07-24-2015, 03:49 PM

What follows is a poor attempt at humor to dress up an extremely boring topic: Probability. But it has to be done. The hobby is just crawling with nerds who do love their math. Insofar as arcane formulas freak out the normal people. Actually applying a sharp mathematical analysis that suggests that math may have led us astray on a lot of beliefs is blasphemy.

A lot of gamers, particularly the finicky ones--who place a premium on the rules themselves (as opposed to the scenarios the rules facilitate) seem to have at least a minor obsession with the bell curve. Sometimes it doesn't stand out as an obsession, because we all learn in high school biology (or worse still, college statistics) that the natural world tends to fit the bell curve. As it happens, reality would disagree. But I digress.

I'd like to make the case that the natural state of game mechanics is always a tendency towards a "U-curve", and that the U-curve dominates the overall shape of probability distribution, regardless of what the core mechanic is. I don't care if it's linear, bell curve, or some other kind of curve, the true distribution is U-shaped.

To get my meaning, you have to take a step back and examine a hypothetical gaming system more universal than any actually in existence (and I'll explain that in a bit). I'm going to use RPG combat as an example just because I think they make for great, easy-to-understand illustrations. But this can apply to any kind of skill check.

What's the absolute worst thing that could happen in the Universal RPG whenever you try to attack? You fumble so bad, you kill yourself (and take your fellow party members with you). What's the absolute best thing that could happen? You instantly slay your foe (or maybe all foes in one mighty sweep or impaling attack). Where actual RPGs differ from the hypothetical universal role-playing system (HURPS?) is that most RPGs restrict skills, challenge ratings, and sit mods so that these extreme results don't happen (unless you're one of the three people still running Arduin Grimoire). That's what makes HURPS universal. Anything that can happen in a real RPG is a subset of the possibilities of HURPS.

I realize this is getting abstract, and I probably explained it very poorly. So take the time to make sure you get what I'm saying so far before reading on.

So back to the U-curve. In HURPS, not only is it possible for these absurd extreme results. It's possible for absurd extreme skill levels (or incompetence levels) that make these extreme results the most probable. Hence at the extreme ends of the skill range, the probability distributions of the results become nigh-vertical. Like the sides of the U. Where all action in the more moderate areas of the universal range--the range where most real RPGs are played--has its place at the bottom of the "U."

It's precisely because real RPGs are bounded that gives the illusion that the probability distribution of the core mechanic can be anything but a "U." By trying to force a bell-curve, you're actually creating some kind of weird, upside-down, twin-peaked bell curve. A sort of a soft-"W" curve. Nature, indeed. This is nonsense on stilts! If the RPG at least used a linear core mechanic, the bottom of the "U" would be a straight line--slightly awkward, but still perfectly recognizable as a U. As usual, the more clever game designers try to be in RPGs, the worse their results.

If the RPG itself is to be "realistic"--or even just aesthetically appealing--it should embrace the U-curve. The core mechanic should aim to be convex, not concave. It should aim to make more extreme results more probable than average results. Crazy, you say?

Understandable. I find most people are afflicted with a mental disorder called Averagomania. Sometimes they're referred to as "meanophiles" or "meanophiliacs". They're sort of crazy cultists, worshiping at the alter of the average. Their outlook leads them to a lot of logical fallacies making them ill-suited for survival. Like believing themselves invulnerable to drowning in a river whose average depth is only four feet. So to someone like that, suggesting the world tends away from average would seem a little crazy.

Let's say your skill is such that you have a 50% chance to hit with a weapon that causes 1d8 damage. The meanophile claims the power of prophecy and foresight. "Your attempt at harming your foe will successfully cause 2.25 damage."

But what really happens? First, you make a hit roll. This immediately causes you to diverge from the average. Either you will miss, causing 0 damage. Or you will hit causing an average of 4.5 damage. You will gravitate towards one extreme or the other. There is in fact zero chance of doing 2.25 damage. Now let's say you hit. That average damage of 4.5? Forget about it. You might do as little as 1, as much as 8. Maybe the dice will be near average, coming in at 4 or 5, but it will move at least a half point away from the average, possibly up to 3.5 points away.

I feel like some famous smarty like George Bernard Shaw must have once said, "The expected outcome is the one outcome one should never expect."

They say nature abhors a vacuum. I say nature abhors averages. There is no room for "average" in the real world. Average is nothing but a snare and a delusion. And so is the bell curve.

A lot of gamers, particularly the finicky ones--who place a premium on the rules themselves (as opposed to the scenarios the rules facilitate) seem to have at least a minor obsession with the bell curve. Sometimes it doesn't stand out as an obsession, because we all learn in high school biology (or worse still, college statistics) that the natural world tends to fit the bell curve. As it happens, reality would disagree. But I digress.

I'd like to make the case that the natural state of game mechanics is always a tendency towards a "U-curve", and that the U-curve dominates the overall shape of probability distribution, regardless of what the core mechanic is. I don't care if it's linear, bell curve, or some other kind of curve, the true distribution is U-shaped.

To get my meaning, you have to take a step back and examine a hypothetical gaming system more universal than any actually in existence (and I'll explain that in a bit). I'm going to use RPG combat as an example just because I think they make for great, easy-to-understand illustrations. But this can apply to any kind of skill check.

What's the absolute worst thing that could happen in the Universal RPG whenever you try to attack? You fumble so bad, you kill yourself (and take your fellow party members with you). What's the absolute best thing that could happen? You instantly slay your foe (or maybe all foes in one mighty sweep or impaling attack). Where actual RPGs differ from the hypothetical universal role-playing system (HURPS?) is that most RPGs restrict skills, challenge ratings, and sit mods so that these extreme results don't happen (unless you're one of the three people still running Arduin Grimoire). That's what makes HURPS universal. Anything that can happen in a real RPG is a subset of the possibilities of HURPS.

I realize this is getting abstract, and I probably explained it very poorly. So take the time to make sure you get what I'm saying so far before reading on.

So back to the U-curve. In HURPS, not only is it possible for these absurd extreme results. It's possible for absurd extreme skill levels (or incompetence levels) that make these extreme results the most probable. Hence at the extreme ends of the skill range, the probability distributions of the results become nigh-vertical. Like the sides of the U. Where all action in the more moderate areas of the universal range--the range where most real RPGs are played--has its place at the bottom of the "U."

It's precisely because real RPGs are bounded that gives the illusion that the probability distribution of the core mechanic can be anything but a "U." By trying to force a bell-curve, you're actually creating some kind of weird, upside-down, twin-peaked bell curve. A sort of a soft-"W" curve. Nature, indeed. This is nonsense on stilts! If the RPG at least used a linear core mechanic, the bottom of the "U" would be a straight line--slightly awkward, but still perfectly recognizable as a U. As usual, the more clever game designers try to be in RPGs, the worse their results.

If the RPG itself is to be "realistic"--or even just aesthetically appealing--it should embrace the U-curve. The core mechanic should aim to be convex, not concave. It should aim to make more extreme results more probable than average results. Crazy, you say?

Understandable. I find most people are afflicted with a mental disorder called Averagomania. Sometimes they're referred to as "meanophiles" or "meanophiliacs". They're sort of crazy cultists, worshiping at the alter of the average. Their outlook leads them to a lot of logical fallacies making them ill-suited for survival. Like believing themselves invulnerable to drowning in a river whose average depth is only four feet. So to someone like that, suggesting the world tends away from average would seem a little crazy.

Let's say your skill is such that you have a 50% chance to hit with a weapon that causes 1d8 damage. The meanophile claims the power of prophecy and foresight. "Your attempt at harming your foe will successfully cause 2.25 damage."

But what really happens? First, you make a hit roll. This immediately causes you to diverge from the average. Either you will miss, causing 0 damage. Or you will hit causing an average of 4.5 damage. You will gravitate towards one extreme or the other. There is in fact zero chance of doing 2.25 damage. Now let's say you hit. That average damage of 4.5? Forget about it. You might do as little as 1, as much as 8. Maybe the dice will be near average, coming in at 4 or 5, but it will move at least a half point away from the average, possibly up to 3.5 points away.

I feel like some famous smarty like George Bernard Shaw must have once said, "The expected outcome is the one outcome one should never expect."

They say nature abhors a vacuum. I say nature abhors averages. There is no room for "average" in the real world. Average is nothing but a snare and a delusion. And so is the bell curve.