Gain from the two-envelope problem via information asymmetry: on the suboptimality of randomized switching

Abstract

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.

An Optimal Threshold Strategy in the Two-Envelope Problem with Partial Information

“…The McDonnell andAbbott (2009) andMcDonnell et al (2011) two-envelope problem framework is the following. Let Y be the random amount of money in the chosen envelope, then…”

“…Gardner (1982) and Kraitchik (1930)). Brams and Kilgour (1995), Christensen and Utts (1992) and McDonnell et al (2011), among others, show that when the player has complete information about the distribution of X, then a deterministic switching strategy is optimal. Consider a one-trial game between a gambling house and one player.…”

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In this paper we propose a strategy that gives an optimal lower bound of the average gain for the two-envelope problem within the McDonnell and Abbott (2009) and McDonnell et al (2011) framework. We obtain this result with partial information about the probability distribution of the envelope's contents.

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An Optimal Strategy for Maximizing the Expected Real-Estate Selling Price: Accept or Reject an Offer?

“…It also touches upon several problems, or puzzles, that have fascinated amateur and professional scientists. In this paper, whose main contribution is an optimal realestate seller's strategy in the motivating problem, we have noted a connection between the motivating problem and the well-known two-envelope problem, in the form of Abbott (2009), andMcDonnell et al (2011).…”

Optimal two-stage pricing strategies from the seller’s perspective under the uncertainty of buyer’s decisions

“…prices that maximize the expected revenue, or minimize the expected loss. Some of the aforementioned works have been influenced by the twoenvelope problem, and in particular by the viewpoint put forward by Abbott (2009) andMcDonnell et al (2011). Furthermore, Titman (1985) considers a twoperiod model to analyze the optimal land prices when the condominium unit prices are uncertain.…”