The Two Envelope Problem: a Paradox or Fallacious Reasoning?

TL;DR

This paper revisits the two envelope paradox, identifying that ambiguity and dependence between envelopes cause the paradox, and proposes a simple resolution by clarifying these issues.

Contribution

It clarifies the source of the paradox by analyzing the ambiguity and dependence, providing a straightforward resolution to the two envelope problem.

Findings

The paradox arises from ambiguity in the envelope's content.

Dependence between envelopes is crucial to understanding the problem.

Resolving ambiguity removes the paradox.

Abstract

The primary objective of this note is to revisit the two envelope problem and propose a simple resolution. It is argued that the paradox arises from the ambiguity associated with the money content $x of the chosen envelope. When X=x is observed it is not know which one of the two events, X={\theta} or X=2{\theta}, has occurred. Moreover, the money in the other envelope Y is not independent of X; when one contains {\theta} the other contains 2{\theta}. By taking these important features of the problem into account, the paradox disappears.