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.