A frequentist’s resolution of the exchange paradox

In this paper, we provide a frequentist’s resolution of the exchange paradox. We propose a probability space with which we can explain both the Bayesian and likelihood-based resolutions. Under the proposed probability space, we find the optimal conditional strategy for the game to maximize the expected gain in money, which equals to that in Bayesian resolution. The probability space also allows us to understand that the paradox is from the additional symmetry assumption which is mistakenly taken. We show that the assumption is not feasible, at least in frequentist view, in the sense that no distribution of the money (inside the envelope) exists to satisfy the assumption. This understanding to the paradox is equivalent to the likelihood-based solution by Pawitan (2001) that the failure of our reasoning arises from treating the above likelihood function as the probability function and taking expectation over the likelihood function.