Die, d20 die! Part 2

 photo CartoonGuideToStatistics_zps68ac7ffa.jpg
When I first wrote “Die, d20 die!” my intent was to keep it math light and fun; in fact it was originally part of another article that it didn’t fit well in so I pulled it out and submitted as a separate article.  I also thought that “There is just so much randomness in a d20 roll (uniform distributions with large spreads give RPGs bad statistics) that it forces you to use large numerical bonuses, which are a VERY bad thing.”  was a self evident statement; particularly since Wizards of the Coast is trying to move away from large numerical bonuses while keeping the d20.  However, I’ve gotten some feedback that questioned the statement I thought was self evident and asked for both a more detailed mathematical treatment and more examples of alternate dice systems.  I was originally going to respond with a comment but I decided that I had enough to say that it could be another article.

I should mention that I have corresponded with the poster known as “Lethal Dose” about the current article, having shown him a draft.  He’s another math/stats guy and the conversation was fruitful.  He’s been encouraging me to get a lot more precise, technical, and “mathy” with this article.  For the most part I’ve resisted, in order to keep the article accessible to gamers without “advanced” mathematical training, but it did become apparent that I needed to define what I meant by the term “randomness” because Lethal Dose and I were using it different ways.  When I say that something is “highly random” I mean it has a large “spread,” where “spread” is the “typical range” which is roughly proportional to the standard deviation.  When I say that something has “little randomness” I mean that it has a small “spread,” i.e. that the distribution is very peaked, i.e. that a lot of the probability mass is in a small region of input space.  It is the size of the “spread,” the typical range, not the entire range (from minimum possible value to maximum possible value) that determines how large a numerical bonus is needed to significantly alter the probability of rolling a success.

So why are large numerical bonuses bad?  Let me count the ways…

The first, most obvious answer is that math done at the gaming table should be simple, and adding large numbers could slow things down.

A second less obvious answer is that it promotes min-maxing and game imbalance; note that I said promotes not requires.  It can also make combat encounters long, boring, and frustrating.

In Star Wars Saga Edition, it was very easy to get a Reflex defense over 40 by level 20.  So what exactly does that imply?  Well let’s think about what it would take to hit such a character.  A maxed out offensive monster of a character would have a Base attack Bonus of 20, point blank shot for +1, weapon focus for another +1, maybe a +4 from Dexterity.  Various other bonuses could apply like the superior weapon focus Elite Trooper talent which is good for another +1, careful shot which is good for another +1 or +2 if should with the double trigger modification; but since a Reflex defense of a maxed out defensive character can very easily be over 40, we’ll simplify matters by saying that the probability of a 20th level offensive monster hitting a 20th level defensive monster’s Reflex defense is about 25% or less.

Most games don’t actually take the players to level 20 and the Big Bad Evil Guy is usually at least a few levels higher than the PCs so he can take on the whole party by his lonesome.  Thus combat monsters in the party are going to hit a well built BBEG maybe 15% to 25% of the time while less optimized (more narratively focused) PCs will only hit on a natural 20.  If a PC can only hit the BBEG on a natural 20, that gets frustrating and boring really fast; heck even only a 15% to 25% hit rate isn’t fun.  So what’s a PC to do in this situation? well there’s an aptly named Saga edition era episode of the order 66 podcast about this entitled “Punch ‘em in the Dump Stats”, but seriously it was very doable to create a 20th level character that had 40 or higher in all three defenses; yes I confess I like optimized characters and I prefer to max-mean rather than min-max.  Min-max implies that the character is good at one thing to the exclusion of all or at least most other things.  I liked building characters that were a jack of all trades and a master rather than a grand master at several… If you’re interested in seeing one, send a PM to EliasWindrider on the d20radio.com forums and ask for the Saga edition build plan for a “Jacen Baurne,” but that is not the point of this article.

So what’s a PC to do?  Well there are always destiny points that allow you to score automatic critical hits on cue; yes that’s fine and dandy but the BBEG has destiny points too and can turn those back into misses, and most GMs I know implement house rules to limit destiny point hording by the PCs so they can’t outspend the BBEG too much.  I can and did write an article on the topic of narrative fiat mechanics which you can read here.  But I think I’ve already made the point that large numerical bonuses promote min-maxing and game imbalance.

Third, when large numerical bonuses are used, it makes a small numerical bonus, read as incremental character improvement, insignificant.

I’m of the opinion that a +1 should mean something; in FATE, which is one of the most narrative RPGs there is, an unmodified roll can take on a value o -4, -2, 0, 2, or 4, and a +1 or +2 is a really big deal.  I think FATE takes it to the extreme in this regard; in my opinion, so as to leave room for incremental character progression, a +1 should be meaningful and a +6 should be a big honking deal.

A user by the screen name of “LethalDose” raised a few other points that I want to address (italicized text indicates a quotation):

A d20 roll is essentially a binomial (success/fail) distribution. The “natural 20″ critical hit result actually extends the sample space somewhat, but I think we can skip that part for the time being.

Actually, I think that’s something we should address rather than skip.  The problem with critical hits occurring on a natural 20 is that it comes up too infrequently, at a 5% rate.  Let me explain why.  To give everyone a chance to act a few times in a combat encounter while still avoiding player boredom, typical combat encounters should last between 5 and 10 rounds, with 5 to 7 being the ideal range.  Narratively, and for player enjoyment, an individual PC should be able to score a critical hit about once every two combat encounters.  This means that the ideal probability of a critical hit (if everything comes together perfectly) is about 1 in 12 or 1 in 14 (and yes a roughly 1 in 14 probability of a critical hit is doable, I give an example later in this article).  This means a d12 is a much better choice than a d20 (with a d12 a bonus of +6 bonus is also a “big honking deal” so the d12 wins on that front too).

There’s really no point in the game where the numerical result of the a d20 roll directly has any bearing on the game. If the numerical result did matter, then the result of a 10 should be different, and twice as great, as the result of a 5. This is simply not the case in d20 systems.

Using a similar system with a d12 instead of a d20, you could add the excess (d12 + attack bonus – defense) to the damage roll; you can’t really do that in a d20 system such as Saga because d20 + attack roll – defense can be huge in some cases.

Actually, In my view, using a d20 is a great choice for action resolution for several reasons. First, its easy for a novice to wrap their mind around the probability of the results and the value of a +1 ( = +5% success), there’s less addition of dice on the fly, and the value of a bonus really is static (not dependent on the difficulty of the task). When the resolution system moves away from a uniform distribution, the value of bonuses becomes dynamic and more difficult to understand.

I disagree with a lot of that paragraph.  I’ll explain why in a moment, but first let me preface this with “It’s my opinion; I’m not saying that you’re wrong.”

I’m going to start with a counter example; the narrative dice mechanic in FFG’s Star Wars RPG is very fun and intuitive to use, but Jay Little intended (and succeeded) at making it difficult for players to wrap their heads around the stats on the fly at the gaming table.  Yes I have communicated with Jay Little about this and I agree with his reasoning.  If players can calculate the odds of success they tend to do whatever is most likely to succeed rather than what’s most narratively appropriate and the focus should be on the story rather than the dice.

All a player needs to know is whether it’s better to roll this die or that die, not how much better.  And there’s a certain intuitive qualitative understanding of a dice roll that comes easily when the typical result is close to an average result.  This is one of the reasons that small variances are a good thing, and uniform distributions with large ranges are a bad thing.

A good friend of mine, who has above average intelligence but is an artsy fellow and not a math guy, once told me that a 1/20 probability of a natural 20 doesn’t mean that a natural 20 comes up about 1 in 20 rolls.  Yes he said that very poorly which makes it a blatantly false statement, and he didn’t understand the math well, and either his long term tracking of dice rolls is seriously lacking (as it is for most people) or he is cursed with bad luck (I suppose this is possible because life has curb stomped him repeated, although I would fault his interpersonal skills or lack thereof) but he was without knowing or fully understanding it, picking up on the reality of the Poisson distribution.  The interested reader can read about the Poisson distribution on Wikipedia.

They say “there are lies, damned lies, and statistics” because statistically true statements can be counterintuitive, deceptive, and seemingly contradictory according to the typical person’s “common sense.”  The point is that in terms of human perception/experience, rolling a d20 (read as any uniform distribution with a large range) is not easy to grasp even though the math is simple (this is one of the reasons that a natural 12 on a d12 is much better for a critical hit).  In terms of human perception/experience, making the typical roll close to the average roll is a good thing; in statistical speak that means small variances are often greatly preferable to large ones.

I agree with you that adding a lot of numbers at the gaming table is a bad thing, but I do not agree that this is the only alternative to a d20.  The central idea that makes dice pool resolution mechanics such a big breakthrough is that rolling a lot of dice does not require that you add a lot of dice; you can interpret the dice individually or as sums of small subsets.  For example, in the 2d10 system I gave an example of rolling 6d10 and summing only the largest two dice; that math is simple, actually it’s simpler than adding a large numerical bonus to a d20 because with the d20 both the die result and the bonus can be larger than 10.  For a lot of people, working with numbers larger than 10 doesn’t come as naturally as single digits; this probably has something to do with humans having 10 fingers.

Another user asked for additional examples of dice systems that are alternatives to rolling a d20.  Here are two more.

Ailowynn from the d20radio.com forums is working on a RPG as a hobby.  He posted a request for feedback online and since then I’ve been suggesting mechanics to him; he’s been choosing from among my suggestions and sometimes modifying them.  The idea he had for his central dice rolling mechanic was that there are 3 dice, where a d100 counts as a single die rather than two dice, of different colors that represent success, difficulty, and fate.  Typically fate is against you and the fate die acts as an additional difficultly die, but when fate is on your side it acts as an additional success die.  When there are two success/failure dice, you use the larger of them.  I personally feel that d100’s (percentile rolls) are vastly more vile than d20’s, so I suggested that he use d12’s instead of d100’s.  Difficulty is subtracted from success, and if that number is greater than or equal to 0 then the attempt succeeds.  Critical hits occur when the success die is 12 and the difficulty die is not 12.  When fate is on your side, the probability of a critical hit is about 1 in 7.  When fate is against you (usually), the probability of a critical hit is about 1 in 14.  As requested, here are bar charts for those rolls, for when fate is against you and on your side.
 photo d12r_zps13d14064.jpg This figure is for the probability of a result without a modifier.

This next figure is the probability of success after a modifier is added; the excess (success die + modifier – difficulty die) of any roll is added to damage.

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The next dice mechanic is the one I use in my 3D RPG, but before I get to it let’s talk about the desirable qualities for an attempt resolution dice mechanic.  I hold these truths to be self evident, but maybe next week I’ll be writing “Die, d20 die! Part 3” to explain them.

  • It is preferable to resolve an entire action with a single dice roll rather than having success determined by one dice roll and conditional upon success rolling an additional set of dice.  Being able to resolve two actions with a single dice roll is even better.  This speeds up game play.
  • It is desirable to roll between 3 and 6 dice at a single time, closer to 3 is preferable, because it means that less time is spent choosing and interpreting the dice.
  • It is desirable to add no more than two dice, when interpreting the roll.
  • It is desirable for the dice roll to result in minor success frequently and major success rarely.
  • It is desirable for the dice roll mechanic to be able to resolve degrees of success and multiple levels of special effects.

The central dice roll mechanic I devised for my 3D RPG accomplishes all of those objectives.  The basic concepts are that you have a die between a d4 and d12 for each attribute and skill, a d4 is bad, and a d12 is awesome.  Equipment also provides dice.  For an individual attempt, you build a pool of 3Dice (hence the name 3D) by choosing your 3 dice from among the ones narratively applicable to the attempt.  Note that most actions have one required skill, but other than that, anything you can justify narratively is fair game.  Think about that for a moment… the mechanism for dice pool assembly encourages narrative justification, i.e. creative story-telling.

Then you roll your dice pool and sort the results in descending order; i.e. the die with the largest rolled result is your “first die,” the die with the second largest rolled result is your “second die,” and the die with the smallest rolled result is the “third die.”

For a single action, the first die is used to determine basic success or failure.  For normal skill checks, additional successes with the second die and third die reduces the time it takes to complete the task.  For attacks, you sum the second die and third die to determine the damage you deal.  Thus no more than two dice are added during the interpretation of the roll.  The damage the target takes from a successful attack is the damage you deal minus their damage reduction (DR) and the stats work out so that on a typical roll you hit and do a few hit points of damage, while on a good roll you can do a bit more damage.  If the damage is reduced to zero or less by the target’s DR, they take no damage and the attack is described as either a near miss or a mild blow that glances of their armor harmlessly.  Note that frequent minor success was a stated goal; this is accomplished by typical damage taken from an attack being only a few hit points after a DR.   Note that the second die and third die determine degree of success for both attacks and normal skill checks.

Since we’re talking about attacks, before I go any further I should say that weapons have a die for damage and a die for finesse; you can use either, neither, or both in your dice pool.  The damage die is about what people are familiar with from D&D while finesse is new.  For example, a long sword has a damage of d8 and finesse of d8 when wielded in one hand; when wielded in two hands it has a damage of d10 and a finesse of d6.  A two handed sword has a damage of d12 and finesse of d4.  A knife has a damage of d4 and a finesse of d8.  A short sword has a damage of d6 and a finesse of d8.  A rapier has a damage of d4 and a finesse of d12.  However, a weapon’s finesse is capped by the character’s skill with the associated weapon proficiency.

For a double action, the first die and second die determine the binary success/fail of the two actions and the third die determines the degree of success for both.  For an attack made as part of a double action (regardless of whether the second action is either a normal skill check or another attack) the third die is counted twice to determine damage for that attack; DR applies separately to two attacks made as double action.   Thus two actions can be completely resolved with a single die roll and the component actions in a double action are individually less effective than a single action.

Also, my 3 dice mechanic has graduated special effects, similar in concept but not implementation to the narrative dice in FFG’s Star Wars RPG.  Each maximal die (die that rolls the maximum possible result, e.g. a 4 on a d4 or a 12 on a d12) generates one advantage.  When multiple advantages are rolled, you can spend them individually for a minor benefit, or together for a more major benefit.  For example a critical hit cots either 3 advantages, or 1 Karma chip, and you can bank advantage using my destiny mechanic to save up for a critical hit on demand.  You can read more about Karma and destiny in my article about narrative fiat mechanics.

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