The “Dutch books” example is not restricted to improper priors. I don’t have time to transform this into the language of your problem, but the basically similar two-envelopes problem can arise from the prior distribution:
f(x) = 1/4*(3/4)^n where x = 2^n (n >=0), 0 if x cannot be written in this form
Considering this as a prior on the amount of money in an envelope, the expectation of the envelope you didn’t choose is always 8⁄7 of the envelope you did choose.
There is no actual mathematical contradiction with this sort of thing—with prior or improper priors, thanks to the timely appearance of infinities. See here for an explanation:
The “Dutch books” example is not restricted to improper priors. I don’t have time to transform this into the language of your problem, but the basically similar two-envelopes problem can arise from the prior distribution:
f(x) = 1/4*(3/4)^n where x = 2^n (n >=0), 0 if x cannot be written in this form
Considering this as a prior on the amount of money in an envelope, the expectation of the envelope you didn’t choose is always 8⁄7 of the envelope you did choose.
There is no actual mathematical contradiction with this sort of thing—with prior or improper priors, thanks to the timely appearance of infinities. See here for an explanation:
https://thewindingnumber.blogspot.com/2019/12/two-envelopes-problem-beyond-bayes.html