An optimal threshold strategy in the two-envelope problem with partial information (original) (raw)
Winning Strategy in the Two Envelope Game with Partial Information
Social Science Research Network, 2012
In this note, we study a winning strategy in the two envelope game. We assume the player only has partial information of the game. In particular, if the players knows the mean and variance of certain distribution, we show a winning strategy that assures a minimum average gain.
The Two-Envelope Problem Revisited
Fluctuation and Noise Letters, 2010
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.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2011
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.
Trying to Resolve the Two-Envelope Problem
Synthese, 2005
After explaining the well-known two-envelope 'paradox' by indicating the fallacy involved, we consider the two-envelope 'problem' of evaluating the 'factual' information provided to us in the form of the value contained by the envelope chosen first. We try to provide a synthesis of contributions from economy, psychology, logic, probability theory (in the form of Bayesian statistics), mathematical statistics (in the form of a decision-theoretic approach) and game theory. We conclude that the two-envelope problem does not allow a satisfactory solution. An interpretation is made for statistical science at large. Synthese (2005) 145: 89-109
Randomized switching in the two-envelope problem
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2009
The two-envelope problem is a conundrum in decision theory that is subject to longstanding debate. It is a counterintuitive problem of decidability between two different states, in the presence of uncertainty, where a player's payoff must be maximized in some fashion. The problem is a significant one as it impacts on our understanding of probability theory, decision theory and optimization. It is timely to revisit this problem, as a number of related two-state switching phenomena are emerging in physics, engineering and economics literature. In this paper, we discuss this wider significance, and offer a new approach to the problem. For the first time, we analyse the problem by adopting Cover's switching strategy-this is where we randomly switch states with a probability that is a smoothly decreasing function of the observed value of one state. Surprisingly, we show that the player's payoff can be increased by this strategy. We also extend the problem to show that a deterministic switching strategy, based on a thresholded decision once the amount in an envelope is observed, is also workable.
An Extremal Inequality Motivated by Multiterminal Information-Theoretic Problems
IEEE Transactions on Information Theory, 2000
We prove a new extremal inequality, motivated by the vector Gaussian broadcast channel and the distributed source coding with a single quadratic distortion constraint problems. As a corollary, this inequality yields a generalization of the classical entropy-power inequality (EPI). As another corollary, this inequality sheds insight into maximizing the differential entropy of the sum of two dependent random variables.
Algorithms for Envelope Estimation
Journal of Computational and Graphical Statistics, 2016
Envelopes were recently proposed as methods for reducing estimative variation in multivariate linear regression. Estimation of an envelope usually involves optimization over Grassmann manifolds. We propose a fast and widely applicable one-dimensional (1D) algorithm for estimating an envelope in general. We reveal an important structural property of envelopes that facilitates our algorithm, and we prove both Fisher consistency and √ nconsistency of the algorithm.
Minimization of a class of rare event probabilities and buffered probabilities of exceedance
Annals of Operations Research, 2021
We consider the problem of choosing design parameters to minimize the probability of an undesired rare event that is described through the average of n iid random variables. Since the probability of interest for near optimal design parameters is very small, one needs to develop suitable accelerated Monte-Carlo methods for estimating the objective function of interest. One of the challenges in the study is that simulating from exponential twists of the laws of the summands may be computationally demanding since these transformed laws may be non-standard and intractable. We consider a setting where the summands are given as a nonlinear functional of random variables that are more tractable for importance sampling in that the exponential twists of their distributions take a simpler form (than that for the original summands). We use techniques from Dupuis and Wang (2004,2007) to identify the appropriate Issacs equations whose subsolutions are suitable for constructing tractable importance sampling schemes. We also study the closely related problem of estimating buffered probability of exceedance and provide the first rigorous results that relate the asymptotics of buffered probability and that of the ordinary probability under a large deviation scaling. The analogous minimization problem for buffered probability, under conditions, can be formulated as a convex optimization problem which makes it more tractable than the original optimization problem. Once again importance sampling methods are needed in order to estimate the objective function since the events of interest have very small (buffered) probability. We show that, under conditions, changes of measures that are asymptotically efficient (under the large deviation scaling) for estimating ordinary probability are also asymptotically efficient for estimating the buffered probability of exceedance. We embed the constructed importance sampling scheme in suitable gradient descent/ascent algorithms for solving the optimization problems of interest. Implementation of schemes for some examples is illustrated through computational experiments.
Buffered Probability of Exceedance: Mathematical Properties and Optimization Algorithms
This paper studies a probabilistic characteristic called buffered probability of exceedance (bPOE). It is a function of a random variable and a real-valued threshold. By definition, bPOE is the probability of a tail such that the average of this tail equals the threshold. This characteristic is an extension of the so-called buffered failure probability and it is equal to one minus inverse of the conditional value-at-risk (CVaR). bPOE is a quasi-convex function of the random variable w.r.t. the regular addition operation and a concave function w.r.t. the mixture operation; it is a monotonic function of the random variable; it is a strictly decreasing function of the threshold on the interval between the expectation and the essential supremum. The multiplicative inverse of the bPOE is a convex function of the threshold, and a piecewise-linear function in the case of discretely distributed random variable. The paper provides efficient calculation formulas for bPOE. Minimization of bPOE is reduced to a convex program for a convex feasible region and to linear programming for a polyhedral feasible region and discretely distributed random variables. A family of bPOE minimization problems and corresponding CVaR minimization problems share the same set of optimal solutions.
A note on fast envelope estimation
Journal of Multivariate Analysis, 2016
We propose a new algorithm for envelope estimation, along with a new √ n-5 consistent method for computing starting values. The new algorithm, which does not 6 require optimization over a Grassmannian, is shown by simulation to be much faster 7 and typically more accurate than the best existing algorithm proposed by Cook and 8 Zhang [7]. 9 of the analysis. Envelopes achieve efficiency gains by basing estimation on the variation 14 that is material to those goals, while simultaneously excluding that which is immaterial. 15 It now seems evident that immaterial variation is often present in multivariate analyses 16 and that the estimative improvement afforded by envelopes can be quite substantial when 17 the immaterial variation is large, sometimes equivalent to taking thousands of additional 18 observations.