The two-envelope problem revisited

Abstract

The two-envelope problem has intrigued mathematicians for decades, and is a question of choice between two states in the presence of uncertainty. The problem so far, is considered open and there has been no agreed approach or framework for its analysis. In this paper we outline an elementary approach based on Cover's switching function that, in essence, makes a biased random choice where the bias is conditioned on the observed value of one of the states. We argue that the resulting symmetry breaking introduced by this process results in a gain counter to naive expectation. Finally, we discuss a number of open questions and new lines of enquiry that this discovery opens up. © 2010 World Scientific Publishing Company.