Two-Envelope Paradox: The Blind Case

Abstract

The present paper addresses the blind case of the two envelope paradox. It first shows that optimal switching strategies, in the worst case sense, must necessary meet some simple conditions in the domain of the log-amount. It derives from that that no positive gain could be expected without due location constraints on the amounts.Assuming then that amounts are all bounded from below by the same number, it is established that harmonic strategy described by David Madore is optimal in absolute gain.In relative gain, assuming that amounts are both bounded from above and below, it is also demonstrated that optimal strategies do exist and their switching function follow a logarithmic staircase profile.Optimal solutions share common features: some are desirable, like slowly decaying behaviour, and some are more unpleasant, like jumps and lack of robustness. As an alternative, a class of smooth strategies is introduced and we prove that optimality still holds but in terms of the relative performance. We also describe a condition on the strategy which enables the player to take advantage of unexpectedly large amounts.