Quote from: Lunamancer;868027You say "sometimes." When wouldn't that be possible?

Obviously when the result depends on the result of the first roll. I didn't mention that because it is or should be obvious. Without much thought, two examples of this come to mind.

• The dice rolled for damage in a number of games depends on the degree of success of the attack roll. In some games (RQ/CoC/BRP, H+I) it depends on a comparison of the degree of success of attack vs. defense.
• In games that use a roll and add mechanics like the wild die in WEG D6 the need for a second, third, or N+1 rolls depends on the result of the wild die in the Nth roll.

QuoteI'm not 100% sure what you're saying here, so I'll just say what I think. You're free to agree or disagree. Damage matters to a shield just as it matters to hit points. Even if there are no wound penalties in the system. Because it affects further decision-making. If I'm walking around with 10 hit points and orcs be doing d8's, I know I can definitely survive one hit from an orc. If an orc then hits even just for 2 damage, that certainty goes out the window.

Whether or not one can definitely survive is system dependent. I don't play systems where definite survival due to having enough hit points to shrug off a blow is likely. In most it isn't even possible for survival to be definite as opposed to being highly probable. That may affect how I look at what is relevant in calculating survival odds.

QuoteI understand the math involved. I just don't find it particularly relevant. There are generalized characterizations of every dice mechanic. They're not always accurate.

Within the limits of the simplification the math is fully generalized and accurate for the system I mentioned (RQ). The formula certainly allows for the case when the defenders armor is below the minimum damage roll. In that case that term in the equation becomes 100%. Perhaps you don't find cumulative probabilities relevant because you say you are solely and narrowly focused on the GM determining modifiers. I am not so narrowly focused. And neither are players. Note that a more complicated formula could be crafted for the unsimplified cases, but it would be significantly more complicated and since you don't seem familiar with RQ it wouldn't be especially informative.

QuoteAs to defense rolls such as parrying, parrying in LA is not an automatic defensive action. It's a choice. So if I want to know my character's odds of hitting Parry Master McGee, it depends whether or not he's going to attempt to parry. I have no way of knowing that until I hit, so the only thing there is to calculate is my own skill roll.

Parrying is not an automatic action in any system I have played. But assuming an opponent isn't going to parry by just ignoring the probability that your blow is parried gives a false idea of the likely outcome. For example, it treats the opponent with a 10% chance to parry the same as an opponent with a 90% chance to parry. Now I assume that attacking is also not an automatic action in your system of choice. (It isn't in any system I am familiar with.) In which case ignoring the parry chance is similar to treating two attackers with a 10% chance to hit and a 90% chance to hit as each automatically hitting since the defender doesn't know if they will or will not attack him.

QuoteAnd by the way, even this example is a little bogus because my desire for transparency of probability is NOT to help players calculate odds.

I didn't know that. Transparency is a term I see used in discussing the visibility of what is going on to two sides in a transaction or activity, e.g. transparency in how governmental decisions are made. Here you are using transparency to describe your ability as the GM to understand what affect a modifier has on the probability of an outcome. There is no other side. That's probably not the word I would have used for that situation, but now I understand what you mean.

In any case, I do care about the transparency of probability for the player since that informs decision making in the game. The point I made was that sequential linear probabilities obscure the probability of the outcomes that are truly critical to player and GM decisions, i.e. Which side is likely to succeed? and Who is likely to survive and who is not? Perhaps you discount this because you are focused on GM assignation of modifiers in what seems to be a D&D derived class/level game where mere ablation of hits points without any definitive result that ends or significantly changes the combat, (e.g. a kill, incapacitation, disabling, penalty, or breakage result) is what occurs in the majority of successful attacks and but you seem to discount the value or importance in decision making of actually being able to assess what will end the combat and how likely that event is.

Personally, I am less concerned about the sort of transparency you describe for me as a GM. I already have a pretty well developed intuition about simple probabilities both linear and nonlinear based on a lot of experience and a bit of study. When picking up a new system, gaining an intuitive understanding of the odds is one of my major areas of attention. As a GM and I run practice simulations to learn the system and to hone my familiarity, especially when the system mechanic is not linear and obvious e.g. a simple d100 or d20 roll.  So for a game like Honor+Intrigue which uses a 2d6 roll I have a good intuition about the effect of a modifier of +N or –N to the chance of getting a success. I don't need to see or compute the exact probability to assign reasonable modifiers during play. In hindsight I can calculate the odds to see exactly what impact a modifier made. But it just isn't particularly relevant in the moment to assigning the modifier.

QuoteNot really.

No really. You mentioned a number of things you don't like

Quote from: Bren;867956wanting to "dumb down math", wanting to "do a billion things in one roll", "Vegas", "nerds", "high rollers", removing "randomness from character generation", or ignoring "race/class restrictions."

These dislikes do not seem related to linear mechanics or to issues of transparency. Some of the things you say you like (or at least you dislike their converse) aren't even better supported by linear mechanics.

• Not wanting to dumb down the math doesn't lead to a preference for linear mechanics. The advantage of linear mechanics is that they simplify the math compared to nonlinear mechanics. So anyone who prefers simpler math should want to use linear mechanics since they come closest to "dumb down math." The simplest random mechanic is the simple linear mechanic known as a coin flip.
• If you prefer random character generation, OD&D/AD&D is the classic random character generation RPG and it uses nonlinear mechanics (3d6) for the majority of the character generation.  
• And "race/class" restrictions is completely orthogonal to linear mechanics.

So not related in any way that I saw.

Quote from: Lunamancer;868027I never really did. I asked one specific question that dealt specifically with probability distributions for degrees of success. So far, nobody's admitted they even care about that, so I haven't asked for the opinion of anyone posting so far.

Well now there appears to be someone who cares.

Quote from: Lunamancer;868034I'm not 100% sure if people realize this is what the effect looks like, and if it's actually what they desire.

Yes I realize the effect. Yes the effect is what I want or I wouldn't use that mechanic or that modifier.

QuoteAgain, I ask, is this really the result people who opt for a bell curve mechanic are expecting?

Again I answer, yes and yes.

It seems like what people say they want, approximately? Gathering from JoeNuttall and nDervishes' description of what they liked about bell-curves.
(I think JoeNuttall said he had a set his system to a fixed -3 =50%, after letting 10s on the 2d10 explode, ...though I don't think most Nd6 systems go through especially rigorous analysis).

Quote from: Lunamancer;868034But if instead of a "flat" 30% modifier, I introduce a successive die roll that cuts probability of failure in half, then Dick Marvel tops out at only 90%.

How would you do that specifically, though?
For instance, if you reduce Dick (80% base) failure chance by giving him a re-roll, with success generated by either roll, then his success chance would actually be 96% (20% chance of failure on either roll = 1 in 25 chance of failure overall).

Quote from: Bren;868037Well now there appears to be someone who cares.

About the probability distribution, at any rate. I think most of the bell-curve people even, probably want a distribution change relating to success-chance, rather than degree-of-success.

Quote from: Lunamancer;868034Well, I often get the feeling that people are caught up on the idea of the "bell curve" more than the reality of the bell curve.

You've not commented on the bell curve in my post, which this doesn't apply to. There's a link in my sig "Open Dice" to the system which talks about the issue in your post, and how I solved it.

One side-effect of the exponential drop-off I mention in other blog posts is the consequence of having a roll to hit followed by a roll for damage. Bren talked about how probability wise this means that to calculate your chances you have to multiply the probabilities for each roll. But with logarithms multiplication becomes addition.

That is, since +3 in either roll has the effect of doubling the chance of success, it (mostly) doesn't matter where you have your bonuses (except for flavour and tenstion in the game). That means you can see how good you by just adding your bonuses, and no-one has to understand probabilities.

So the end result is actually more transparent than with two linear rolls.

Quote from: Bloody Stupid Johnson;868051How would you do that specifically, though?

Roll twice. Take best roll of two. Dick has .2 x .2 = .04 chance of failure. Joe has a .6 x .6 = .36 chance of failure. Johnny has a .8 x .8 = .64 chance of failure. So for success, Dick = 96%, Joe=64%, Johnny=36%.

But as you point out, the numbers he gave don’t match what is calculated by taking the best of two rolls or by using a rule like you have to fail two rolls to actually fail. So either he’s using a different method or his math is wrong.

Quote from: JoeNuttall;868058But with logarithms multiplication becomes addition.

That's the nice thing about logarithms alright. It's almost like they were created* just so people didn't have to do difficult multiplication by hand.


* If you thought I should have used a different verb than "created" like say, "discovered" then as far as the ontology of math you are Platonist.

Quote from: JoeNuttall;868058You've not commented on the bell curve in my post, which this doesn't apply to. There's a link in my sig "Open Dice" to the system which talks about the issue in your post, and how I solved it.

It's hard to make heads or tails of it without seeing the exact probabilities. I mean what I get based on the graph and estimating probabilities is that the cumulative probability still has an inflection in the curve. So that means both extreme high and extreme lows of the scale would have modifiers change probability of "success" by only a small percentage relative to points closer to the peak of the bell (or inflection point in the graph that link shows).

So if you have the exact probabilities, would you mind looking at what TN is associated with a 40% chance of success, how much of a bonus would that character need to have a 30% chance. Then apply that same bonus to someone whose base TN is such that the person has a 20% chance of success. What is that person's modified probability? And then repeat for someone whose base skill is such that normal chance for success is 80%. What is that final probability?

QuoteThat is, since +3 in either roll has the effect of doubling the chance of success, it (mostly) doesn't matter where you have your bonuses (except for flavour and tenstion in the game). That means you can see how good you by just adding your bonuses, and no-one has to understand probabilities.

So here I'm not following at all. I see in the link a 3 point adjustment is supposed to double/half probability of success. I can see how that might work for higher base TNs. It definitely does not work on the lower TN end of the scale.

I know for certain the only way to get a "2" is rolling 1 on each die right out of the gates. Which means if your TN is 3, you have a 99% chance of success. If you apply a -3 modifier to the dice, the odds of success (I'm guestimating) goes to the very high 80's. That is neither dropping the probability of success by half, nor doubling the probability of failure.

In essence, while you may have solved the problem for one end of the curve, the curve is still has the characteristic I pointed out about a regular bell-curve. Modifiers have the biggest percent-wise impact on the mid-range (or peak range or inflection range, whatever you want to call it) and relatively lower impact on BOTH high ends and low ends of the skill range.

Again, I say it isn't bad or wrong. I'm just not sure that's how all gamers who choose bell curve intend things to work.

Quote from: Bloody Stupid Johnson;868051How would you do that specifically, though?
For instance, if you reduce Dick (80% base) failure chance by giving him a re-roll, with success generated by either roll, then his success chance would actually be 96% (20% chance of failure on either roll = 1 in 25 chance of failure overall).

You generally don't make the exact same skill check twice. If I want to cut in half probability of failure, I flip a coin.

The concept actually arises completely organically in certain situations. Suppose a character is playing a game of chance where you place a wager and have a 1 in 3 chance of winning (you have to pick one of three boxes where one of them, determined at random, has the prize). But I have a character with Luck ability at 25%.

Do I make the character roll luck to win? Failure means loss? That would be silly. Your reward for having Luck is getting only a 25% chance to win rather than 33%.

Instead I could say if you pass your luck check, then by luck you choose the right box. But if you fail your luck check, you are left with the same chance anyone else would have. What this is doing is diminishing your chance of failure (from the perspective of a luck skill check) to two-thirds that of normal.

So if you have zero luck ability, your odds of getting the right box is 33%. The logical amount any person should have. A boost of 33% relative to 0% skill.

If you have 25%, your odds of getting the right box is 50%. A boost of 25% relative to skill 25%.

If you have 70% Luck, your odds of getting the right box go up to 80%. A boost of just 10% relative to skill.

Another situation where this comes up organically is I had borrowed the "firing into melee" rule from AD&D 1st Ed, where (adjusted for size) each person has an equal of being hit. I modified this for the Lejendary Adventure system, which has an Archery ability to say random selection only occurs when the ability check fails. You might still get your exact target by random selection, though. (I also added Archery as a proficiency to my AD&D games, it has this same effect.)

So if there are 4 potential targets, Johnny B Bad's probability of getting the right one (with his 20% skill) is 40%. For Joe Average (with his 40% skill) is 55%. Dick Marvel (with his 80% skill) is boosted to 85% of hitting the right target. Again, we see the percentage effects of positive adjustments have diminishing returns the higher in skill you are. This is consistent whether you're on the low, average, or high end of the scale.


I should also point out, because I feel this starting to get lost the deeper these discussions go on, that I still like linear modifiers. For some situations, maybe even a lot of them, I find them more appropriate.

Quote from: Lunamancer;868104Another situation where this comes up organically is I had borrowed the "firing into melee" rule from AD&D 1st Ed, where (adjusted for size) each person has an equal of being hit. I modified this for the Lejendary Adventure system, which has an Archery ability to say random selection only occurs when the ability check fails. You might still get your exact target by random selection, though. (I also added Archery as a proficiency to my AD&D games, it has this same effect.)

So if there are 4 potential targets, Johnny B Bad's probability of getting the right one (with his 20% skill) is 40%. For Joe Average (with his 40% skill) is 55%. Dick Marvel (with his 80% skill) is boosted to 85% of hitting the right target. Again, we see the percentage effects of positive adjustments have diminishing returns the higher in skill you are. This is consistent whether you're on the low, average, or high end of the scale.

It looks like you are saying that if Johnny is shooting at one single opponent with no one else in the way he has a 20% chance to hit the person he is aiming at. But if there are three other people around his chance of hitting the person he is aiming at is effectively 40%. Is that correct?

Quote from: Lunamancer;868097It's hard to make heads or tails of it without seeing the exact probabilities.

100%   99.00%   97.00%   94%   90%   85%   78%   71%   62%   53%   44%   35%   28%   22%   16%   12%   9%   6%   5%   4%   3.30%   2.60%   2.00%   1.50%   1.20%   0.90%   0.70%   0.60%   0.50%   0.40%

Quote from: Lunamancer;868097I mean what I get based on the graph and estimating probabilities is that the cumulative probability still has an inflection in the curve. So that means both extreme high and extreme lows of the scale would have modifiers change probability of "success" by only a small percentage relative to points closer to the peak of the bell (or inflection point in the graph that link shows).

You're missing what I said in my first post:

Quote from: JoeNuttall;867706For anything with a <50% chance of success, +3 means you double your chances of success, -3 means you halve it. So +1 and -1 always have the same effect.

It's not logarithmic for things you have a greater than 50% chance of success at. That's by design.

Quote from: Lunamancer;868097So if you have the exact probabilities, would you mind looking at what TN is associated with a 40% chance of success, how much of a bonus would that character need to have a 30% chance. Then apply that same bonus to someone whose base TN is such that the person has a 20% chance of success. What is that person's modified probability? And then repeat for someone whose base skill is such that normal chance for success is 80%. What is that final probability?

13+ is 35% chance of success, 14+ is 28% chance of success, so +1 makes you only 80% as likely to succeed.
16+ is 16% chance of success, 17+ is 12% chance of success, so +1 makes you 75% as likely to succeed.
As you can see from the graph the precise value of +1 varies slightly, but it's always around 80%.

As I said before, the system doesn't behave like that if your success rate is 80% as that's above 50%. The value of a +1 drops gradually to almost nothing.

Quote from: Lunamancer;868097In essence, while you may have solved the problem for one end of the curve, the curve is still has the characteristic I pointed out about a regular bell-curve. Modifiers have the biggest percent-wise impact on the mid-range (or peak range or inflection range, whatever you want to call it) and relatively lower impact on BOTH high ends and low ends of the skill range.

No, it doesn't happen at one end and at the other end it is there by design. I don't talk about why in that blog post as I thought that would overcomplicate the issue.

I originally had a pure logarithmic system but I didn't like that +1 changed you abruptly from 100% success to 80% success. I thought about that for a long time before deciding that wasn't what was wanted, so I replaced that with a 2-ended open system which was logarithmic for success in one direction and failure in the other. That was definitely a move in the right direction, but in practice no-one cared about the subtle difference between 98% success and 99% success, so I went for this third system as it was simpler.
 

Quote from: Lunamancer;868097Again, I say it isn't bad or wrong. I'm just not sure that's how all gamers who choose bell curve intend things to work.

What I've found out is that whether players like a mechanic or not is a combination of several things, and when writing a game you have to be prepared to chuck the coolest idea you ever had because it just didn't survive a playtest.

@Lunamancer: Thanks for clarifying.
Its working in those situations. I have my doubts that it can be applied across a range of situations though, without the extra die roll seeming fairly tacked-on in many of them.
I get that on the linear vs. nonlinear adjustments, with your method -assuming its justifiable - of course you can apply either form of modifier, whereas for the multiple-dice-roll its more difficult to go the other way and apply a 'linear' modification - short of having specific rolls that revert to d20 rather than 2d10, or the equivalent. And transparency is clearly better for linear rolls as well.

Quote from: Bren;868109It looks like you are saying that if Johnny is shooting at one single opponent with no one else in the way he has a 20% chance to hit the person he is aiming at. But if there are three other people around his chance of hitting the person he is aiming at is effectively 40%. Is that correct?

I'm assuming the roll to hit the correct person in melee would have to be on top of the normal to-hit roll? Or, at least there'd need to be some sort of to-hit penalty for firing into melee as well.

Quote from: Bloody Stupid Johnson;868125I'm assuming the roll to hit the correct person in melee would have to be on top of the normal to-hit roll? Or, at least there'd need to be some sort of to-hit penalty for firing into melee as well.

One would hope.

Based solely on tactile aesthetics, I have always preferred the feel of a pair of dice (of any sides) in my hand to that of a single die.  This probably explains my fondness for 2d6 systems.

Quote from: Bloody Stupid Johnson;868125@Lunamancer: Thanks for clarifying.
Its working in those situations. I have my doubts that it can be applied across a range of situations though, without the extra die roll seeming fairly tacked-on in many of them.

One thing about having a wealth of options for modifying probabilities for a given situation is it makes you think about WHY you're assigning a modifier. And I think if you stop and think about it, there are plenty of applications.

20 years ago when we were playing CyberPunk, we were afraid of driving. Every time we decided to get into a car, the GM made us make driving checks. On the one hand, it's reasonable. Accidents happen every day in even mundane travel. There should be some probability of that in the game.

On the other hand, it was absurd. He wasn't the greatest GM. Even if you maxed out your driving skill, you still had a 1 in 10 chance of failing the check. Race car drivers don't crash THAT frequently.

A more reasonable approach would be to have "wandering monster" checks while you're on the road. These could be things like traffic jams, rude drivers, etc. There's a small chance of especially dangerous road conditions that would actually require a check. This would be another organic example of factoring in a second probability check to reduce the rate of failure.

You are in essence reducing the opportunity for failure. And that is the case in the magic box game or firing into melee. There's a natural chance of just getting the right one by dumb luck, even before it becomes a question of skill.

Consider an attack where there is an active attempt at a parry. Only do something radical. View it from the perspective of the defender. If the attacker has a 50/50 hit probability, the probability that the attacker misses diminishes the opportunity to fail at your defense roll.

This sort of thing is virtually everywhere.

QuoteI get that on the linear vs. nonlinear adjustments, with your method -assuming its justifiable - of course you can apply either form of modifier, whereas for the multiple-dice-roll its more difficult to go the other way and apply a 'linear' modification - short of having specific rolls that revert to d20 rather than 2d10, or the equivalent. And transparency is clearly better for linear rolls as well.

Another way to get "curve" results out of a d20 mechanic is to instead roll 3d20 and take the middle result.

QuoteI'm assuming the roll to hit the correct person in melee would have to be on top of the normal to-hit roll? Or, at least there'd need to be some sort of to-hit penalty for firing into melee as well.

Yes. In AD&D, due to the chaos of melee, firing into melee calls for the target to be chosen randomly. That would mean the apt comparison is 40% likelihood of the intended target being the one selected in a group of 4, vs 100% likelihood in a group of 1. The hit roll still has to be high enough to hit the AC for whoever the target is. The idea of bringing in a proficiency is so instead of getting even odds, you skew them in your favor.

Quote from: Lunamancer;868176Yes. In AD&D, due to the chaos of melee, firing into melee calls for the target to be chosen randomly. That would mean the apt comparison is 40% likelihood of the intended target being the one selected in a group of 4, vs 100% likelihood in a group of 1. The hit roll still has to be high enough to hit the AC for whoever the target is. The idea of bringing in a proficiency is so instead of getting even odds, you skew them in your favor.

Assuming the targets are equally likely, the random probability of the intended target being chosen in a group of 4 is 25% not 40%.