Proc. R. Soc. A (2011) 467, 2825–2851 doi:10.1098/rspa.2010.0541 Published online 11 May 2011 Gain from the two-envelope problem via information asymmetry: on the suboptimality of randomized switching BY MARK D. MCDONNELL 1 ,ALEX J. GRANT 1 ,INGMAR LAND 1 ,BADRI N. VELLAMBI 1 ,DEREK ABBOTT 2, * AND KEN LEVER 1,3 1 Institute for Telecommunications Research, University of South Australia, Mawson Lakes, SA 5095, Australia 2 School of Electrical and Electronic Engineering, The University of Adelaide, Adelaide, SA 5005, Australia 3 Cardiff School of Engineering, Cardiff University, Cardiff CF10 3XQ, UK The two-envelope problem (or exchange problem) is one of maximizing the payoff in choosing between two values, given an observation of only one. This paradigm is of interest in a range of fields from engineering to mathematical finance, as it is now known that the payoff can be increased by exploiting a form of information asymmetry. Here, we consider a version of the ‘two-envelope game’ where the envelopes’ contents are governed by a continuous positive random variable. While the optimal switching strategy is known and deterministic once an envelope has been opened, it is not necessarily optimal when the content’s distribution is unknown. A useful alternative in this case may be to use a switching strategy that depends randomly on the observed value in the opened envelope. This approach can lead to a gain when compared with never switching. Here, we quantify the gain owing to such conditional randomized switching when the random variable has a generalized negative exponential distribution, and compare this to the optimal switching strategy. We also show that a randomized strategy may be advantageous when the distribution of the envelope’s contents is unknown, since it can always lead to a gain. Keywords: two-envelope problem; two-envelope paradox; exchange paradox; game theory; randomized switching; information asymmetry 1. Introduction The two-envelope problem and the associated two-envelope paradox (also known as the exchange problem, and the exchange paradox) are intriguing conundrums that have captured the attention of mathematicians, economists, philosophers and engineers for over half a century—see a brief review in McDonnell & Abbott (2009). Further areas of potential impact to applications areas and open questions arising are discussed by Abbott et al. (2010). There are many versions of the two-envelope problem/paradox (Nalebuff 1989; Nickerson & Falk 2006). However, it can be quickly demonstrated that any apparent paradox seen when analysing such problems is the result of incorrect *Author for correspondence (dabbott@eleceng.adelaide.edu.au). Received 20 October 2010 Accepted 11 April 2011 This journal is © 2011 The Royal Society 2825