This has the property that if even a single person answers uniformly at random, then the final number you compute will be uniformly random, regardless of how everyone else answers.
Paradoxically, this may be the best answer because it is not the best answer.
In a room of 8500 people, it is likely that there is someone in the room smarter than you and knows a better way to pick random numbers. Come into the room and tell everyone that your objective is to pick a random number from 1-10. Maybe even go as far as to tell them your mod 10 idea. Wait a bit to let them to think about it, then start asking for the random numbers. It is likely that someone will have come up with a better way to pick a random number than your mod 10 solution, and thus their answer will be more uniformly random that what your solution can produce. And thus your solution becomes at least as uniformly random as theirs.
I'd love to see something written up about how quickly this converges, assuming the variables are i.i.d and distributed according to the distribution in the article.
Empirically, from running a quick script on the data in the article, it seems like you only need to sample 5 numbers to get a distribution that looks uniform using this "sum mod 10" method.
If you're going for empirical, why not ask for between 1 and a very large number? May as well extract as much entropy per person as possible for post-processing.
Mod10, no matter how large the number you’re only keeping the last digit (assuming they choose base 10 numbers), which... is probably no more random than asking them for a single digit in the first place.
> This has the property that if even a single person answers uniformly at random
Has any human ever proven they're capable of this? Generating truly random sequences is more or less impossible for humans AFAIK. (E.g., see https://news.ycombinator.com/item?id=19336754 which challenges you to do just that. Spoiler: you will probably fail miserably.)
It's an interesting idea, but in practice I think relying on the assumption that "even" one person is truly answering randomly (let alone uniformly at random) is a non-starter. But perhaps if there are enough people, the resulting sequence blends enough entropy together to get something that looks almost like a uniform random variable anyway? It would be interesting to test empirically.
Philosophically, no it is not possible, though neither can any natural phenomenon for which we reasonably rely on for randomness.
Practically I suppose your goal is just to generate numbers such that the next number cannot be predicted given only the previous numbers but also without considering any outside knowledge. Potentially possible. Potentially impossible to test. If you can beat the test, it probably just means your method beats that specific test
There are algorithms that beat most humans, because humans don’t play randomly. Conversely there are humans that can beat the human-beating algorithms reliably because the algorithms don’t play randomly either.
No free will needed. You see the predictions. Let the model learn and then change your sequence. It takes some time for the model to adjust. This way you can get the prediction rate to 50% or below.
I think you need at least one person who is completely independent of all others. For example if the same random is added up twice (say if the room is just you and your evil clone), you can get stuck with just even numbers. Independence is a pretty strong property to ask for since we all share the same biology...
But if you do have independence, the proof is easy! Let S be the sum of everyone else and X be the discrete uniform random in [0, 9]. Then:
Pr(X + S mod 10 = i)
= \sum_j Pr((X + j) mod 10 = i | S = j) Pr(S = j)
= \sum_j Pr(X = i | S = j) Pr(S = j)
= \sum_j Pr(X = i) Pr(S = j)
= Pr(X = i) \sum_j Pr(S = j)
= Pr(X = i)
Due to total probability, symmetry, independence, and total probability again respectively. The handwaved part is the mod where you can imagine the histogram columns in the pmf getting rotated/shifted around but ending up looking exactly the same afterwards since all columns are symmetrical.
In a room of 8500 people, it is likely that there is someone in the room smarter than you and knows a better way to pick random numbers. Come into the room and tell everyone that your objective is to pick a random number from 1-10. Maybe even go as far as to tell them your mod 10 idea. Wait a bit to let them to think about it, then start asking for the random numbers. It is likely that someone will have come up with a better way to pick a random number than your mod 10 solution, and thus their answer will be more uniformly random that what your solution can produce. And thus your solution becomes at least as uniformly random as theirs.