Variations on the Two Envelopes Problem

TL;DR

This paper reviews and compares key variations of the Two Envelopes Problem, analyzing correct decision strategies, mathematical justifications, and common errors, especially focusing on calculations based on revealed amounts.

Contribution

It provides a comprehensive comparison of the main variations, clarifies correct decision-making, and addresses common mistakes in the mathematical analysis of the problem.

Findings

Correct decision strategies are identified for each variation.

Mathematical errors in previous analyses are highlighted and corrected.

Using revealed amounts improves the accuracy of calculations in the problem.

Abstract

There are many papers written on the Two Envelopes Problem that usually study some of its variations. In this paper we will study and compare the most significant variations of the problem. We will see the correct decisions for each player and we will show the mathematics that supports them. We will point out some common mistakes in these calculations and we will explain why they are incorrect. Whenever an amount of money is revealed to the players in some variation, we will make our calculations based on the revealed amount, something that is not achieved in other papers.