Friday, February 22, 2013

Stupid Dice Tricks: The Return Of Central Tendency

It's funky dice Friday, which means it's time for more dice math!

Last time I visited the concept of central tendency, I spent some time talking about why we using it for some things (like rolling ability scores) and not others (like to-hit rolls). Just to reiterate what we talked about back then, rolling multiple dice to achieve a single result creates a strong tendency toward the mean result (the average) since that number can be made out of more combinations of dice results; we use this sort of roll for things like ability scores because they're most likely to be middling scores, but have at least some chance of being extraordinary (good or bad). The other side of the same coin is that when you roll a single die, every face is equally likely as the others, so for situations where we want a simplified randomized result where the probabilities are easy to intuit, such as in a success/failure sort of situation (like to-hit rolls), single-die rolls (like a d20) make for a very simple way to generate a result with very little (and easily intelligible) math.

So, after that last post, it's a virtual certainty that someone is going to come along and mention that some game systems use multiple-dice, central tendency-influenced resolution systems. GURPS, for example, uses 3d6 for rolls and the venerable Tunnels & Trolls uses handfuls of d6s at a time. Some folks have suggested to me using a 2d10 resolution system for attack rolls in D&D-like or -inspired games. Hell even the 4dF dice pool of FATE is an example.

The problem with using multiple dice for task resolution is that you have to account for central tendency.

Since most resolution systems employ a "target number or higher" success standard, the probability of rolling any target number (or better) gets lower and lower the further away you get from the range's mean, but not in a linear fashion (for most distributions at least). If success on a 2d6 roll required an 11 or 12, this would be far less likely than for any other 2 number pair within the range (except for 2 & 3); there'd be a higher chance, for example, of rolling a 7 or 8 than an 11 or 12. So how do we determine success or failure in an environment like this?

Answer One: Mean Plus One

Fig 1: Likelihood of rolling the mean or better
Here's a thought: in a system where you are more concerned with a degree of success rather than just a binary pass/fail result, what if we aim close to the mean of the expression's range, then aim just a little higher? The chart here (how cool is it that I made a chart?) demonstrates the probabilities of rolling the mean result or higher for any particular dice expression. For a 2d8 roll, for example, you have  a 56.25% of rolling the mean result of 9 or better. That's pretty good odds. Here's the thing, though: most resolution systems will have some sort of influence from character abilities (ability scores, skill ranks, talent points, whatever), so you're likely to be getting +1s or +2s to that 2d8 pretty early on, which skews the "target number" probability.

The Moongoose version of Traveller (and possibly other versions of Traveller; I'll be honest, I'm still a newbie on the Traveller front) uses a 2d6 roll for everything, counting a success as anything higher than an 8. If an adjustment for something being harder or easier is needed, it's made not to the target number, but rather to the dice roll (six of one, half dozen of the other, really) and success or failure beyond pass & fail is determined by the distance of the result from 8. The mean of the 2d6 distribution is 7 (58.33% of this or better on 2d6), but they chose 8. Why?

Fig 2: Likelihood of rolling the mean +1 or better
They chose the number 8 precisely because it's not the mean of the distribution, but damn close to it. You're 41.64% likely to roll an 8 or higher. Those are pretty good odds, right? Particularly in a system with relatively low skill modifiers. The idea behind this decision is that an untrained person has a 41.64% chance of success in a normal environment, whereas someone with training is rewarded for it with significantly higher chances of success and with a greater impact when those successes occur. Furthermore, a positive modifier to this roll from skill will represent an improved ability to cope with adversity (negative dice modifiers).

This strategy (which I'll call "Mean Plus One") does a very good job of providing a simple standard of success that works very well with a "degree of success" mechanic. For all you folks who love the "no and," "no but," "yes but," "yes and" sorts of success logic, the Mean Plus One mechanic will work really well for you. For folks who insist on rolling odd numbers of dice, however, it will not be as successful as those rolling even numbers. This is due to the fact that odd numbers of dice produce means that are halfway between numbers (like 10.5 in a 3d6 distribution), so the Mean Plus One ends up actually being Mean Plus One And A Half.

Answer Two: The Mean As Hump

Some folks out there like "roll under" systems. I can dig that, even if it seems counter intuitive to me (dice are for rolling HIGH damnit!), and here's why: in these systems, the mean acts as a benchmark for when you become 50% (or more) likely to accomplish the thing.

Colonial Gothic, a game which I've only recently picked up (since the Second Edition came out), uses a very simple "roll 2d12 low" number, using as the target a number equal to the ranks in the skill being used plus the relevant ability score (with adjustments). Thus, since we're trying to roll that target number or lower, a target number of 13 (the mean) will result in a 54.17% probability of success. That "13" is the big fat hump in the middle of the probability distribution that the target number wants to get over. The funny thing about the distribution is that the further away from 13 you get, the less contribution to the overall success the increase will have. There ends up being diminishing returns on success rates.

Does this game logic work? Sure! Does it soundly use the central tendency to good effect? Hell yes! If anything, this sort of success mechanism encourages diversification in skill choice due to the effect of the diminishing returns and thus suits the sort of game that Colonial Gothic is trying to be with real (non-hyper-specialized) characters.

[EDIT] It turns out that GURPS runs this way as well, using a 3d6 dice roll. I knew that GURPS used 3d6, I just couldn't remember the "roll low" aspect of it since it's been since the early, early 90's since I've played GURPS. To tell the truth, I'm just not interested in the system, which might be because SJG gave their game the worst freaking name for a system imaginable. GURPS. Sounds like intestinal distress.

Answer Three: Pre-Calculated Objective Difficulties

Right, so, let's say we define a task with an Average difficulty of having a 50/50 chance of success. If we're rolling 3d6 (let's say we're rolling 3d6 for this imaginary resolution system), then our target number (on a straight unmodified roll, assuming a "roll high" standard) would be 11, since you have a 50% chance of rolling an 11 or greater. We can set similar benchmarks for other difficulties. We might set an Easy task at 75% likely, a Hard task at 25% likely and an Extremely Hard task at 10% likely. Again, assuming the 3d6 distribution, our target numbers will  be 9 (Easy), 13 (Hard) and 15 (Extremely Hard).

A word to the wise about this strategy: First, it works best (and by best, I mean "is easiest to calculate") using odd numbers of dice. Second, it requires an interpretation of the nature of task difficulty that unhinges the difficulty of the task from the person undertaking it. It just is this difficult to do this task. Climbing this wall is an Easy task, but surviving that 100 foot dive is a Hard task, and the numbers associated with the difficulty of those tasks never change, even if the skill and ability of the characters undertaking them does.

Final Thoughts

Any of these strategies toward game mechanics are equally viable and each influences the way the game is played in specific ways. Before you pick one of these methods, think about how you want the task resolution method to interact with the rules. Do you want a system where the degree of success (or even failure) is important? Then use a Mean Plus One system. Do you want to subtly encourage skill diversification to approximate a "jack of all trades" system? The Mean As Hump mechanic's low-roll diminishing returns will push your PCs in the right direction. If you're planning on using few, low modifiers to dice rolls and want a simple system that easy for you, the GM, to keep track of, use the Pre-Calculated Objective Difficulties method (man, we've got to come up with a shorter name for that).

That's it for this week's Stupid Dice Tricks. Let's hope a new and exciting topic in dice math pops into my brain before next Friday. Until then, thanks for reading and I hope this clears up a bit.