Why do we change whatever amount we found in the first envelope: the Wikipedia "two envelopes problem" commented

TL;DR

This paper examines the philosophical and probabilistic confusions in the Wikipedia 'two envelopes problem', clarifying when it is rational to switch envelopes based on whether the initial amount is known or unknown.

Contribution

It provides a detailed analysis of common misconceptions and clarifies the conditions under which switching envelopes is rational, resolving ambiguities in the problem.

Findings

Switching is rational if the first envelope is opened.

No point in switching if the first envelope remains unopened.

Clarifies the role of probability and knowledge in the problem.

Abstract

We analyze the main arguments that attempt to explain why there is no point in changing the envelope. Most people confuse estimation and calculation, conditional and unconditional probabilities, random and non-random variables, modelling and reality, chance and uncertainty (or credence), whatever X known and whatever X unknown, personal and interpersonal estimates. Such confusions suggest that there would be no point in swapping the envelope. Our analysis is that if the first envelope is opened, whatever the amount, it is rational and consistent to think that it is worth changing it. If the first envelope is not opened, it is rational and consistent to believe there is no point in swapping the envelope.