Variations on the Two Envelopes Problem

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.

💡 Research Summary

The paper “Variations on the Two Envelopes Problem” provides a systematic comparative study of the most frequently discussed versions of the classic two‑envelopes paradox and derives the optimal decision rule for each case. It begins by recalling the paradox’s origin: the naïve argument that swapping the envelope you hold always yields a higher expected payoff because the other envelope is either twice or half the amount you see. The authors point out that most of the existing literature reaches a contradiction by treating the problem with an unconditional prior distribution and by ignoring the information revealed to the player.

Four principal variants are examined.

1. Standard symmetric version – Both envelopes contain amounts that are in a 2:1 relationship, and a prior distribution over the smaller amount is assumed (uniform, geometric, or any normalizable distribution). Using Bayes’ theorem the conditional expectation of the unseen envelope given the observed amount x is derived as
    E(Y|X=x)=½