A Mathematician Speaks: Rolling dice

[Vaxalon]

Okay, this is me trying to be helpful and offering what I know.  I tried to see if it was covered before, elsewhere, and I don't see it, but I could easily be wrong.  Also, for some of you, this will be "Well, duh" but I have found that some people can be remarkably math-phobic, even in the gaming community.  So here's what I got, edited down for the mathophobe.

One of the questions asked of mathematicians is, "If the most likely outcome of an event is X, how far do you have to go from X before you have encompassed most of the possible events?"

This is a question gamers want the answer to.  Let's say you're playing old-fashioned way-back Tunnels and Trolls.  You and your party are getting seven dice when you fight, but the troll is getting eight.  How often do you roll higher than he does, so you can do damage to him?  

This is where the standard deviation comes in.

One standard deviation from the mean (that's the fancy mathematician word for average) in both directions covers roughly 68% of the results.  Two standard deviations covers 95%, and three standard deviations covers 98%.

I'm not going to bore you with how to calculate standard deviations, that's not really useful to the game designer.   The point I'm going to make here, is that the more dice you use, the LESS random the result is!  That may sound counterintuitive, but it's quite true.  This is true whether you're summing the dice, averaging, taking the best, counting the 10's, whatever.

How many is too many?  That depends on your game, and the mechanics you derive from the dice, but it should be noted that the folks over at Steve Jackson games decided that anything over six dice just didn't have enough variation.  Above that score, you roll four, five, or six dice, and multiply by some factor.  60 dice isn't rolled; instead you roll 6d6 and multiply by ten.

And now we'll get to the point.  When creating a resolution mechanic, you need to consider the variability of the dice you're using.  A roll of a single die is much more likely to create "wild" results than a roll of several dice.

[Ron Edwards]

Hi Fred,

More good topics.

I've fully with you on the discussion of distributions and probabilities, and you'll find another enthusiastic participant in Mike Holmes.

If I'm not mistaken, what you're calling attention to is the difference between a flat and bell (or other curve) distribution. My favorite gaming example is, indeed, D&D of any period. You're rolling a single die with quite a range: 5-100%, broken into 5% increments.

Now, despite horrible eye-rolling and teeth-gnashing to the contrary, we all really know that there is an equal chance to roll a 20 as a 3, or any other single value, on a single die. The difference between a 2 or less and an 18 or less cannot be observed if each character is given only one chance to hit. It can only be observed across multiple instances of trying to hit (in which case the guy with 18 is better, duh).

But here you are, with a low-to-mid level D&D character. The quick way to get an effective character is not by increasing one's chance to hit - that comes too slowwww! And it really isn't going to make a difference in those crucial situations when you have 4 hit points left and simply must kill the troll right this turn. No, what you needed was a +12 Monster-Dicer sword, not for its increased chance to hit, but mainly for that monstrous damage. As you essentially "swung wild" per hit, making each one count is the key to success.

Same goes for the real meat in any rules-set which relies on a flat line distribution: getting multiple attacks. When you get those three attacks in one turn, in D&D, watch out! Because effectively, you didn't just get three attacks - you got one attack, using a 3d20 dice pool, with your damage dice multiplied by the number of successful dice in the roll.

Fred, is this along the lines of the issues you wanted to discuss in this thread? I'll do some hunting for the discussions from a couple years ago about sinusoid curves (bell curve offense vs. bell curve defense).

Best,
Ron

[newsalor]

A good point Vaxalon. Though it was clear to me already, I feel that many folks propably hadn't thought about it. This is a classic karma / fortune issue. People designing, tinkering with and playing games need to be aware of the choices they are making.

Many of the games here at Forge seem to be dice pool based games. When characters get more competent, fortune doesn't effect them as much.

One thing that I like about HeroQuest, is that you can pretty much tweak your drama/karma/fortune balance by yourself.

EDIT: Ron, an interresting point you make. D&D does have a bell curve, because each situation really needs several rounds of action, possibly with multiple attacks or something per round. On the other hand, HeroQuest has truly has a flatline, because a conflict is a conflict. Well, at least in simple contest that is. :) My players are all too aware that it's much harder to beat the odds in an extented contest. =)

[Ron Edwards]

Hiya,

Newsalor, that's right: consequence is also an issue. Most players in HeroQuest, when faced with a conflict which might include character death or anything else that's pretty undesirable, will shoot for an Extended Contest. In that case, the degree of consequences of single rolls are determined by bidding points, modified to be sure by the dice, but not wholly dictated by them.

One of the frustrating elements of low-level D&D play is that one's entire "imaginative entry into play" (the character) is at often risk based on the outcome of a single flat-curve roll.

Best,
Ron

[Vaxalon]

If I'm not mistaken, what you're calling attention to is the difference between a flat and bell (or other curve) distribution.

Yes, but it's not just that; it's also that the MORE dice you're using, the more "spiky" the bell curve gets; the more dice you roll, the closer the results cluster around the average.

[LordSmerf]

One thing that i find interesting is the shape of a probability curve generated by unmatched dice.  Rolling 2d6 gives you a triangle with its highest point found at the value 7.  Rolling 1d6+1d4 results in a triangle with a flat top centered on 6.

I am not sure that this is direcly relevant to the discussion at hand, but it is a rarely used option for probability curves...

Thomas

[Vaxalon]

Heh.  It's not.  But you're right that it's an option that creates subtle differences in probability distribution that people should be aware of.

[GB Steve]

Yes, but it's not just that; it's also that the MORE dice you're using, the more "spiky" the bell curve gets; the more dice you roll, the closer the results cluster around the average.

And this means that any contest that takes a lot of rolls to play out (such as a low-bidding Hero Wars contest) will tend towards the expected result more than one with fewer rolls (such as a high bidding HW contest).

The upshot is that with the smaller pool in HW, bid higher! You need those random kinks to win.

[Callan S.]

I'm going to make here, is that the more dice you use, the LESS random the result is! That may sound counterintuitive, but it's quite true.

I'm a bit dyslexic myself, but I thought this is intuitive. Indeed, its one of the hidden tricks of HP in D&D...who care's if someone rolls low or high at a lower level...the more you level, the closer you get to to having a rock solid average amount of HP.

Also, and I don't know if it's appropriate to bring up here, but it would be interesting to note the effect of to hit rolls on damage.

For example, if your rolling lots of D20's to attack and only hitting half the time, if your damage was a flat 10 points (not random), you'd find the average of the damage your doing on each attack is five points. Because your hitting half the time, your effectively doing half the follow up damage, really. Each attack essentially delivers five points of damage.

This means that systems that take account the quality of the hit are double dipping in a way. If your skilled, you will hit more often, which means you will be delivering a better damage average per attack. Now if you do more damage when your roll is beyond the target number required, not only are you hitting more often (with a better damage average delivered per attack), but adding onto the damage from that with more damage.

It's sort of like a death spiral from the other end...if the dudes got more skill than you (attack skill Vs defense skill), your screwed at a geometic rate as the difference increases.

[Ron Edwards]

Hiya,

Fred, the point about more dice yielding a more consistent average is clear. It directly relates to my own concern in the game Over the Edge, in which the crucial shift from 2d6 (most people at most things, and characters at most things) to 4d6 or 5d6 plays, in my opinion, an overly-intrusive role in the system.

I'm not sure if you're familiar with multiple-dice systems that do not rely on adding up the dice, or for that matter, even target numbers. The basic parent of these systems is probably Prince Valiant, which uses d2 (coins), but its prolific spawn include Story Engine, Sorcer, InSpectres, and many others.

Or conversely, consider "add-up" systems which do not rely on quantity but rather rank. I think the ultimate rock-solid "one curve to rule them all" game is Fudge, which relies on d3's (actually d6 with two plusses, two minuses, and two blanks). You roll four of them and superimpose the results on a seven-step ladder of competence (mediocre, fair, good, etc), starting from the point your character's skill or whatever is rated at.

The point of Fudge is to make sure that all characters are subject to the same standard deviation no matter what their basic competence is, and it it succeeds very well. Whether this is a good design consideration is a matter of local judgment and priorities.

Thomas, one game I've always wanted to play in order to examine the dynamics of the point you're making is Alternity. Wasn't there a thread or two about Alternity in Actual Play ...? Gotta check.

GBSteve, yeah! One of the things I like about HeroQuest is that bidding is literally the degree of investment, which also means the degree of risk, which also means the degree of unpredictability. I was very influenced by the dynamics of HeroQuest in designing Trollbabe.

Best,
Ron

[Vaxalon]

I'm familiar with OTHER systems that don't rely on adding up dice, such as WoD.  The thing is, it doesn't really matter what you're doing with them; adding them up, counting the number that come up with a particular value or more, etc., the fact remains that the more dice you use, the less variability you end up with.

Fudge has taken a bold step by always rolling what amounts to 4d3-8 to get a range from -4 to +4, clustered around zero.  I personally don't think it's a good design consideration, because some endeavors have an inherently greater amount of "chaos" in the outcome.

I've never played Alternity.

[Valamir]

Understanding probability curves and dice pools is a very powerful tool for game designers.  Understanding the implications of what seems to be a straight forward choice can make all the difference.

For instance, few people would consider D&D (and we'll talk 3E here though its true of all editions) a die pool system.  After all you roll a single d20 with a linear curve right?

Not in combat you don't.  No combat in D&D (save at the lowest levels) is resolved with 1d20 roll.  Its resolved only after a series of d20 rolls, the number of which is set by HitPoints/average damage per roll suffered (where per roll counts misses as 0 damage in the average).

If a combat lasts 8 rounds, than in terms of probabilities its the equivelent of rolling 8d20 (before accounting for multiple attacks).

Essentially D&D combat is a die pool system where the dice in the pool are compared to a target number (Armor Class) and successes are counted.  The only difference between D&D and White Wolf in this regard (other than die size) is that in D&D the dice in the pool are rolled 1 at a time and in White Wolf they're all rolled together.


What causes a major breakage in the d20 system is that ONLY combat is handled this way.  Everything else you might want to do in the system is handled by a single d20 vs Difficulty Class...usually with just 1 single roll.

So what you have is a normally distributed (mostly) bell curve for combat resolution, paired with a linearly flat curve for everything else.  This is not good for many reasons, not the least of which is the different scale by which Difficulty must be measured on.  Having a 50% chance to succeed on a single d20 is not that bad when you'll be rolling 8-10 d20s over the course of a battle.  Having a 50% chance to succeed on a single d20 is hugely risky when you have only 1 shot to make the roll.  One can't apply the same standard to determine DC for skill checks as you do for combat DCs.

I can't say for sure how much thought the 3E designers put into recognizing the difference in curves between the two systems.  In fact, I'd say very little since one of the huge selling points of the game was a "unified system" for everything, and most players would quickly point out that the "system" for handling skill checks is identical to the "system" for handling combat.

Fact is, however, its not.  Its very very different in a way that aspiring game designers should spend a lot of time being aware of in their own designs.  There may be a very good game enhancing use for using two seperate probability curves in your design.  But its something you should do on purpose, not by accident.

[Vaxalon]

THat's true, as long as you see a whole combat as one "event" to be rolled.  Many gamers don't see it that way.

Consider, for example, a climb.  In DnD, this is not handled by one skill roll.  This is handled by multiple skill rolls (one per 30' of climb, roughly) in which any one failure means the failure of the whole endeavor.

There are other examples of multiple skill rolls being required in 3.0 and 3.5 DnD.  Crafting, for example.

[Valamir]

Many gamers don't see it that way.

Exactly.  But they should, which is my point.  Because in probability terms that's what it is (or is closest to).

Consider, for example, a climb. In DnD, this is not handled by one skill roll. This is handled by multiple skill rolls (one per 30' of climb, roughly) in which any one failure means the failure of the whole endeavor.

Yeah, there are a few.  Basically a combat is broken into smaller pieces.  My point would be that if they really wanted to claim that skills and combat use the same system, they all should be.    All of the Skill uses should be broken down into smaller pieces...even if its just roll 3d20 representing the beginning, middle, and end of the task.

But, climbing.  Climbing isn't really.  Not as long as the penalty for failing any single climbing roll is falling to your death.  Then its just a series of unrelated linear tests.

[Vaxalon]

The penalty for failing any one climb roll is not necessarily falling.  Unless you fail by five or more, you just fail to make progress.