On the Two-Box Paradox

Abstract

On a game show, you are presented with two identical boxes. Both boxes contain positive monetary prizes, one twice the other. You are allowed to pick one box and observe the prize x > 0, after which you can choose to trade boxes. In terms of simple expected value, it is always better to trade since (2x) + () 5 > x. That is the paradox. Simple thought experiments suggest that a sufficiently large observed prize would cause a player not to trade, despite the mathematical computation of expected value. In individual cases, this creates some threshold, which depends on the observed prize, for ceasing to trade. A player may have in mind prior probabilities about what prizes the game show would offer, so that an observed prize of $10,000, for instance, would not yield equal judgmental odds of $20,000 or $5,000 in the unobserved box. The judgmental probability approach to the two-box problem seeks to develop optimal threshold strategies in terms of prior distributions on the set of possible prizes. Recent articles in this MAGAZINE have focused on the judgmental probability approach, although they have also discussed the second line of attack on this problem, expected utility [2, 3]. In expected utility theory, it is assumed than an individual has an underlying utility function for wealth. This utility function is increasing because it is presumed that an individual will always prefer more wealth to less wealth. In addition, the utility function is concave because it is presumed that an individual will have nonincreasing marginal utility for wealth. The utility function u is thus an increasing, concave function from the positive half line into the real line. The scaling on this function is unimportant because a positive linear transformation a + bu, with b > 0, is equivalent for individual REM 2. When n > 2 is even, integers a, b, and c satisfy a2 + b2 = cn if and 302 MATHEMATICS MAGAZINE