15 = 12 -> hit. That would be cool, but yeah, as you mention, it comes with the same caveats: it's still tough to do in the player's head, so you'd need a calculator or graphs. Plus, with this mode, only the GM can do the rolls unless you reveal monster stats to players.

I've toyed with ideas involving just rolling more dice as your attribute increases, maybe with an opposed roll from the opponent rather than just comparing vs their number directly. But nothing comes to mind that would keep the proportionality. And this method would be a lot more interesting if you could increase your die size or die quantity separately, but most players will not have 6d8 on them.

*Titan*, the board game, is also nagging at my mind. I feel like there might be a solution there, but I'm not sure how. In Titan, creatures have two stats: power and skill. Power indicates the number of d6es the creature rolls to attack as well as how many hits it can take. Skill is compared to the opponent by a simple chart that is easily memorized, since skill can range from 2-4.Code: Select all

` 2 3 4`

-------

2| 4 5 6

3| 3 4 5

4| 2 3 4

So, a 6 power, 2-skill monster attacking a 5 power, 3-skill creature would roll 6d6 and get a hit for each roll of 5 or 6. The other creature would roll 5d6 to attack and get a hit for each roll of 3, 4, 5, or 6.

It's an interesting mechanic, but I can't see how it really applies. Maybe my intuition sensor is miscalibrated. :P

- Lachlan the Sane
**Location:**I come from the land down under, where women blow and men chunder

### Re: Need Mechanics/Math Help

Here's my very bare-bones suggestion on how to make this system work in a way that can be done through the simple rolling of dice -- lots of addition, but no multiplication or division. Let's say that a character is making a roll to hit a monster, with their attack stat (X) vs a monster's evade stat (Y). The character rolls Xd6+X, and the monster rolls Yd6+Y. If the character rolls higher, they hit; if the monster rolls higher, they dodge. If their stats are equal, this means that each character has an almost even chance to succeed. Chances of success begin dropping fairly rapidly as the monster's evade state exceeds the player's attack stat, but success is only impossible if the monster's stats substantially exceed the player's (a player could hit a monster with an evade stat of up to 3.5x their own attack stat with phenomenally lucky rolls). Using dice which are larger than a d6 would increase the stat which you could hit with phenomenally lucky rolls, but would also make those phenomenally lucky rolls far less likely.

Problems with this system:

- The chances aren't quite even if the characters have even stats; what you decide to do with ties will skew the probability one way or the other.

- The stats you're talking about, which apparently can make their way up to 100, do not work well with this system -- who wants to add 100d6+100? One possibility could be that a character rolls 1 die for every 10 points of stat they have -- e.g. a character with a stat of 69 rolls 6d6+69, while a character with a stat of 71 rolls 7d6+71. This slightly reduces the probabilistic beauty of the system, and makes raising your stats through multiples of 10 much more important than other raises (although that's hardly unprecedented -- consider that in D&D raising your stats through even numbers is much more desirable than odd ones).

I'm not sure that I really like this system which I've just written, but I have been known to be very picky about my preferred RPG mechanics, and all this talk of using

Problems with this system:

- The chances aren't quite even if the characters have even stats; what you decide to do with ties will skew the probability one way or the other.

- The stats you're talking about, which apparently can make their way up to 100, do not work well with this system -- who wants to add 100d6+100? One possibility could be that a character rolls 1 die for every 10 points of stat they have -- e.g. a character with a stat of 69 rolls 6d6+69, while a character with a stat of 71 rolls 7d6+71. This slightly reduces the probabilistic beauty of the system, and makes raising your stats through multiples of 10 much more important than other raises (although that's hardly unprecedented -- consider that in D&D raising your stats through even numbers is much more desirable than odd ones).

I'm not sure that I really like this system which I've just written, but I have been known to be very picky about my preferred RPG mechanics, and all this talk of using

*tables*and*dice pools*really rubs me the wrong way. Give me an additive system any day of the week.### Re: Need Mechanics/Math Help

Instead of proportionality and complicated math,why not try using the "more dice" systems?Basically,you roll a bunch of dice equal to your skill,the defender rolls a bunch of dice equal to their skill,and the one with the greater number of successful dice(say the top half of the numbers,or as in risk compare the dice to each other)wins.The better your skill,the more dice you have.This way,you can still lose even if you outclass someone 20 to 1,but you most likely wont.Granted,this can become a bit annoying to do if you use only physical dice,but with various rngs that exist for computers and cell phones these days,its not that hard to simulate even 100d100.But,if you prefer regular dice,and dont have a huge number of them,you can scale them up like this:A skill of one means you roll 1d4,two means you roll 1d6,then 1d8,then 2d4,then 1d10,etc.

### Re: Need Mechanics/Math Help

Certainly this sort of thing would be trivial in a video game

I'm playing Endless Legend right now and what Cuthalion describes is the combat system it uses. It has a cap of relative attributes at 1:2 and 2:1 with it described here and the combat system represented graphically here. Yours doesn't have the cap but it should scale and look like that.

I don't know if this helps you translate it into a system that would be easy to use without a computer or not. Seeing the graph and knowing how it works in a real game hopefully will help.

### Re: Need Mechanics/Math Help

@Steve C: Ah, cool, so they do that for damage. (I just use additive for damage, while I do the ratio thing for accuracy). Neat to see an example from another game!

I've definitely played this game with this rule many times, so it's not that it's untested. In fact, it's the testing that makes me wonder if I can have my cake (proportion-based accuracy) and eat it too (without requiring a chart or calculator). Most players view the chart as a weakness and prefer the GM use it instead of them or just have a hard time learning it.

@DL and Lachlan: I lol'd at "all this talk of using

Hmm... I wonder if the math works out any differently if I do it Risk-style, comparing highest rolls of each player's dice down the line until one player runs out of dice and seeing who won the most matchups.

Or perhaps if I did set-style, where you need to get sets of the same number. So, you're looking to get the most dice showing the same number out of your roll (e.g., if you rolled 6d6 and got 2,3,3,3,6,6 you'd have 3 of a kind) instead of just the highest total. I know there is at least one system out there that does something like this, where you roll Xd10 and the amount of dice in the set determine one thing while the number shown on the dice in the set determines another.

I'm pretty sure straight adding multiple dice doesn't scale any differently than just adding a flat bonus to a single die does, though it would curve more strongly toward the one with the advantage winning. Maybe that curve would do what I want if I kept the numbers in a narrower range (say, 2-7 dice instead of 20-200 accuracy) so that it's always possible for the underdog to roll higher than the other guy?

Perhaps if the chart wasn't how high you can hit based on your stat and roll every time you roll, but instead was how many dice you can roll based on your stat? Then I could set up the dice tiers to be logarithmic instead of linear. 3d6 at 8, 4d6 at 16, 5d6 at 32, 6d6 at 64, 7d6 at 128, etc. (Maybe not powers of 2, but you get the idea.) I wonder if I can make that come out scaling the odds by attacker/defender ratio instead of attacker-defender difference?

I'd have to think about those or plot them out to see how they respond when the matchup becomes less even.

I've definitely played this game with this rule many times, so it's not that it's untested. In fact, it's the testing that makes me wonder if I can have my cake (proportion-based accuracy) and eat it too (without requiring a chart or calculator). Most players view the chart as a weakness and prefer the GM use it instead of them or just have a hard time learning it.

@DL and Lachlan: I lol'd at "all this talk of using

*tables*and*dice pools*really rubs me the wrong way." Yeah, a dice pool thing might get me something that curves well enough to still give the GM some slack and allow different players (or different monsters) to have widely varying stats while still being in the same battle. I would definitely have to tier it though (you get an extra die every 5 or 10 in a stat) or pull the numbers back (typical stats range from 1-10 instead of 15-50) if I went that route.Hmm... I wonder if the math works out any differently if I do it Risk-style, comparing highest rolls of each player's dice down the line until one player runs out of dice and seeing who won the most matchups.

Or perhaps if I did set-style, where you need to get sets of the same number. So, you're looking to get the most dice showing the same number out of your roll (e.g., if you rolled 6d6 and got 2,3,3,3,6,6 you'd have 3 of a kind) instead of just the highest total. I know there is at least one system out there that does something like this, where you roll Xd10 and the amount of dice in the set determine one thing while the number shown on the dice in the set determines another.

I'm pretty sure straight adding multiple dice doesn't scale any differently than just adding a flat bonus to a single die does, though it would curve more strongly toward the one with the advantage winning. Maybe that curve would do what I want if I kept the numbers in a narrower range (say, 2-7 dice instead of 20-200 accuracy) so that it's always possible for the underdog to roll higher than the other guy?

Perhaps if the chart wasn't how high you can hit based on your stat and roll every time you roll, but instead was how many dice you can roll based on your stat? Then I could set up the dice tiers to be logarithmic instead of linear. 3d6 at 8, 4d6 at 16, 5d6 at 32, 6d6 at 64, 7d6 at 128, etc. (Maybe not powers of 2, but you get the idea.) I wonder if I can make that come out scaling the odds by attacker/defender ratio instead of attacker-defender difference?

I'd have to think about those or plot them out to see how they respond when the matchup becomes less even.

### Re: Need Mechanics/Math Help

There are 2 major mathematical problems which come to mind with a "more dice" system, it's a good idea and may well be the best option available but it will require some thought.

- It makes crit pass and crit fails a bit more complicated, with 6d20 the player has a one in 64 million chance of getting all 1s. If you say any 1s are a crit then they've got a 0.3 probability of getting a crit which is way too high. If you say that crits are now dependant on if the sum of the dice is below a certain value you have to start analysing probabilities, you then risk either making your players do some heavy maths or make a system which doesn't scale well at all.
- You enter the messy world of probability, with multiple dice rolls it is much more likely you'll get mid range values. The more dice the more likely mid range values become and the less likely extremities become. This may or may not be fine for you depending on how random you want your game to be, where less dice is more random.

### Re: Need Mechanics/Math Help

@Steve C: Ah, cool, so they do that for damage.

No. It is only for attack vs defense. Damage is a static attribute.

Let's say for a unit it's damage stat is 100. There will be a random roll for attack. That is compared to the defense according to the chart above. The four possible results will be damage of 150, 100, 50, or 0. There is no roll that could deal 75 or 30 or 140 damage for instance. Instead of a binary result of hit/miss, there are two extra possible results of critical hit and critical defense. Those can be pushed off the chart by vastly outclassing the opponent.

Note I'm just describing their system. I'm not making suggestions as I have none to give. It's simple and easy in a video game. I do not know how to make that system ergonomic for a human with dice.

### Re: Need Mechanics/Math Help

Did some stuff with http://anydice.com/ and LO Calc to see if multi-dice total vs. total would work, maybe with specific thresholds at which your stat gets you another die.

Probability of row d6 >= column d6. Please pardon the weird slant from pasted tabs:

Sadly, I don't think that's what I'm looking for. The ranges where it makes sense are much too narrow to allow much of a sense of progression. You'd basically have to start at 7d6 and end at 10d6, which breaks characters and monsters into pretty distinct low, medium, high, and expert chunks. I'd like something a little smoother than that.

EDIT: d20 + Xd6 vs d20 + Yd6 gets me a little closer. I'd still have to tier things weird though, giving you 2 dice with 10 stat, 3 with 15, 4 with 20, 5 with 30, 6 at 60... and the odds would still get weird if you had 3d vs 6d instead of 2d vs 4d. Not sure I like that either. http://anydice.com/program/606b

Probability of row d6 >= column d6. Please pardon the weird slant from pasted tabs:

Code: Select all

` 1d6 2d6 3d6 4d6 5d6 6d6 7d6 8d6 9d6 10d6`

1d6 58% 16% 3% 0% 0% 0% 0% 0% 0% 0%

2d6 91% 56% 22% 6% 1% 0% 0% 0% 0% 0%

3d6 99% 85% 55% 26% 9% 2% 1% 0% 0% 0%

4d6 100% 96% 81% 54% 28% 12% 4% 1% 0% 0%

5d6 100% 99% 94% 78% 54% 30% 14% 5% 2% 0%

6d6 100% 100% 99% 92% 76% 53% 31% 16% 7% 2%

7d6 100% 100% 100% 97% 90% 74% 53% 33% 17% 8%

8d6 100% 100% 100% 99% 96% 88% 73% 53% 34% 19%

9d6 100% 100% 100% 100% 99% 95% 86% 71% 53% 34%

10d6 100% 100% 100% 100% 100% 98% 94% 85% 70% 53%

Sadly, I don't think that's what I'm looking for. The ranges where it makes sense are much too narrow to allow much of a sense of progression. You'd basically have to start at 7d6 and end at 10d6, which breaks characters and monsters into pretty distinct low, medium, high, and expert chunks. I'd like something a little smoother than that.

EDIT: d20 + Xd6 vs d20 + Yd6 gets me a little closer. I'd still have to tier things weird though, giving you 2 dice with 10 stat, 3 with 15, 4 with 20, 5 with 30, 6 at 60... and the odds would still get weird if you had 3d vs 6d instead of 2d vs 4d. Not sure I like that either. http://anydice.com/program/606b

### Re: Need Mechanics/Math Help

JJR wrote:It makes crit pass and crit fails a bit more complicated

Actually it makes them just as simple,yet more fair(you wont get a critical hit/miss 5% of the time forever).

One way to get critical hit is "any X*number of successes".If X=2,that means you need twice the number of successes.This way,the more dice you have,the higher the chances to get a critical hit,which makes sense.And some very difficult tasks mean no way you get a crit.Other way would be "all successes are 6s".Meaning if you need 5 successes,and you get 7 successes,5 of which are 6s,its a crit.Again,the more dice you have,the higher chances you have to get a crit,but this way you can get a critical success every time you can succeed.Unlike the first method however,you cannot scale the difficulty of getting a crit.

For failures,you can say "all 1s means a critical failure".This means the more dice you have,less chances of a critical failure.Other would be "no successes at all means a critical failure".Same as the previous one,but with more chances to fail miserably.

### Re: Need Mechanics/Math Help

Good ideas! The way I used to do crits was, if you hit and did at least average damage (it used to be weapon + (1d6-1d6)*S, where S was a "swing" value that typically was 10% of the weapon), you rolled a d12. If the number you rolled on the 12 was at or above your critical requirement (started at 12 and went down by 1 every time your weapon skill improved), you got a crit.

Currently, since we were constantly forgetting the d12 part, and it seemed like that was just one step too many to attack somebody, I've given in and folded it into the accuracy roll. If your accuracy d20 is high enough (just the die number, modifiers ignored), you get a crit. Starts needing 19 or 20, requirement still goes down 1 every time weapon skill improves. (And damage was changed to be simpler: weapon + d6*10 for most weapons, weapon + 2d6*10 for two-handed melee weapons and possibly certain spells. d6 -> d8 -> d10 -> d12 as your str/dex/mag attribute goes up. The only clunky bit left now is really the accuracy roll.)

Important thing to note in this game is that criticals do different things depending on the weapon. Spears typically ignore armor, swords typically add ~50% damage, blunt weapons typically stun, axes typically cause bleed, etc. They're meant to occur more often than in D&D, so the odds start at 10% and go up from there. So it's intended in this game for crits to get more likely as one of your stats improves.

Right now, I'm trying to decide whether to officially say, "GM, you need to learn this chart. Don't worry about getting the players to use it unless they want to," or to sacrifice some of the elegant scaling for the sake of more fluid play and scaring fewer players away (players, in my experience, do not like charts).

Currently, since we were constantly forgetting the d12 part, and it seemed like that was just one step too many to attack somebody, I've given in and folded it into the accuracy roll. If your accuracy d20 is high enough (just the die number, modifiers ignored), you get a crit. Starts needing 19 or 20, requirement still goes down 1 every time weapon skill improves. (And damage was changed to be simpler: weapon + d6*10 for most weapons, weapon + 2d6*10 for two-handed melee weapons and possibly certain spells. d6 -> d8 -> d10 -> d12 as your str/dex/mag attribute goes up. The only clunky bit left now is really the accuracy roll.)

Important thing to note in this game is that criticals do different things depending on the weapon. Spears typically ignore armor, swords typically add ~50% damage, blunt weapons typically stun, axes typically cause bleed, etc. They're meant to occur more often than in D&D, so the odds start at 10% and go up from there. So it's intended in this game for crits to get more likely as one of your stats improves.

Right now, I'm trying to decide whether to officially say, "GM, you need to learn this chart. Don't worry about getting the players to use it unless they want to," or to sacrifice some of the elegant scaling for the sake of more fluid play and scaring fewer players away (players, in my experience, do not like charts).

### Re: Need Mechanics/Math Help

Right now, I'm trying to decide whether to...

Your system has to compete against every other system out there. Any small barrier to entry is huge. Don't go with a chart for anything that needs to be done frequently.

- Charnel Mouse
**Location:**England, UK-
**Contact:**

### Re: Need Mechanics/Math Help

OK, this is purely written with regard to theory rather than the practical stuff others are coming up with, so take this with a pinch of salt. I tried tweaking the system given the inital goals you outlined, and here's what I got. There's one big caveat, which I mention at the end. I am also completely ignoring criticals.

Say the attack value is a, the defence value is d, and the probability of success is p(a,d). Then the main thing it appears you wanted to have is p to be such that p(ca,cd)=p(a,d) for any constant c. The easiest way to get this is to look at functions of z=a/d, say p(z).

I'm now going to assume that ideally we'd like p to have the simple form p(z) = (Az+B)/(Cz+D), where we have to pick the values for A,B,C and D.

Now, you also wanted, under this notation, to have p(1)=1/2. I'm also going to assume that we have f(b)=0 and f(c)=1 for some values c and d. Originally, you had b=0 and c=infinity. The values for b and c gives a set of equations to solve for A,B,C, and D, that I'll spare you.

If we assume r is the rolled number, and assume, for the moment, that it can take any value between 0 and M -- not necessarily a whole number -- b and c give the following success criteria:

If c=infinity, r/M >= (1-b)/(z-2b+1) = d(1-b) / (a+[1-2b]d).

If 1<c<infty, r/M >= ([b-1]z+c)/([b+c-2]z+c-2bc+b) = ([b-1]a+cd)/([b+c-2]a+[c-2bc+b]d).

We'd like to choose b and c so that the fraction on the right hand side to be something easy for someone to calculate at the table, regardless of the value of a and d. The most obvious one is to take b=1/2,c=infty, so that a success happens when r >= (d/a) (M/2), i.e. ra >= d(M/2).

If we now switch to r being a whole number, this can be roughly approximated as succeeding if ra > d(M/2), where M is the number of sides on the die.

For a twenty-sided die, we thus have the reasonable case that the roll is a success if (roll)*(attack value) > 10*(defence value). In other words, roll a d20, times the result by your character's number. If this is higher than the opposing number times 10, you succeed. This is basically what JJR was talking about, but multiplying d by 10 is a lot easier than working out ra/d.

This comes with the main caveat that, although the success chance still scales, having an attack value of half the defence value or less gives an automatic failure, which is not really what you were looking for. It also means that the success chance is not symmetric between the attacker and the defender. But the maths is more friendly for people at the table if you do that.

Say the attack value is a, the defence value is d, and the probability of success is p(a,d). Then the main thing it appears you wanted to have is p to be such that p(ca,cd)=p(a,d) for any constant c. The easiest way to get this is to look at functions of z=a/d, say p(z).

I'm now going to assume that ideally we'd like p to have the simple form p(z) = (Az+B)/(Cz+D), where we have to pick the values for A,B,C and D.

Now, you also wanted, under this notation, to have p(1)=1/2. I'm also going to assume that we have f(b)=0 and f(c)=1 for some values c and d. Originally, you had b=0 and c=infinity. The values for b and c gives a set of equations to solve for A,B,C, and D, that I'll spare you.

If we assume r is the rolled number, and assume, for the moment, that it can take any value between 0 and M -- not necessarily a whole number -- b and c give the following success criteria:

If c=infinity, r/M >= (1-b)/(z-2b+1) = d(1-b) / (a+[1-2b]d).

If 1<c<infty, r/M >= ([b-1]z+c)/([b+c-2]z+c-2bc+b) = ([b-1]a+cd)/([b+c-2]a+[c-2bc+b]d).

We'd like to choose b and c so that the fraction on the right hand side to be something easy for someone to calculate at the table, regardless of the value of a and d. The most obvious one is to take b=1/2,c=infty, so that a success happens when r >= (d/a) (M/2), i.e. ra >= d(M/2).

If we now switch to r being a whole number, this can be roughly approximated as succeeding if ra > d(M/2), where M is the number of sides on the die.

For a twenty-sided die, we thus have the reasonable case that the roll is a success if (roll)*(attack value) > 10*(defence value). In other words, roll a d20, times the result by your character's number. If this is higher than the opposing number times 10, you succeed. This is basically what JJR was talking about, but multiplying d by 10 is a lot easier than working out ra/d.

This comes with the main caveat that, although the success chance still scales, having an attack value of half the defence value or less gives an automatic failure, which is not really what you were looking for. It also means that the success chance is not symmetric between the attacker and the defender. But the maths is more friendly for people at the table if you do that.

### Re: Need Mechanics/Math Help

Whoa, math. I'm going to need to try and process all that lol. Thanks!

Hm. If you had the person with the upper hand always roll, that would fix the auto-fail below 50% thing, and it would be symmetrical again, I think. In fact, I think it would be the same as my original math, the two main differences being:

1) GM must reveal whether the player or the opponent has a higher number and have that one roll, which introduces another (brief) step, gives away some information that may not be obvious in character, and means that in tough situations (like a boss battle), the players would mostly be watching the GM roll, since the opposition would have the better stats. This would have the effect of encouraging challenge via quantity of opposition rather than quality of opposition. (Fight the horde vs fight the boss.)

2) Instead of needing a chart to convert your attack and roll into the maximum opposition defense you can succeed against, you now need to multiply your attack by roll instead. (You technically need to multiply opposition defense, but that's just adding a 0, and if I went with this rule, I'd just bake that into the defense stat for purely defensive attributes anyway.) I'm not sure 17x34 is actually easier than looking at a 17,34 chart.

This is where I get lost. What are b and c? I assume it's not the same as in your earlier equations.

Hm. If you had the person with the upper hand always roll, that would fix the auto-fail below 50% thing, and it would be symmetrical again, I think. In fact, I think it would be the same as my original math, the two main differences being:

1) GM must reveal whether the player or the opponent has a higher number and have that one roll, which introduces another (brief) step, gives away some information that may not be obvious in character, and means that in tough situations (like a boss battle), the players would mostly be watching the GM roll, since the opposition would have the better stats. This would have the effect of encouraging challenge via quantity of opposition rather than quality of opposition. (Fight the horde vs fight the boss.)

2) Instead of needing a chart to convert your attack and roll into the maximum opposition defense you can succeed against, you now need to multiply your attack by roll instead. (You technically need to multiply opposition defense, but that's just adding a 0, and if I went with this rule, I'd just bake that into the defense stat for purely defensive attributes anyway.) I'm not sure 17x34 is actually easier than looking at a 17,34 chart.

I'm also going to assume that we have f(b)=0 and f(c)=1 for some values c and d. Originally, you had b=0 and c=infinity.

This is where I get lost. What are b and c? I assume it's not the same as in your earlier equations.

### Re: Need Mechanics/Math Help

What about using a variant of the Arkham Horror system (I'm sure it's been used elsewhere but AH is where I know it from)? You roll a number of dice based on your stat and then every die that comes up a 5+ is a success (or some other static number you prefer). Your number of successes needs to beat your opponents successes (which can either be rolled or simply their dice pool divided by three if you want to reduce rolling for monsters). I'd need to think about how to run the probabilities on this though, I'm not sure it would allow for the sort of meaningful progression you're after.

To be honest I think JJR's suggestion to use ratios is probably the way to go. In order to get what you want you'll NEED ratios in some form, the question is really how to accomplish it in a manner that is relatively easy to calculate.

To be honest I think JJR's suggestion to use ratios is probably the way to go. In order to get what you want you'll NEED ratios in some form, the question is really how to accomplish it in a manner that is relatively easy to calculate.

- Charnel Mouse
**Location:**England, UK-
**Contact:**

### Re: Need Mechanics/Math Help

Cuthalion wrote:Hm. If you had the person with the upper hand always roll, that would fix the auto-fail below 50% thing, and it would be symmetrical again, I think.

It would, yes.

Cuthalion wrote:In fact, I think it would be the same as my original math, the two main differences being:

It is similar. I think the original maths uses the probability 1-1/2z for z>1.

Cuthalion wrote:1) GM must reveal whether the player or the opponent has a higher number and have that one roll, which introduces another (brief) step, gives away some information that may not be obvious in character, and means that in tough situations (like a boss battle), the players would mostly be watching the GM roll, since the opposition would have the better stats. This would have the effect of encouraging challenge via quantity of opposition rather than quality of opposition. (Fight the horde vs fight the boss.)

If you're set on having symmetry without worrying about who's rolling, then the easiest option is to use success chance a/(a+d) = z/(z+1), because it's equivalent to using log(z) as the input to the logistic function (graph here). That will probably need a chart.

Cuthalion wrote:2) ...I'm not sure 17x34 is actually easier than looking at a 17,34 chart.

Yeah, that's fair. My hope was that most of the time it wouldn't be a close-enough call to need to calculate it exactly.

Cuthalion wrote:I'm also going to assume that we have f(b)=0 and f(c)=1 for some values c and d. Originally, you had b=0 and c=infinity.

This is where I get lost. What are b and c? I assume it's not the same as in your earlier equations.

Ah, I didn't notate these properly. Yes, these are different from earlier. b and c are new on this line, and we haven't decided on their value yet. Basically, we suppose that the success chance p(z) is zero if z=a/d is no greater than b, and is one if z is no less than c.

So b and c determine the ends of the range of z over which there isn't an automatic success/failure. This is before we account for a die not being a continuous RNG.

The rest of the equations are then about finding values of b and c to keep the formula for p(z) easy to calculate.

I originally picked b=1/2,c=infinity, in order to prevent any terms where a and d are added together.

The logistic version I mentioned above uses b=0,c=infinity.

How high do you expect the ability scores to go? That'll determine how unwieldy any charts would turn out to be.

### Re: Need Mechanics/Math Help

Charnel Mouse wrote:How high do you expect the ability scores to go? That'll determine how unwieldy any charts would turn out to be.

Thanks for the explanation! b and c as "the attack/defence ratios where you always fail and always succeed, respectively" makes sense.

It was fun to try to follow the math and suggestions you posted. I enjoyed it.

Ability scores typically start at 15 and increase from there. A level 10 character (as high as you can go in a single class right now without repeating it) can have a maximum of 50 in any one attribute. (Max is 3*level + 20, just so that I don't get min-maxers so intense that the game becomes unplayable.) Derived stats can be higher: weapon skills start at 15 or 20 and improve slowly, and accuracy and evade are each an attribute multiplied by the skill for the relevant weapon (calculated ahead of time of course). Thus, they are typically 3 digit numbers, but the charts are easier to read and the odds are the same if I have players divide by 10 before writing it down, so I do.

In other words, we're talking mostly 2-digit numbers here. Occasionally 3, but by that point you can probably safely switch to a chart that covers a higher range.

What do you think of the pair of charts from my OP?

- Charnel Mouse
**Location:**England, UK-
**Contact:**

### Re: Need Mechanics/Math Help

Cuthalion wrote:It was fun to try to follow the math and suggestions you posted. I enjoyed it.

Cool, glad it was more or less intelligible.

Cuthalion wrote:Ability scores typically start at 15 and increase from there. A level 10 character (as high as you can go in a single class right now without repeating it) can have a maximum of 50 in any one attribute. (Max is 3*level + 20, just so that I don't get min-maxers so intense that the game becomes unplayable.) Derived stats can be higher: weapon skills start at 15 or 20 and improve slowly, and accuracy and evade are each an attribute multiplied by the skill for the relevant weapon (calculated ahead of time of course). Thus, they are typically 3 digit numbers, but the charts are easier to read and the odds are the same if I have players divide by 10 before writing it down, so I do.

In other words, we're talking mostly 2-digit numbers here. Occasionally 3, but by that point you can probably safely switch to a chart that covers a higher range.

Hmm. I did some plots for scores in the 1:200 range against a defence of 50 for some of the different tests discussed when I wrote my original post. I could upload them and link, if those sound like those would help for comparison.

Cuthalion wrote:What do you think of the pair of charts from my OP?

I thought they were fine, although I personally preferred the one listed as the old chart. Repeating the roller's score between every single roll column feels like overkill to me, and makes the chart look cluttered. Maybe one on the left, one on the right, and one in the middle, at most? That's more something to test on your players, though: I'm more used to looking at charts in rulebooks than I am to having to regularly use them. It looks a better choice than the other option - where the row and column are the a and d, and the cells show the roll required - because that chart would get rather wide.

### Re: Need Mechanics/Math Help

Charnel Mouse wrote:I thought they were fine, although I personally preferred the one listed as the old chart. Repeating the roller's score between every single roll column feels like overkill to me, and makes the chart look cluttered. Maybe one on the left, one on the right, and one in the middle, at most? That's more something to test on your players, though: I'm more used to looking at charts in rulebooks than I am to having to regularly use them. It looks a better choice than the other option - where the row and column are the a and d, and the cells show the roll required - because that chart would get rather wide.

I'll have to try that! The new chart is faster to use, but makes it busier again, which makes it more intimidating. An older version was designed to work with 3-digit numbers, which meant there was very little whitespace. I also had to make the numbers on the row and column into ranges, which made it take longer to find your place. So, I had those 3-digit stats just divide by 10 as part of the calculation for determining them. So the character sheet now had them as a 2-digit number for any low- to mid-level player, which takes up less space and doesn't require ranges because you can fit 10x as many on there.

And an even older one did exactly what you said - the row and column were a and d, and the cells showed the roll required. That didn't work in practice because players tended to roll first and check their sheet next. So, the GM would be waiting for the player's a so they could use the d to look up the roll, but they'd get these numbers in the wrong order and have to start over and ask again for the roll after the player already announced it, because the GM wasn't ready for it until they heard the a. Plus, only the GM could really use the chart because it was designed to be used by someone who knew the monster's d.

### Re: Need Mechanics/Math Help

Revised chart. It's not an entirely fair comparison, as I changed the font and color, and it's in a more zoom-friendly pdf, but the main thing is that the player's attribute column doesn't repeat as often. Hopefully that makes it less busy.

https://drive.google.com/open?id=0B7rlo ... authuser=0

https://drive.google.com/open?id=0B7rlo ... authuser=0

### Re: Need Mechanics/Math Help

For the bored, I've found a dice mechanic that gets a fairly similar scalability and doesn't require a chart, at the cost of a little bit of bumpiness in the curve and steepness on the edges.

Every X points in an attribute, the die for that attribute increases size. Once you pass d12, you add +1 every time. Since I hate rolling d4s, even though they look cool, I start at 6, and since my stats currently start at 15-20 (I might compress them now that there aren't a lot of mechanics that need the raw number), it increases every 10 points. So, at 1-29 dexterity, you have a d6. At 30, a d8; at 40, a 10; at 50, a d12; at 60, a d12+1; etc. Most play is expected to stay within the d6-d12 range, with exceptions for high-level play and big monsters. This (less the d12+X rule, which I hadn't needed to think about until now) is a mechanic I'm already using for damage (strength, dexterity, or magic, depending on the attack) and recovery (endurance and magic), so really all I'm doing is giving another use to the dex die and adding one for the agility die. I might change the MP recovery to use aura, so I can make aura have a die, since it's the only attribute without one now.

So, you have a die for your dexterity and agility. Next, your weapon skills also give you a die that gets bigger as the skill goes up. Again, d6 -> 8 -> 10 -> 12 -> 12+X. These are specific to weapon type, but previously they affected both accuracy and evade. Dex had affected only accuracy, but for all weapons, and agility did the same for evade. Instead of multiplying the two numbers (blunt weapon skill x dexterity, for example) and recording them on you character sheet to get accuracy or evade and then comparing them with a roll on a chart, you now just roll both the dice for your attribute (dex) and relevant weapon skill (unarmed/blunt/edged/pole/throwing/archery) and add them together. The defender does the same, but using agility instead of dex and the skill for their own weapon instead of yours. Defender wins ties.

To my happy surprise, I found this gives pretty similar odds as the old method. It's not exact, and it assumes that the dice you use are proportional to the attribute behind them (d12 requires about twice as much dexterity -- 50 -- as the typical d6 -- <30). It gets a bit steep when you and your opponent are pretty far apart (ex: 2d6 vs d12+4+d12+4, which would indicate that the second guy has more than triple the attribute

It does assume that pretty much everybody will be playing in the d6 to d12 range, so it cuts into the theoretically infinite scalability, but I can live with that I guess.

======

For critical hits, I used to have them start at a 19 or 20 on the d20 rolled to see if you hit, and then as weapon skill increased the required number would go down by 1 each time. You still had to get a hit by the chart; rolling high enough to crit wasn't an auto-hit. Crits are meant to happen much more often in this system, since they trigger weapon effects that are meant to be part of the strategic choice of what weapon to use. So, the old odds would have been 10% at low level to 35% at high level, assuming that any roll high enough to crit was also high enough to hit (which, again, isn't strictly the case).

The new way is that you crit if your weapon skill die rolls 6+

Every X points in an attribute, the die for that attribute increases size. Once you pass d12, you add +1 every time. Since I hate rolling d4s, even though they look cool, I start at 6, and since my stats currently start at 15-20 (I might compress them now that there aren't a lot of mechanics that need the raw number), it increases every 10 points. So, at 1-29 dexterity, you have a d6. At 30, a d8; at 40, a 10; at 50, a d12; at 60, a d12+1; etc. Most play is expected to stay within the d6-d12 range, with exceptions for high-level play and big monsters. This (less the d12+X rule, which I hadn't needed to think about until now) is a mechanic I'm already using for damage (strength, dexterity, or magic, depending on the attack) and recovery (endurance and magic), so really all I'm doing is giving another use to the dex die and adding one for the agility die. I might change the MP recovery to use aura, so I can make aura have a die, since it's the only attribute without one now.

So, you have a die for your dexterity and agility. Next, your weapon skills also give you a die that gets bigger as the skill goes up. Again, d6 -> 8 -> 10 -> 12 -> 12+X. These are specific to weapon type, but previously they affected both accuracy and evade. Dex had affected only accuracy, but for all weapons, and agility did the same for evade. Instead of multiplying the two numbers (blunt weapon skill x dexterity, for example) and recording them on you character sheet to get accuracy or evade and then comparing them with a roll on a chart, you now just roll both the dice for your attribute (dex) and relevant weapon skill (unarmed/blunt/edged/pole/throwing/archery) and add them together. The defender does the same, but using agility instead of dex and the skill for their own weapon instead of yours. Defender wins ties.

To my happy surprise, I found this gives pretty similar odds as the old method. It's not exact, and it assumes that the dice you use are proportional to the attribute behind them (d12 requires about twice as much dexterity -- 50 -- as the typical d6 -- <30). It gets a bit steep when you and your opponent are pretty far apart (ex: 2d6 vs d12+4+d12+4, which would indicate that the second guy has more than triple the attribute

*and*more than triple the weapon skill, so the odds of the weaker hitting the stronger would already have been worse than 1:18 in the old system), but that would be an unusually unfair fight anyway.It does assume that pretty much everybody will be playing in the d6 to d12 range, so it cuts into the theoretically infinite scalability, but I can live with that I guess.

======

For critical hits, I used to have them start at a 19 or 20 on the d20 rolled to see if you hit, and then as weapon skill increased the required number would go down by 1 each time. You still had to get a hit by the chart; rolling high enough to crit wasn't an auto-hit. Crits are meant to happen much more often in this system, since they trigger weapon effects that are meant to be part of the strategic choice of what weapon to use. So, the old odds would have been 10% at low level to 35% at high level, assuming that any roll high enough to crit was also high enough to hit (which, again, isn't strictly the case).

The new way is that you crit if your weapon skill die rolls 6+

*and*your opponent's weapon skill die does not. (If your weapon skill die and your dexterity die are the same size, you have to designate which is which before rolling or just use whichever lands on your right as the weapon one.) So, weapon skill increases grow the die size, which increases your chance to crit and also your ability to parry enemy crits. I did some math with AnyDice (odds for this roll available at http://anydice.com/program/61fe), and the overall chance to crit goes up with power level. 6v6 is 14%, while 12v12 is 24%, assuming that you didn't miss by rolling really bad on your dex while your opponent rolled really well on their agility. This is good. It also means you're more likely to crit versus weaker enemies (12v6 is 49%) than stronger ones (6v12 is 7%), which is a new effect that I'm ok with. I think this new rule kind of puts a bigger spotlight on your weapon skill and weapon choices, and I kind of like that.### Who is online

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