Creating fair dice from random objects
arstechnica.com40 points by epipolar 4 days ago
40 points by epipolar 4 days ago
The question I have is how stable are the probabilities over time? My guess is traditional dice are more physically robust to wear and degrade more gracefully.
It does not seem to be so useful and practical to use strange shapes for dice; the common shapes, with numbers (or other symbols that are applicable for the game you are playing) on each side, will probably be more useful, anyways. However, it might be interesting.
Another reason to use dice for tabletop games is so that the game can be played without the use of a computer.
When I play GURPS, I generally use different dice with each dice roll in order to try to mitigate some of the bias. (I don't know quite how much effective this really is, though.)
The Roman rock crystal icosahedron die in the Louvre would be nice:
https://archimedes-lab.org/2021/07/15/amazing-roman-rock-cry...
Hey hey, it's Keenan Crane again :)
For those that don't know, he is a HIGHLY respected researcher and well known for effectively communicating complex topics. He really makes it fun. Often as visually entertaining as 3B1B while diving into more depth. I'd highly recommend people poke through his site and YouTube channel
https://www.cs.cmu.edu/~kmcrane/
Keenan Crane is legendary
How to create a fair coin from an arbitrarily biased coin:
1. Toss the coin and remember the answer.
2. Toss the coin again, if it is different from your previous toss then your result from #1 is fair. Otherwise, go back to step 1.
If p is the probability of getting heads, there are four possible outcomes with their associated probabilities:
TT -> (1 - p)^2 (rejected)
HT -> p * (1 - p)
TH -> (1 - p) * p
TT -> p^2 (rejected)
"arbitrarily" is doing some heavy lifting!
I'm not sure that two concurrent harmonious answers constitutes a "fixed" coin or a diagnosis of a fixed coin.
This scheme will be rubbish with a one sided coin ie the limit for "arbitrary fixed coin".
How is that "heavy lifting"? It's perfectly reasonable for any real-world "coin".
That's cute. intuitively, if two flips give different outcomes, it's fifty/fifty which would be first.
You are assuming an unbiased coin.
Imagine I glue a poker chip to a washer. There's a clear bias in the outcome of this "coin".
This method resolves that bias.
But also, you might have to flip the coin an arbitrarily large number of times before you get a "heads tails" or "tails heads" roll (if I can arbitrarily pick how biased the coin is).
The opening scene of "Rosencrantz and Guildenstern Are Dead" springs to mind.
And that coin wasn't even biased... although Tom Stoppard was a confounding factor.
VN extrator is a specific case of a more general idea: When you independently (hard assumption of VN extractor) draw M times with N possibilities then you can extract entropy from their permutation.
Assign some scheme for converting permutations to an index.
Then get uniform bits out, maintain two variables: one is the product of the number of permutations, the other gets multiplied by the number of permutations and the index added. Whenever the number of possibilities is divisible by two, output the LSB of the index accumulator and halve the number of possibilities.
Size up your groups and accumulators and you can get arbitrarily high extraction rates.
Doing it efficiently and in constant time (e.g. without divisions) is the more exciting trick. A colleague and I managed an extractor for the binary case that packs takes 10+3N multiplies and N CTZs to pack N bits (giving an exact invertible encoding when bits choose ones is < 2^64).
the title is a classic quant interview problem
the basic idea is that, because multiplication commutes, probability of A then B is the same as probability of B then A, so long as they are independent events (rolling objects typically meets this criteria)
so instead of using just A or just B, which might neither have 0.5 probability, you only count "A then B" and "B then A" as rolls
and this trivially extends to constructing a fair N-sided die out of any arbitrarily biased die for any N
That isn't what the article is about at all. It's not even what the first paragraph is about.
What they are doing is designing physical shapes that will have a specified probability of falling in different positions.
What you are talking about is post processing a biased random signal to get a less biased signal.
And yet the person you replied to was quite clear that they are responding to the title.
That isn't the title either: the title is “Creating fair dice from random objects”, while what they are responding to may be something like “Creating fair coins from biased coins”. So they're only responding to the “Creating fair _ from _” part of the title. Responding to three out of six words in the title isn't bad I guess.
> and this trivially extends to constructing a fair N-sided die out of any arbitrarily biased die for any N
They wrote something interesting, even if it only tangentially matches the topic.
Pointing out that it doesn't exactly match the topic also adds to the conversation, I guess, but I think we've now exhausted any interest (so I won't be arguing further).
This technique is formally known as the Von Neumann extractor (1951), a foundational concept in randomness extraction.