Wednesday, June 24, 2015

Unlucky dice? SkipBo as a die

Most players at some point will swear at their die and start to doubt the laws of physics. Having a random output is half the fun of it, as is clear when NPCs deal damage by the set output or when the encounter is in the plane of Mechanus (the latter is fun due to madness of the plane, but that's a different point). One way to stop complaints about accursed dice is to not use dice and use cards.

Skip-Bo is a really rubbish game. It is like Uno, but with tedium replacing treachery. However, a Skip-Bo deck (with the 18 skip-bo cards removed) can double as a random number generator. It has twelve series of numbers form 1 to 12, which has two advantages over poker cards: the numbers go up to twelve and cards of a same value are identical (no suits, no background distractions). So if you used between a single and all twelve sets, once you burn through the whole deck you can rest happy knowing you had no bad luck (or good luck).

Statistically this benefit is not large, but is not neglegible. My best guess on how to express luck is to use the 'standard error of the mean'. This is the average value of how off can the average of the roll results be from the mean dice roll value after a certain number of rolls. A die rolls a uniform distribution and its standard deviation can be obtained from Wikipedia.
Doing some maths, it is clear that 'luck' dissipates rather fast:

So if you used a deck of a single set, you would have a standard error somewhere under 1.5 damage point depending on the damage dice.
A rapier is a d8 and after 8 rolls the error is of 1 damage point (technically 0.81). So after eight d8 rolls an bad average of 3.5 is common and it means that instead of doing 36 points of damage, one did 28 (12% less). Expressed differently, the distribution of the obtained averages can be calculated (I cheated and used the anydice website for the data so I didn't have to do much math). A d8 on average does 4.5, but you have after 8 rolls got a 23% chance of it being between 4.25 and 4.75 , while a 19% chance between 3.75 and 4.25, 10% between 3.25 and 3.75, which is the last bin within the 1 point of the standard error. Below which the chances are 4%, 1% and lower.

I should note I am talking here about fluctuations in a small number of rolls that in the long run cancel out. In another post, it is shown that there are lots of equivalent dice combinations that in the long run (i.e. luck average out) do identical damage (e.g. d12 ≡ d10+1 ≡ 3d4 - 1 ≡ d2 + d3 + d5). These dice do have different levels of 'luck', some quite dramatic, but it doesn't at all matter on average and didn't result in a significant about of lucky victories in dire and contrived matches.

In conclusion, using a single set of a deck of cards followed by reshuffle is no different than a dice in the long run, but on a small scale between each reshuffle it does make a difference as it avoids the ten or so percent fluctuation in damage given, but doesn't alter the standard deviation as much as changing the dice combination. With more than one set of numbers in the deck, the benefit is annulled, not to mention that accusations of bad shuffling will be more likely.

PS. If a royal-free poker deck where used instead of a dice, the suits could be a marker for some interesting dynamic, say the weapon does fire, slashing, acid or blugeoning damage.

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