May 7, 2009

Towards more Cinematic Gaming, Part 1 - RPG Statistics

The d20...

So easy, so iconic. So flat.

Ah, but what I am talking about it the statistical distribution of outcomes. I love the d20 for its ease and its iconic fun-ness, but I hate it for the fact that getting a 1 is just as likely has getting a 20, or a 7 or an 11 for that matter.

Thus the likelihood of success is easy to calculate, but is always linear. You simply add up the numbers from the chart above that would be considered a success. Need a DC15? Six possibilities - 30% chance.

Although this is fine for quickly and easily determining difficulty and outcome, there are several problems with it.
  1. The outcome is not predictable, since a 3 is just as likely as a 12. A 1 just as likely as a 20.
  2. The outcome is always binary: i.e. you either succeed or fail. You survive the Death/20 poison or you die. Outside of a critical you can't hit 'really well'. Your damage dice are completely independent of your hit roll. It is IMPOSSIBLE outside of a coup-de-grace for James Bond to kill any non-mook with his Walther PPK, no matter how skilled the shot.
This can result in a break from disbelief of the players, because when that really skilled +20 player rolls a 3; he still hits but we all still see the 3 and cant help but imagine him clumsily pulling off a hit by the skin of his teeth. We feel let down by the 3, Ultimately our minds want to see a high to-hit roll result in a "better hit" than a low one. Similarly when a player rolls a 19 and then a 2 on his damage dice. Pity the letdown of the less-hardcore gamer who sits patiently waiting for their turn with the expectations of helping, then roll a 4 and then dejectedly sit out of another 20 minute pass through party actions.
How many times have you scored a Natural 20 only to roll a 1 on the damage dice?
Can it be made better? Can it be made to better fit mental expectations? Does it even matter?

Honestly, I don't think it matters much with novice gamers. The novelty of playing D&D sustains them alone. However many experienced gamers depend on "feeling like they are there", and if suspension of disbelief is broken they quickly become bored.

In college, I was fortunate to have a great group of gaming buddies who were all in engineering and good at math. We played multiple campaigns and constantly tweaked and adjusted the rules set until it was nearly unrecognizable. Our usual modus operandi was that if the outcome of a rules set did not fit what we would imagine, then we would change it until it did. The mathematical gymnastics were still easy for us then, and the result was such a gripping sense of 'reality' in our games that even simple missions were extremely memorable and a whole hell of a lot of fun. Unfortunately, it typically took a new player months to learn the rules, until then they just acted their characters and the GM would do the necessary dice mathematics. (ironically, this made it closer to a 'pure' RPG). I dug up my old house rules changes and they are... ready... 60 PAGES long. Not doable by any means for those with a full time job.

Ok enough rambling.

Lets take reality first. Many would agree that the vast majority of results of chance events in life fit into two distributions:
The Bell Curve

and the Poisson Distribution (when outcomes are uncommon, or when deviation from the mean is biased)

(I am leaving out bimodal and other polynomial distributions)

The Poisson Distribution is far more common than you would expect, especially once bias is taken into account. For example a skilled marital artist is far _less_ likely to make a mistake than he is to pull off a move extraordinarily well. That is because his training lets him feel when things are not quite right and correct his balance, etc. The 'success' tail of the distribution would be broader, and the 'fail' curve would be more truncated and abrupt. For an unskilled or fatigued fighter the outcome would be more normalized and flat.

Both situations definitely do not fit the flat curve of the d20, although god-bless-it, the d20 is easy, fast, and looks cool. So what I ask is: Are there any other easy and fast options? I'll share what I have experienced in some of my games, though please, comment and share how you have tackled this issue in yours.

  1. The most obvious simple answer would be to roll 2d10 rather than a d20. this generates a bell curve of outcomes, and is thus a bit more predictable. Downside is that critical success is still just as likely as critical failure, and the outcome is still binary. FYI: this is a very easy change to incorporate into a campaign but be advised, high numbers are about half the likelihood, so in general you need to half the magnitude of all situational modifiers. You only have a 10% chance on 2d10 to get a 17+. EDITORS NOTE: Using 3d8 approximates the same effects as a d20 (~5% chance to get a 20), but still leave a nice bell curve for the rest of your rolls. Of course, you can also roll a 24 this way too...
  2. One could just stick with the d20 but make the degree you beat the targets reflex defense affect the number of damage dice rolled. A basic form of this is used for the barrage rules in d20 Star Wars for banks of laser weapons on large capital space ships. Basically the weapons bank had a huge + to hit (like +30) but only did low base damage (2d8). But for each couple of points it beat your AC by, it did another 1d8 cumulative. SO most of the time it would do a little damage to you, but sometimes it would do a whole lot, and taking evasive measures to raise your AC directly affected how much damage you took.
  3. Mix the two and you get a decent Poisson distribution. One could roll 2d10, but depending on character skill, have broader crit ranges, and re-rolls of low numbers. Critical hits will add an additional d10 to the roll. (Ex: an expertly trained swordsman might crit on 15+ as well as reroll any individual 2's or 3's on the d10's. While untrained swordsman crits on 19+ and does not reroll anything.) this generates a fairly nice curve without too much mental gymnastics. (BLUE: Roll 2d10. RED: Roll 2d10, Reroll individual 1's, If roll 20, then add 1d10, shown in yellow). Thus although the expertly trained swordsman still mostly rolls an 11, fairly rare for him to get less than a 6, common for him to get in the teens, and has a low but real likelihood of getting 20+ (~1.2% of the time)
  4. Some old-school systems such as Shadowrun (and more recently Vampire: The Masquerade) build asymmetrical distribution into their core mechanic. They roll skill # of dice against a target number, with number of success as the outcome (iterative dice model). Very similar to the skill challenges of D&D4E. For example you might have swords skill 5, so you roll 5 d10's against a target number reflecting how difficult your target is to hit. You therefore generate a # of success as the outcome. Usually 1-2 with the occasional 0, 3,4, or 5. By adjusting the target number you can easily change your distribution bias from left to right. In the hands of a very mathematically inclined GM this method can be made to duplicate very 'real' feeling games. Unfortunately it needs either a ton of trial and error, or a someone with a graduate level statistical background to wrap their head around the probabilities when generating rules on the fly. It is very difficult to write rules in this system on your own.
  5. Old school Marvel Super Heroes basically just mapped out a Poisson distribution to percentiles and then made this big chart on the back of the players guide. You would roll percentile for everything, then cross with your skill mastery level on the chart and determine outcome. It actually worked pretty well, but you basically spent your whole game reading small numbers on charts. It was also fairly difficult to generate rules on the fly to fit a novel situation as in #3.

So where does all this leave me?

I'm not really sure where it leaves me. I am currently playing a d20 Star Wars SAGA campaign and I am enjoying it. But on the other hand I still find the core d20 rules extremely arbitrary. The 2d10 method is a nice quick change, but attribute bonuses then become very dominant, and the assist actions becomes very powerful. These are not necessarily bad things.

I crave my old statistical knowledge that allowed quick and agile use of the iterative dice model. I must admit those were the best adventures we ever ran in terms of mental satisfaction of both social and combat aspects of RPGing. Im not sure what it was; those rule sets felt very alive. They were hard to learn, but god dammit those were some fun-to-play rules. We played pre-made adventures then, and now matter how boring thought the adventure was going to be, the rule set never failed to keep us on the edge of our seats.

Recently I have played a small one-shot d20 adventure where we made beating the AC by 5 double damage dice, and beating it by 10 triple damage dice, etc. Natural 20's ignored AC. We also rerolled 2's, 3's, and 4's depending on if you were skill focused or mastered. The results were OK, and still played fast. I think were going to try method #2 in our next one-off and see how it goes.


  1. I was going to complain that you never actually explained the Poisson distribution, but I went to Wikipedia and they don't really explain it either. The idea I'm getting is that it's what a binomial distribution looks like when it's just getting started, and only the right tail is showing.

  2. I'll jump in here; since I'm not sure how long it will take for Tom to reply.

    A Poisson Distribution approximates a number of events over a time interval. It's a descrete probability distribution; think of it like a tally of counts over time... it has a skewed tail. Here's what they look like:"To be fair, I added some of the graphics to the post above (Tom's still learning the Blogger platform). But, in my opinion, I would rather not see a Poisson for gaming and prefer a "long tail" distribution such as something called an extreme value distribution. In an EVD, there IS a chance that you can get a rediculously huge outlyier in the results (like... the arrow hit the orc in the eye, or the Fireball spell fed off the campfire and literally exploded the tree).

    This is what EVD's look like:
    (look at the second distribution shown) to get a "long tail" distribution for dice rolling, you would have to have 1) a pool of dice that are summed (like 3d8; which is my preferance, alone these would provide a Normal distribution); and 2) and an extreme success mechanic - like; if you roll an 8 on any of the dice, roll another d8 or something. I think shadowrun used to have a system like that.

  3. Fire arms, in the Skill and Tactics AD&D 2nd edition book, also used 'exploding' dice, if you rolled maximum damage then you got to roll again for additional damage. I never used firearms in any of my campaigns though so don't really know how this felt when it was in use.

  4. The example Poisson graphs all have three or four different shapes depending on lambda. Which shape do you want?

  5. Any chance of posting those house rules, or at least the interesting dice mechanics in them?

  6. This was a great article. I like the "higher math" and the exploration into the probably of using different dice combinations. However, I think I'd have a really difficult time selling this to my gaming group. It just seems like an additional level of rules for them to learn with little additional benefit (from their point of view). Although I'm sure these changes work great once you get to know and understand them, it's getting over that initial hump of new rules that my group will frown on.

    I think this will work great for advanced players who are looking for something a little different. But for players who just want to kill monsters for a few hours a week, I think this is way more detail then they'll even need.

  7. I think that you are missing the part where rolling a d20 was just a pass/fail test, albeit with a different percentage than 50/50. You seem rather blinded by the idea that rolling a 20 is inherently better because it is higher than a 12. Not so for attacks and saves. One simply rolls higher than their target number, which becomes easier as the target decreases with level. There were no critical hits in the beginning, and there have never been critical saves. With the earlier editions, there were no degrees of bypassing the enemies' defences, you either did or you didn't. The damage roll determined how effective that was.

    So, you are starting with something of a misguided premise; if your target number is 11, then 20 is no better than 12. In fact, as you mention, the odds of getting either are exactly the same, hence, there is no solid reason to make rolling a 20 anything extraordinary.

  8. It's odd how blog post topics seem to sometimes sync up:

    I'm working on a different method to create an alternative system for the original D&D systems. I think I've just about got the kinks worked out so hopefully I'll get a chance to describe how it works (and start playtesting it) in the not so distant future.

    The problem with messing with the d20 with the D20 system (3rd & 4th ed) is that so much of the system is built upon that horrible, flat probability curve. For example, higher DCs become much more difficult to hit once you move away from the linear increase you see with a regular d20.

    2d10 works to create a pseudo-bell curve (2d20 is fascinating to me because it generates these two linear curves that meet at the mode which my gut tells me could be cool to fool with) which is nice but as Jonathan mentions the modifiers suddenly become very powerful. Hence the whole system gets thrown out of whack. The beauty of the system though is that suddenly critical hits (and misses) are actually meaningful because they occur much rarer than any other result.

  9. Sorry I couldnt reply earlyer, my job is ... extreme... at times; I promise I will get to all those comments tomorrow when I have a day off - I am really flattered at all the discussion!

    Anyway, Poisson distribution is a asymmetric distribution of # occurances on repetetive measurements, when the events have an inherent bias as opposed to purely random. ie: rare events, or when the operator/subject is doing somethign about it. It can predict the likelyhood of a given level of occurance. The sqewed-ness or bias is reflected by the lambda. The more bias, the more assymettrical the distribution is. It is by far the most common epidemilogical distribution in living life. In my grognard RPG experiences, dice models that duplicate these asymmetrical distributions feel much more 'alive'

  10. @ stormbringer - you are right, d20 disconnects hitroll and damage; thats why I dont like it. Instinctually I always wanted a 'good roll' to be a 'better hit' The d20 disconnect ends up making the game *feel* more arbitrary *to me*. I am curious if I am wierd of if anyone else feels the same. My post is all about changes that can be done to re-intertwine hitrol and damage to make the game feel more alive and less empty, if you feel like I do. But in no way do I think everyone *should* feel that way!

  11. OK this is going to be a long one but I do want to get to everyones questions now that I know I am goign to have a day off tomorrow.

    @ TheRecursionKing
    I loved 2nd edition combat and tactics; that book was awesome. It was the seed which got me messing around with rules in the first place. Totally worth picking up off ebay if anyone ever gets a chance.

    @ Noumenon
    You ideally want a mechanic that can adjust the lambda based on skill: a untrained fighter would be closer to a bell curve, thile a expert woudl have a heavy lambda skeq to the curve. This can be accomplished by letting a higher skilled character reroll numbers on EITHER d10 of a 2d10 roll: untrained, no rerolls. Basic: reroll 2's. Expert: reroll 2's and 3's. Master: Reroll 2's 3's and 4's. That shifts the average over. Then to lengthen the tail, extend out when the dice explode: ie: [Approximate] basic crit woudl be 18+ (6%), expert crit woudl be 17+ (10%), master would be 15+ (21%). Unfortunatley Jon did the graph with only rerolling 2's and only exploding the dice at 20 which would be less than basic training in the last example, so thats why it basically just looks bell curved. not his fault though, I was essentially unavailable all week.

    @ Jalepeno dude: ooh, they are kind of a big file and not all aspects are down in print, some are still in our noggins (such as out magic rules); but I'll pretty them up. Back in the day we used the iterative dice model starting with 2d10 and using exploding dice first used in 2nd edition combat and tactics like the firearms, but we used it for all weapons. Ended up very much like white wolf or shadowrun, though we were playing it way before vampire came out. It rocked. pick up an old 2nd edition shadowrun use them to play D&D. its fun) Right now were getting ready to start a simpler campaing closer to the d20 ruleset that uses the following [relatively] lighter variations: d20 where you add extra damage dice for every 5 you beat the ac by; AC is just based on size and dex; Armor is damage reduction. Your con bonus + 4 is your wound threshold; for every mutiple of this after your damage reduction it results in a mild, serious, or critical wound. Called shots can be declared to further add wounding effects. Hit points are now just "stamina points" and when gone you are simply two tired to fight on/unconcious, but you can take second winds just like in d20 Star wars Saga, and everyone starts out with a base of approximately 30 hp and level 3. Thus defeating an enemy quickly is more based on achieving serious and critical wounds, whether by high dmage weapons, rolling high hit numbers, or by called shot wound effects. We have playtested it and it runs smoothly once you practice a few combats; makes battles with big guys very tense and dramatic, with lots of assists to get hit rolls up high and achieve serious and wounds (cant fight with a broken femur). Peons go down rapidly in grisly fashions since their wound thresholds are fairly low. Gives a very cinmatic dichotomy of the large troll with the huge club (high dmaage weapon) vs the nimble assassin who can do very high damage or elevate the woudn category with a crefully placed called shot. DANG, I could go on and on but then this post woudl be 20 pages.

    @ Ameron - you are absolutely right: but note this - the 4E rule book is a couple hundred pages long... no matter how you dress your house rulset up it will still likely be less than 50 pages. Nevertheless, if your group isnt bored with the standard d20 ruleset, then keep having fun and save all this stuff for later!

    @ stormbringer
    If your TN ws an 11, what if getting that 11 got you d8 on your longsword, getting 16 got you 2d8, and getting 21 got you 3d8, etc? then three things would happen: 1. You really could be james bond and drop that huge badass with your walther p47 if you rolled high enough. 2. You would have to be careful even with peons, because if they get lucky they can still do some serious damage; things like cover and planning an assault become more important.

    @ MJ Harnish
    Yeah but modifyers are easy to change. just change em to fit the probability that woudl be appropriate! THis is what I did to approximate target numbers: as the GM make thee NPC's: a basic troop, a special ops one, and a grandmaster badass; see how high you can built their skills up to then set your easy/challenging/difficult target numbers such that you get 90%/60%/30% success rates that make intuitive sense.

    remeber! the whole goal is to make the mechanic more cinematic! Imagine a scene in a movie, or one you make up yorself that matches the level of reality/superhumannes you want in your campaign; then imagine how it should all work out ahead of time, then playtest/tweak/playtest till the rules can reproduce the outcome with the appropriate amount of reliability.

  12. I don't think you can reproduce a poisson distribution with any dice or combination thereof.

    I use 2d10, with no threat rolls. Some additional mechanics of my system mean I have open-ended rolls (so if you roll 20 you roll again and add) but for basic d20 system you don't need it.

    For basic d20 system, 2d10 with no threat rolls means:

    criticals occur 1% of the time
    fumbles occur 1% of the time
    the dice are stacked for middle value rolls

    the key thing when changing dice is to remember that DCs need to change to meet them. A DC of 20 is easier, and a DC of 25 harder, in 2d10 than in d20.

  13. @ faustnotes
    you cant reproduce one with d20 itself, but you can approximate one. If you want to have one exactly, thats basically white wolf's and shadowrun's system. DC's are no big matter, since you can change them to anything you want. the crit range can be 18+ (4% on 2d10) or 3d8 as Jon mentioned above.

  14. Very nice statistical analysis, the more I read your work the more I think you're really going to appreciate the game I'm working on. Keep it up.

  15. I read this post with great interest, and will have to go through it again. I have no background in statistics and probabilities but I am working on my own game design and really want to try to find a playable version that models things effectively while remaining relatively easy to resolve and master.
    I think I have my mechanics in a workable format and I am using Ueberdice to chart it out and see if it seems consistent enough. Do you think I might post you a summary of what I am working on and get some comments on it? I would like to hear your thoughts...

  16. @Warren - While I can't speak for Tom (he guest authored this post) - I'd be happy to take a peak at whatever you're working on Warren! Recently; I've been playing a lot of Savage Worlds - it's dice system is really nice, not flat at all... and with "Aces" (when you roll the max number on a die) you roll again and add it to the previous result (aka exploding dice). It adds a huge amount of randomness to things; for better and... for worse. =D


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