How curved distributions can drive you mad

Posted by on January 21, 2017

Let me tell you the story of a tabletop RPG designer…a designer driven mad.

This designer, let’s call him “Chris,” was tired of the standard RPG mechanic of rolling a die, adding or subtracting a few modifiers, and comparing it to a single target difficulty number. He didn’t like how swingy it is. He didn’t like how boring it is.

Then he discovered the distribution curve.

See, if you roll a d20, you have an equal chance of rolling any number from 1 to 20.

However, if you roll 3d6, the odds change dramatically. You’re much more likely to roll a 10 than a 2. In fact, 10s and 11s come up frequently, while lower and higher numbers come up less frequently. See this chart, courtesy of anydice:

3d6 chart

3d6 chart

This made eminent sense to Chris! Most of the time, a character will perform at an average rate. Sometimes she’ll do poorly or well, but the more remarkable the results, the less likely they are.

So Chris merrily continued to calculate bonuses for his game, giving players a few points to put into various attributes and skills.

And this is where the madness set in.

Chris couldn’t find bonuses that fit right. If rolling against a target difficulty of 12, a bonus of +1 meant success 50% of the time, but a bonus of only +3 meant success 74% of the time, and +5 meant success 91% of the time.

He started scaling back his bonuses. But then he couldn’t make characters progress well; a single +1 had a big effect, so he could only hand out small increases as characters leveled up. Worse, target difficulties were confined to a very small range.

And things got worse in playtesting. Players complained because the fighter only had 2 points higher strength than the ranger. He tried to explain that it’d make a big impact in the game, but the dice didn’t roll well and it didn’t look that way during the session. Worse, players with low attack modifiers simply couldn’t hit enemies, and something in the back of Chris’s mind told him that, statistically, they wouldn’t.

So Chris went back to the numbers and kept fiddling with numbers, trying to find the balance that would let those folks with slightly lower combat abilities contribute. But the distribution curve was working against him now; characters on the wrong side were just so unlikely to hit the game became un-fun. He heard the curve laughing in his dreams.

Sadly, Chris is now confined to the Game Designer’s Madhouse, where we hope he will eventually recover and realize that math is more complicated than it looks.

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