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Will_Sawin comments on Techniques for probability estimates - Less Wrong
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I think you could perform the dice rolling experiment without any need for security against tampering. To generate a random number from 0 to N-1, have every interested party generate their own number (roll their own die), then everybody reveals their numbers together and the group adds them all up and takes the remainder after dividing by N.
With that procedure everybody should be convinced that the result is at least as random as their own number.
This provides an excellent demonstration of E.T. Jaynes's point that making something more random really means making it more complicated.
The point isn't to make it more random, the point is to make it more trustworthy. You can participate in the process and be confident that the result is random without having to put any trust in the other participants.
Regardless, your original language claiming it made it more random was correct, because it does make it more hard-to-predict-but-with-clear-symmetry, aka random.