The Ringing and The Singing of the Bell's curve

How the danger sinks and swells,
By the sinking or the swelling in the anger of the bells -
Of the bells -
Of the bells, bells, bells, bells,
Bells, bells, bells   -- Edgar Allen Poe, "The Bells"

(This is a sorta-followup to this post, just so you know.)

I've been knocking about the idea of replacing a d20 "to-hit" roll + damage with 3d6 + static modifier. Suppose a sword does 1d8 damage and a 1st-level fighter with no BAB needs to hit an AC 16 target. What is the expected damage done? Assuming an average roll of 4.5 points of damage x 0.25 (25%) chance of hitting, you have 1.125 points inflicted per round.

Now, suppose we use 3d6. Alone, 3d6 have a 4.63% chance of rolling 16 or higher. So that translates, with an average d8 die roll, to .208 points per round. But here's the trick: rather than rolling damage, the sword adds a static number to the 3d6 roll, and additional damage is determined by subtracting the AC from the eventual result. Assuming the sword gives a static 5 point bonus, we get a 25.92% of hitting, on average, 4 points, or 1.0368. Fairly close.

The idea behind this is, people are often going on about how high rolls on a d20 should somehow represent that their attack was extra special. These people do not understand the difference between a bell curve and a linear distribution, but if we don't want the fun experience of having a high "to hit" roll to be spoiled by rolling 1 for damage, we need to swap out systems.

In d20, people like to have at least the possibility of doing extra damage on an unmodified, or "natural" 20. How would we model that in 3d6, you might ask? One answer would be to swap something from the latest edition of Tunnels & Trolls: the concept that doubles and triples add and roll over. So if you roll 5,1,5, you take the two fives, re-roll them, and add that number to the total; say the second roll is a 2 and a 4, you have a total of 17, plus the static +4 bonus for the sword. Now you've done 5 points of damage. If those had been 2 and 2, you would roll them yet again.

(To be accurate, in T&T you only add and re-roll doubles when rolling two dice, and triples with three, but the principle isn't that far off this way.)

The numbers seem to work out if you take half the maximum die roll and add one, so a d4 gets 3, a d6 gets 4, etc.

Now, some purists would accuse me of tainting our dear retro-D&D play with such blasphemous bell-shaped results, but I think this is perfectly within the spirit of the OSR, since this is a simple mechanism that can be swapped out with the existing one without changing any other element of the game. (Magic swords that have different bonuses "to hit" and for damage might be a problem, but consider reversing the adds: a sword, +3 to hit and +1 damage becomes +1 to hit, but +3 damage. So our average damage becomes 3 points. A 40% chance of getting 5.5 points with the traditional system equals 2.2 points. Tweaking the numbers, +1/+2, gets us closer, at 2.72 points. Taking doubles and triples re-rolls into account gets us closer still.)

cheers,
Adam

Lines and Curves

(D'oh. Found this in my "drafts" folder. This post is part of the background for this one.)

A lot of times people will post, on reddit or elsewhere, that 3d6 is a "better" resolution roll than a d20 because the results fall along a bell curve. Whether or not this makes it better is a matter of choice,  but it never helps anyone's case when they follow up their opinion with a statement like "With a d20, a roll of twenty is just as likely as a one." That tells me they don't understand probability, because with a 3d6 roll, an eighteen is also just as likely as a three. I've even read this phrased as "with a d20, you're just as likely to roll under your target as you are over", which is patently untrue for any target number other than ten (Okay, "equal to or under".) But alas, some people on the internets are wrong, and you can't correct them all.

What they should be saying is, "with a d20, a roll of twenty is just as likely as a nineteen", which makes more sense; with 3d6, a roll of eighteen is not as likely as a seventeen, not even close: a seventeen is three times as likely. And a sixteen is twice as likely as that. But any number is equally as likely as any other number on a d20. If you need to roll, say, 16 or higher on a d20, that means you have a 25% chance of success, about what you'd have if you needed to roll a 13 or higher on 3d6. But if you make the d20 roll, any of the successful numbers is as likely as any of the others.

To me, this means that a linear roll is best used only to determine flat, binary, yes-or-no questions: Did I hit him? Do we find anything? Did I die from the poison? Conversely, it shouldn't be used to determine degrees of success. If a question ever contains the words "by how much" or "by how many", a 3d6 (or other bell curve friendly) roll is usually more appropriate. D&D decouples the question of "how much" from the success roll in most cases; whether this is a good thing or a bad thing is mostly a matter of taste, but there's no reason that the one roll must, as a matter of course, influence the second.

I mention this because I'm at best ambivalent about rewarding a "natural" twenty with a critical hit or a "natural" one with a fumble in my OSR game. There simply is no logical reason why, assuming an attack was successful, one equally likely number should be rewarded over any of the other equally likely numbers.

Of course, if you find that the arbitrary inclusion of massive damage or horrible fumbles increases the enjoyment of your game, who am I to say otherwise? But that's my reasoning.

cheers,
Adam

Tilting at the Lists; or, The Utility of Tables

Although I will occasionally refer to them, I prefer not to use pre-published random monster tables. If I've decided that Orcs, for one reason or another, don't exist in my world, a load of 1st-3rd level charts, from the 1st Edition DMG to the Pathfinder Bestiary, suddenly have gaping holes in them. You can substitute brigands, say, when that happens, but at that point, you're just making things up anyhow, unless you pencil in a substitution for every occurrence of the word "orc" in the table.

Needless to say, such behavior is beneath decent, civilized folk like us. So maybe you just look up and down the table until you see something that strikes your fancy. Nothing wrong with that; we've all done it before.

Nonetheless, having a table you can roll some dice against keeps you honest, and also keeps you from returning to the same kind of encounter by force of habit. Also, by not being tied to a specific location, wandering monsters give the illusion that the game world goes on even when the PCs take the day off; those Bugbears obviously don't live in the Ghoul's Crypt, they're probably looking to loot it themseves. (That, or they've made a very bad choice of shelter coming in from the rain.)

The solution to the problem of inappropriate results is to come up with your own tables. And there are two parts to each basic table: the "stuff", and the "numbers".

The "stuff" is just that, whatever's going to show up after you've rolled on that table. A treasure hoard, perhaps. Or a monster. Or an NPC personality trait. This part is not particularly hard to come up with; you just start listing things that you want to see in your game. A table is nothing more than a list with some weighted probabilities attached to it. You list the things that are going to be interesting, not necessarily what is most likely to be there. An NPC may have a secret fear of snakes, but if it's not going to come up in the game, why waste a line on the table?

The trouble is with the numbers. You want something that you can easily roll on dice, and that reflect your (subjective, approximate) feeling of how likely it is to come up. (Note how I said, just a paragraph before, not to list things by how likely they are to come up; that's true for the list part of creating a table, but not for the numbers part.) There are countless ways of doing this, but for the beleaguered DM, it boils down to two old friends: the Bell Curve, and the Linear Distribution.

If you're fine with Linear Distributions, then you're good if your list contains 4, 6, 8, 10, 12, 20, or 100 objects. Just roll the requisite die. Most of us know you can also simulate, say, a d16 by rolling 1d8 and a 1d6; if the d6 comes up 4-6, add 8 to the d8, otherwise just read it as it is. This process can be generalized to get linear distributions of 16, 24, 30, 36, or 1200 results, but I'll leave those as exercises for those so inspired.

But what about Bell Curves? Are we stuck with mimicking D&D stat rolls? We all know that 3d6 create a Bell Curve, and that's great if you have sixteen possibilities that you want to select from; but if you have twenty, it seems sad to relegate your score of encounters to a Linear Distribution just because you have a d20.

The trick is to realize that there are more ways to generate a Bell-Curve-like distribution than just 3d6; rolling 2d8+1d6-2 gives you a slightly distended bell curve in the range of 1-20. Rolling 3d4-2 gives you a curve between 1 and 10. 2d4+1d6-2 gives you 1-12. (There's no reason to subtract that two, if you don't want to; I just like to start from 1 for purposes of illustration.)

If you don't mind doing the math in your head, d3+d4+d6-2 gives you a curve from 1 to 11. The advantage of using an odd-numbered die (either by halving a d6 or a d10) is that the middle value will be uniquely more common than the rest. With regular polyhedrals, the middle two values will always be equally likely. The disadvantage is, unless you have a random number generator handy, you will need to do a slight bit more arithmetic in your head.

Three dice that are close to each other in terms of number of sides (like a d4 and a d6) work best when you want the probabilities fairly close to each other, but there's no reason you can't do, say, 6d4-5 to get a curve where the middle value (ten) is really more likely than the next two. It's probably best in most cases, though, not to obsess with how much more likely one value is to appear, and just realize that values closer to the middle will be more common than those farther away.

So now you have, say, twenty items on your list. And you have a table with lines numbered from 1 to 20. You assign the two most likely (or most/least desirable, it's your call) encounters to slot #s 10 and 11; you take the next two and put them in slot #s 9 and 12; and so on, until all 20 slots are filled. Optionally, you can put the more desirable encounters on the lower half (10 instead of 11, 9 instead of 12, etc.); then a successful  recon strategy or a good Search/Spot/Survival roll can adjust the die roll down a point or two, rewarding the cautious player.

When you need to pull out the chart, roll 2d8+1d6-2 and check that number against the table. (I recommend maybe noting the required dice and math at the top of the sheet, just to make it easy to remember.)  When you roll a 20, your players will know you're honest, and aren't just siccing Orcus on them because they handily dispatched your previous encounter.

cheers,
Adam