Anime Exponential Score Roll System

[Alex]

Hello everyone! I have been considering an idea lately for a random system that deals with exponential scores.

By score here, I mean any numeric value for a trait (whether it is an attribute, a skill value, a special trait or whatnot). Exponential scores aren't something new in RPGs. TORG in particular, I remember, uses them, and I remember seeing them used in various other games, especially those based on super-heroes. Exponential means that for every point a score is raised, the value it represents should be multiplied by some number. For instance, if the exponent was the fifth root of 3, that would mean that for each 5 points, the score represents a value that is thrice the old one. For instance, someone with a strength of 12 would be thrice as strong as someone with a strength of 7. Exponential values have two interesting aspects that make them useful in an RPG. First, a static difference is always equally important. The difference between a character of strength 10 and strength 13 is as important as that between one with strength 97 and 100. In particular, this means that static modifiers don't need to suffer inflation due to high scores. Another useful aspect, especially for campaigns with a large variety of powers, is that you don't need to have attributes that reach very high numbers to represent very high power.

One problem with exponential scores, however, is that for things that don't have a directly measurable representation, they can be a bit meaningless. What does it mean to say that someone is twice as intelligent as the average person? Or twice as dexterous? Or twice as good driving a car? But recently, I had an idea! We could define it by the chance roll! If someone with average intelligence has 1/3 of a chance to solve a problem, then the person who is twice as smart has double that chance. No, I don't mean he would get a chance of 2/3; I mean he would get two 1 in 3 rolls. Which, of course, can be calculated as a static chance of 1 rolls. Since the chance of failure in this case is 2/3, then the chance of failure for rolling twice is that squared, or 4 in 9. Which means the chance of success is 5 in 9.

Now, this is nice, but if each point of the score means that we need to double the represented value, that is, if our ratio is 2, then we don't have a lot of leeway to represent intermediary values. Besides, how do you deal with people who are below average? If you are only half as intelligent as the average person, what is your chance? Well, I realised we could leave all that to the exponential function! To make things less abstract, I will set a scale before explaining how this could be calculated. Also, since this calculation is a bit complicated, rather than dice rolling, it would be much simpler to make rolls using an electronic roller. I know this can be a let down for some people, but ultimately the result, as far as failure or success is concerned, is the same.

Anyway, for our example, we can say that 10 is the human average. Someone that has a strength of 10 is as strong as the average man. Our ratio is the fourth root of 2, which is close to 1.19. This may sound a weird number, but it means that for every 4 points a score is risen, the underlying characteristic is twice as powerful. A man of strength 18 is four times as strong as the average man. I like this scale because it doesn't stray too far from neither GURPS nor AD&D. If you were adapting stuff from either, as long as we weren't talking about something with super-strength, you could more or less use attributes as written. The scale also works very well for skills. Someone with a skill of 10 represents well someone that has just received basic knowledge of the skill. A skill of 12 is a starting professional. A skill of 14 is an accomplished professional, 16 someone who mastered something and 18 someone who is a leading authority in a field.

Anyway, to do a roll, the GM should come up with a chance for a certain base score. 10 in this scale is a very simple base number to use, since it is the average person. But especially for skills, it may make sense to use others. The GM might decide that a certain operation is really high risk and would be 1 chance in 3 even in the hands of a very good professional (skill 16). He could decide it is 50/50 for someone normal, or, in a supers campaign, he could decide to use 50 as the base for incredible feats. It doesn't matter what the base score is. To calculate the chance at our score, we can do the following calculation:

BS is the base score. BSC is the base success chance, that is the chance the action will succeed if attempted by someone with the chosen BS. BFC is the base failure chance. It is the opposite of BSC, and thus, 1 - BSC. CS is the character score. The actual score of the character we are considering. SD is the step difference. it is how many times the character in question is twice as good as someone with a score of BS. If SD is 3, for instance, the character in question is 8 times (2^3) as competent as the base score. SD = (BS - CS)/4, FE is the failure exponent. It it the power to which we will elevate the BFC. FE = 2^SD. AFC is the actual failure chance. AFC = BFC^FE. ASC is the actual success chance and the number we really want. ASC = 1 - AFC.

Sorry if this was a bit overly-complicated. The interesting aspect of this calculation is that it gives us a success chance that follows the aspects of what we wanted for our system (having a dexterity of 18 means that you have four chances that a character at dex 10 would have of doing something related to the score) while being able to deal with any value. In fact, it should be noted that the SD we calculated in the above algorithm could be a 0, negative or a non whole number. The calculation will work regardless. Exponential is continuous on rationals. If SD is 0, 2^0 will become 1 and AFC will be equal to BFC, as it should be (that is the case where your score is the same as the base score). As long as SD is greater than 0, even if it is not a whole number, 2^SD will be greater than 1. Which means that BFC^FE will be less than BFC. In other words, as long as your score is greater than the base, your chance of failure is less than the BFC.
If SD is less than 0, 2^SD is going to be less than 1, but greater than 0, which means that AFC^FE will be greater than BFC, but less than 1.

What do you guys think? Am I missing something about this that I should have realised? I've tried running some numbers against GURPS's 3d6 and the scores aren't dissimilar as long as we keep the numbers low, and the higher numbers don't break things, as long as they can be used to do actions that you would never try with the lower ones. I have also thought of a way to draw a "degree of success" from the rolls, but I will talk about it next post, this one is pretty long already.

 

[Alex]

If anyone is interested in seeing how the probabilities work, I made a table on google sheets calculating various probabilities on base 10 for various other bases. Worth noting:

6 has trouble with stuff that is simple for 10, which is what we want. 3d6 < 14 should be an easy roll (which you don't want to rely on saving your life everyday, but still something you should be capable of accomplishing most of the time). For 10, that is ok 83.8% of the time, but for a score of 6 you don't quite reach 60. For a 4 (a very low score, but what you might need to face given penalties), you are already more likely to fail than to succeed.

12 is not such a huge jump over 10, but it is statistically significant. A score of 12 can still almost succeed half of the time on a roll of 3d6 < 10, while a score of 10 is already at 37%. 14 can do it 60% of the time and a score of 18 can turn even a roll of 3d6 < 8 (a roll that would succeed only 16.2% of the time) into something more likely to succeed than not. I rather like that, basically, a very renowned surgeon (for instance) has a real shot at something that a doctor still in school knew he would likely be murdering his patient if he tried. But it still is far from safe. A 22, someone that would either be the best surgeon in the world or just fiction (more or less where the border would be) could do it 75% of the time.

Very high values don't trivialise mundane chances too fast. 3d6 < 7 (a roll that has a little less than 10% chance of succeeding) can only be reliably done (chance of 95% or greater, basically, you would only fail this on a d20 if you rolled a 1) on a score of 30 or above. A score of 50 is almost always going to succeed even at the hardest "mundane" stuff. The 3d6 rolls were used because they work well in GURPS. A 3 in that game is a critical success. Rolling a 3 on a 3d6 only happens in 1 out of 216 rolls. A character with a score of 50 (which is where I would put a powerful super's scores at what he is good, so basically superman) can do that 99% of the time. Note that a score of 50 means you are 1024 times as good as the average person.

"Impossible" rolls, rolls that wouldn't happen in 1 in a million or worse, require really high scores (probably outside the scope of most games, but it is interesting seeing it work). A score of 90 (not surprising, 1 million times better than the average person) is more likely than not by then. 1 in a billion still is really unlikely by 100, but would be more likely than not by 130 and one in a trillion by 170. As such, a score of 200 (about 200 quadrillion times better than the average) can easily beat even such unlikely odds. Of course, by then it makes much more sense to use a different base than to try to split hairs over what is 1 in a million chance and what is a 1 in a trillion. And, of course, this kind of number probably only makes sense if you are taking Dragon Ball more seriously than anyone ever should. Still, nice to know that even with luck 100, you aren't going to win the lotto every time.

 

[deuxhero]

Mutants and Masterminds (and I think a lot of super hero games in general) have attributes be exponential except for chance of success

 

[Alex]

Mutants and Masterminds (and I think a lot of super hero games in general) have attributes be exponential except for chance of success

As far as I know, trying to model the chance of success like this, where character A having an attribute twice as good as character B means A has the same chance as two attempts of B, is something new. I can't be sure, especially because some games that use attribute tables might have replicated that idea in how their total chance is generated. I particularly liked it because it feels a lot less arbitrary than other systems where the numbers are just thrown around but never given any kind of base for their value. Someone with intelligence 14 in gurps is very smart, but other than how the 14 relates to the 3d6 dice throw, it is not ever made clear how they compare to someone with intelligence 10. A system like D&D is even more arbitrary, and while going to AD&D 1e and 2e does make different score mean different things by making it give a variety of different bonuses, it still is not quite clear how having an intelligence of, say, 18, should compare to 10, except that you are in the top 0.5% of the population.

 

[Mr. Magniloquent]

I don't understand the value of exponential curves. It's unwieldy and doesn't lend well to modification. What advantage does this have over something like, "Roll XdY, take highest"?

 

[Alex]

I don't understand the value of exponential curves.

There are two properties of exponential numbers that I consider useful. One is that you don't need numbers that are too big to represent forces that are in very different scales. This is useful in some styles of campaigns and useless or even bad in others. For a game like GURPS that tries to be generic enough to represent very different power scales, I would consider it a boon. For a game like Call of C'thulhu, on the other hand, at least assuming you won't fight C'thulhu with a mech, it probably is unnecessary complication.

The second property is that a "+1" is always relevant. The difference between strength 10 and 11 is not more or less significant than the difference between strength 100 and 101, or 0 and 1, or -20 and -19, etc. This means that things like bonuses and penalties for creatures with really high (or really low) attributes don't need to suffer through inflation or deflation.

It's unwieldy and doesn't lend well to modification.

Could you explain this point better? If you are talking about the specific system I've created for generating success or failure, I do suppose it is a bit unwieldy. But the formula used and the rationality behind it are explained and open to the GM to modify if they so desire. I consider the most important aspect of my post not so much the use of exponential scores themselves (thought that is what gave me the idea in first place) but what is the meaning of the score of not easily measurable attributes like dexterity or intelligence or charisma or skill levels.

What advantage does this have over something like, "Roll XdY, take highest"?

Well... at least once you have it in program format it is faster (well, you could have a program roll XdY for you too, though). The X in this format would allow you to account for the same idea I mentioned earlier. Someone that has an intelligence of 3d8 is three times as smart as someone with 1d8 (it is a bit unclear how he compares with someone who has intelligence 2(d12+2), though). But if this is taken to a game that has very high attributes, the number of dice can become a liability. In my experience, this was never a problem in Deadlands, but it apparently was sometimes a problem in Shadowrun.

Another advantage of the system I proposed is that it would allow for intermediary numbers, and in fact, arbitrary scales. Using XdY, a 1dY has to be the smallest possible value, and every other value must be a whole multiplier of it.

 

[Mr. Magniloquent]

The +1 always being relevant is good. Being inflation proof is very valuable. I overlooked that. I find this system not good for modification because you're only ever able to modify it via an integer which always produces a set significant effect. This system is good at representing magnitude, but not simulating natural ability vs skill/experience. Maybe I am mistaken.

Also, 1dY isn't the smallest value, per se. Below 1dY take highest, is XdY take lowest. The flaw here is that someone with larger dice will be more affected than someone with small dice when their pool goes "negative". Scaling doesn't become too much of an issue, because if you're dealing with demigods or the like than you can do something like XdY, take highest 2 if you don't like simply increasing the dice size.

 

[Alex]

(...)I find this system not good for modification because you're only ever able to modify it via an integer which always produces a set significant effect.

Do you mean modifying the roll? Or the system itself. If the later, could you provide an example of what you might want to modify so I can better envision what you are talking about? If the former...

You can modify it via an integer bonus or penalty. As you pointed out, this is a bit unwieldy because the +1, like mentioned before, is always significant. This means that if you want to add up a bunch of different modifiers you will end up with ridiculous numbers. For instance, if you can get 8 different sources of a "+1" to your roll, they are making you four times better than normal.

But there is a different variable as well, which is the base chance. The base chance requires a chance variable (which could be a percentage, a fraction or even a set die roll) and a baseline, which is the attribute of what you are trying to compare to. For instance, the base chance could be 50% at base 10, which has the meaning of: "something the average person could do, but would be as likely as not to fail". This means that various circumstantial modifiers, rather than being processed as a bunch of bonuses and penalties to be added together, could be summed up instead as "this is so and so hard for this kind of person". For instance, a player wants to do a daring maneuver driving his car at night, in the rain, right were the mountain road takes a sharp turn. The GM could go: "well, this is a really hard one. Even the best people in the stunt business would probably only manage it one in four tries. So he sets the baseline as 18 and the chance at 25%. Or perhaps, rather than saying "one in four", he prefers to think in GURPS terms, and says the effective skill would be 6 (3d6 < 7). He could input that too, and the random generator would calculate the percentage chance for him (around 9.26%). It would also be possible to add up bonuses and penalties if we keep the "3d6 < Difficulty" thing as an integral part of the system. In fact, it would be compatible to how GURPS does this. For instance, if we used GURPS' tables for shooting distance, size and speed and just added the penalties and bonuses to difficulty, it would make sense! So bonuses to the attribute itself could be kept for things that literally boost your performance while the small stuff could be added to the difficulty. Although this binds us to the 3d6<X format somewhat.

This system is good at representing magnitude, but not simulating natural ability vs skill/experience. Maybe I am mistaken.

This is something I've been considering how to do, how to model skills in this system. One aspect of how GURPS does it that I dislike is that DX and IQ in that system (dexterity and intelligence) form a base value for your mental and physical skills respectively. A DX 15 person will become a working swordsman, gymnast, runner, swimmer and several other capacities with just a bit of training. For some skills, this is actually feasible, but not others. And the difference between someone with a lot of raw ability and someone who lacks that but has a lot of training is minimal to non-existent in that system.

One way to fix this, at least for mental skills, is to have int not work as a "baseline" for the skill at all, but rather as what is tested when you are learning. A high IQ person can learn a bunch of skills quickly because he won't fail the downtime learning rolls. If he is daring, he can tackle a harsher learning regimen to learn even faster, but at a higher chance of failure. The skill of the teacher, both in what is being taught and in didactic, could affect this somehow as well.

But this doesn't work so well with dexterity. In theory, I believe DX as a baseline for physical skills is at least partially true. Someone who is learning fencing has to learn what movements are good when using the fencing sword. What works and what doesn't, what to pay attention to in their stance and what they expose to an enemy. But once that is done, the difference is all about committing this to muscle memory and reflexes (or, at least, that is how I understand it. I am open to different views here). Perhaps for physical skills, rather than making the value of the skill be open ended, it could be limited to the specifics of the skill. For instance, a swordsman has to learn positioning, the different kinds of parrying, of feinting and attacking he can use. Once he has learned these well enough they are part of muscle memory, the only way to become a better swordsman is either: a- develop an even better style, with better moves and positioning. Or b- become more dexterous (which is probably not really feasible, but in an RPG it might be).

Also, 1dY isn't the smallest value, per se. Below 1dY take highest, is XdY take lowest. The flaw here is that someone with larger dice will be more affected than someone with small dice when their pool goes "negative". Scaling doesn't become too much of an issue, because if you're dealing with demigods or the like than you can do something like XdY, take highest 2 if you don't like simply increasing the dice size.

Yes, but I meant, preserving the idea of "someone A being "twice as dexterous" as someone B means that if B's chance of doing something is "2d8, take the highest and see if it is 7 or more", then A's chance is 4d8 vs difficulty 7. Inverting the mechanic (roll two dice, take the worst) somehow doesn't work as the opposite by this logic. The chance of "2d8, take the worst > 6", if done twice, won't have the same chance of succeeding as simply "1d8 > 6". My point is, of course, you can use dice pools like XdY in a lot of cool and interesting ways. But if you are committed to keeping the idea I mentioned about "twice the skill means you have two chances where the baseline would get one", those variations don't conform to the idea.

 

[Mr. Magniloquent]

I think we understand each other. Your system is much like AD&D. I think it had been originally designed as 3d6 rather than d20. Its nice and tidy as long as you don't do much to it or make too many demands of it. I think you're going to end up with a lot of tables, just like AD&D. I personally don't like that.

I enjoyed reading all this though. I have tried to get people here to nerd out on RPG dice mechanics before but haven't gotten any takers. Both surprising and disappointing. Www.anydice.com is a fun tool and has lots of comparisons to different RPG dice mechanics. Check it out.