Everyone has to name a guess (mean) and range (standard deviation) of a normal distribution. Whoever’s pdf takes the largest value at the true answer wins. Bonus: you may opt to invertibly transform the input first in one of several acceptable ways, most notably taking the logarithm. Now take that and simplify it. ;)
Something like that could work, but it seems like you would still need to have a rule that you must guess before you know the other players guesses.
Otherwise, player 2 could simply guess the same mean as player 1 - with a slightly larger standard deviation—and have a PDF that takes a higher value everywhere except for a very small interval around the mean itself.
Alternatively, if 3 players all guessed the same standard deviation, and the means they guessed were 49, 50, and 51, then we would have the same problem that the opening post mentions in the first place.
Everyone has to name a guess (mean) and range (standard deviation) of a normal distribution. Whoever’s pdf takes the largest value at the true answer wins. Bonus: you may opt to invertibly transform the input first in one of several acceptable ways, most notably taking the logarithm. Now take that and simplify it. ;)
Something like that could work, but it seems like you would still need to have a rule that you must guess before you know the other players guesses.
Otherwise, player 2 could simply guess the same mean as player 1 - with a slightly larger standard deviation—and have a PDF that takes a higher value everywhere except for a very small interval around the mean itself.
Alternatively, if 3 players all guessed the same standard deviation, and the means they guessed were 49, 50, and 51, then we would have the same problem that the opening post mentions in the first place.
This seems fairly nice, except that actually doing the math is far too demanding for most settings where I’d play this game.