A semi-technical question about prediction markets and private info
There exists a 6-sided die that is weighted such that one of the 6 numbers has a 50% chance to come up and all the other numbers have a 1 in 10 chance. Nobody knows for certain which number the die is biased in favor of, but some people have had a chance to roll the die and see the result.
You get a chance to roll the die exactly once, with nobody else watching. It comes up 6. Running a quick Bayes’s Theorem calculation, you now think there’s a 50% chance that the die is biased in favor of 6 and a 10% chance for the numbers 1 through 5.
You then discover that there’s a prediction market about the die. The prediction market says there’s a 50% chance that “3” is the number the die is biased in favor of, and each other number is given 10% probability.
How do you update based on what you’ve learned? Do you make any bets?
I think I know the answer for this toy problem, but I’m not sure if I’m right or how it generalizes to real life...