Probability and Algebra

On Saturday, one of our pre-game conversations came to the topic of THAC0, that old 2nd Edition abbreviation.

For those who don't know, THAC0 meant "To Hit Armour Class 0" - this was the number that had to be scored (by d20 roll + modifiers) to hit a target with an AC of 0.

What this meant in practice was that you rolled your attack, added your modifiers, and then subtracted that total from your THAC0 to get the AC that you actually hit. Which actually isn't all that difficult mathematically - it's just an addition and a subtraction.

(Incidentally, THAC0 is almost but not quite the same (in effect) as the 1st Edition rule, which had attack tables for each AC. The key difference there, though, was that the attack tables would have 6 twenties in a row, followed by a '21' - to hit such a target you required a natural 20 on the die, and had to be using a +1 magic weapon.)

Now, while THAC0 actually isn't all that complex, a little algebra gives us an objectively better method for achieving the same results:

AC hit = d20 + BAB + other modifiers

That is, the 3e/4e/5e system.

(For a certain breed of old-school players, that's an actively heretical statement, by the way, because the old ways are sacrosanct. And there's another breed of online debaters who would likewise take exception to my use of the word 'objectively', claiming that that's just my opinion. Such people are, quite simply, wrong. Moving all the known values and calculations to one side of the equation, and eliminating the subtraction in favour of an addition means we get exactly the same result with reduced complexity. Mathematically, computationally, and pedagogically, that is better. Fact.)

Anyway, that's all a lengthy tangent around what I'm actually wanting to talk about, which is innovations in RPG systems, which ties into a comment I made over on Brindy's blog. There, I noted that most new RPGs are actually developed not because the designer has a new and exciting mathematical construct to reveal, but rather because the designer has a new setting that he wants to present.

Which is cool - I have absolutely no problem with that. What I take exception to is the creation of new RPG systems to support these new settings when an existing ruleset (either used 'straight' or modified) would do the job just as well.

See, here's the thing: with very few exceptions, RPG systems fall into a handful of moulds:

Single Die vs Target: This is the method familiar from D&D - you roll a d20, add modifiers, and compare with a target number. Your competence in a task is expressed as a modifier to your roll, which is added to the single die rolled.

Dice Pool, count successes: This is the method used in World of Darkness, Shadowrun, and Ubiquity - you roll a number of dice, compare each die with a target to get a number of 'successes', count those successes, and that detemines overall success or failure. Your competence in a task is expressed in the form of a larger number of dice.

Dice Pool, accumulate vs target: This is the method used in the Star Wars d6 game - you roll a number of dice, accumulate the total, and compare it with a target number. Your competence in a task is expressed as a bigger number of dice.

(A key variant of this is seen in Serenity - here you roll two dice, total, and compare with a target number. Your competence is expressed in the size of the dice.)

Roll-under: This is the system used in most percentile-based games, such as Call of Cthulhu or the Warhammer games (excluding WFRP 3e). Here, you roll a fixed dice pool, hoping to score under your individual skill threshold. Competence is expressed in an increase in your chance at a given task.

(I'm not sure I've ever seen a version of this system that wasn't percentile-based, though I daresay they're out there. The key cognitive problems I have with these systems is the question of where difficulty modifiers should be applied, and that values of greater than 100% are easily attained.)

(However, one could readily envisage a game where you rolled different dice based on the difficulty of the task (d4 for trivial, d6 for normal, etc), and competence was given as a taget - with normal people having '3' in most things.)

It's worth noting that converting from CoC's percentile system to a d20 roll is easy - divide the %ages by 5, subtract 1, and roll d20 + this modifier vs DC 20. Of course, that's not quite what CoC d20 did, nor is the change in dice type the reason CoC d20 didn't work out.

Other Systems: There are some few systems that do something different, with WFRP 3e being a notable instance (with a 'luck' and 'skill' axis on successes). Such efforts should be applauded, of course.

Anyway, the thing is, having boiled RPG systems down to a handful of basic cases, I have to question the value in developing new ones. In virtually every case, it's just a matter of using algebra to rearrange the terms, and tinkering with the probabilities, probably in ways you don't understand terribly well. The World of Darkness system is almost exactly the same as the Shadowrun system (one uses d6 and a fixed threshold of 5, the other uses d10s and a fixed threshold of 8... but reroll 10s. In both cases, each die has a 'value' of one-third), so why bother with both? Why create a new system for Numenera, when instead you can reuse the d20 system (or Savage Worlds, or whatever), and instead spend those energies on either optimising the system or on further developing the setting? Either way, rather than reinventing the wheel, you've instead made a better product.

(And, as a corollary to this, why do we not have implementations of all the various forms already available via the OGL? Copyright doesn't apply to game rules, only to their specific expression, and the various retro-clones have shown the way in this light. So, where are the kernels of the four key systems for people to tinker with?)