Sunday, January 24, 2016

Alternative Mechanics: Alternative Distributions

Interact with results
Yesterday, I looked at changing the spread of values in GURPS by using differently shaped dice. Today, I'll look at different distributions along the series of integers 3:18.

The Standard Distribution

All of these distributions are the "normal distribution" or "bell curve" (Perhaps technically, not the normal distribution because it is a discrete series, but blah.) This is called a bell curve normally because it has an even rise and fall when looking at a frequency chart, with the most common values being close to the mean (average). The second graph illustrates this. If you look at the key in the graph, you will see two numbers, the mean (average) and the standard deviation. The standard deviation is kind of a way to tell how random the numbers are. If it is a very big standard deviation, the numbers will vary all over the place; if the standard deviation is very small though, it means that the numbers will generally be closer to the mean.

GURPS Default: 3d6

From Wikipedia
GURPS, by default uses a 3d6 dice roll for almost all throws besides a few random lookups, damage, and social encounters. When throwing 3d6, the standard deviation is 2.96. The average for all combinations is always 10.5. The standard deviation is a special number that says, when using a normal distribution, that you will have a 68% chance of rolling the average, plus or minus the standard deviation. For 3d6 this means a range of 7.54:13.46, or probably since we are dealing with dice, 8-14. Playing GURPS, you probably notice you get these numbers a lot. A smaller standard deviation more greatly rewards higher stats, and a larger standard deviation lends leniency to lower stats.

No Curve: D&D style

If you look at the chart, you can see a flat line that is matched to "1d16 +2". There is technically no (normal) 16-sided dice, but it can be simulated by using 2d4 or 1d8 and a 1d2 among other things. My personal favorite when doing this is to use a normal six-sided die with an 8-sided die. The value of the 8-sided die is as shown, but add 8 more to that number if the six-sided die has an even number (I hate flipping coins!) and you can do these both at the same time, You get the biggest possible standard deviation with this method: 4.61, which gives a 68% chance of a number from 5.89 to 15.11, or rounding to 6-15. Each number has exactly a 1/16 chance of occurring, which almost matches what we got when we round. Though, a 6.25% chance makes for lots of critical failures and successes. If you really want to play this way, you might want to change critical success to only occur on 3, and failure only on 18... and it will still be more common than the normal 3d6. Having no curve can be seen as more exciting because things feel a little less predictable. It becomes possible to get lucky and do impossible feats on the regular, or catastrophically fail on the most mundane tasks.

The Smallest Standard Deviation: 15 pennies

We have a few more shown, but today has been busy, so I'll skip straight to the other extreme: throwing 15 pennies, counting the number of heads found, and adding 3. This has a standard deviation of 1.94, or a 68% range of 8.56 to 12.44, or rounding, 9-12. More than half of the results you can... flip, I guess instead of roll, consist of the 4 numbers in the middle. 4 numbers are twice as likely as any of the other numbers that outnumber them 3:1. This small standard deviation makes skill checks and contests extremely reliable, and increases the value of high attributes much quicker... at the expense of diminishing returns kicking in much earlier as well. Critical successes and failures will be extremely rare at this rate. To be close to the same ratio of chance, Critical failures should occur when rolling a 15 or higher, and a critical success should occur when rolling a 6 or lower.

Other Notes and Wrapping Up

I'm not sure how practical using any other dice rolling scheme would be. Like I said earlier, a large standard deviation favors low skill levels and increases the odds of criticial failures and successes; while a smaller standard deviation favors higher stats and decreases the odds of critical failure and success. I don't know if these should cost anything as a quirk or a perk, but practically speaking, at least, I'd be annoyed by anyone who wanted to shake 15 pennies every skill check. I didn't detail all the other possible curves which are just more or less gradients from a smaller to a larger standard deviation with less pronounced effects compared to the two poles I did discuss, but might be easier to handle. 5d4-2, in fact, comes close to the distribution in Fate if that is something you like... I personally hate 4-sided dice though... so, that's that.
Precis - A look at other dice distributions in GURPS without changing the 3-18 spread.

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