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Switch Award Paradox

The prior tip praised the book, Why Flip a Coin?: The Art and Science of Good Decisions by H. W. Lewis. Chapter 9, "A Paradox," reveals a most interesting example.

Lewis defines a paradox as "an apparent self-contradiction." Let's see what you think of his problem.

A benefactor decides to award money to two people, we'll call them Adam and Bob. The benefactor decides to have some fun with his bequest. A check for one-third of the total amount goes in one envelope, and a check for the other 2/3 goes in a second envelope. He seals the envelopes and gives one each to Able and Bob.

Consider some possibilities:

  1. Adam and Bob open their envelopes and are delighted to receive the windfall. However, the person with the smaller amount feels less fortunate.
  2. Before they open the envelopes, they agree to switch. Each then opens his envelope and learns whether he would have been better off to have kept his original envelope.
  3. Suppose Adam opens his envelope and sees that his check is for $X. Bob has either $2X or $X/2. Should Adam approach Bob and offer to exchange envelopes?

Take some time to carefully consider this situation. The players are risk-neutral but want to maximize their probability of having the larger award. Which statement(s), below, is/are correct?:

I would like to test the calibration of your intuition. Please email me at john's email address with which of the statements {A, B and/or C, above} that you think are correct. I'll post the distribution of answers (if enough readers participate) and a solution this fall.

—John Schuyler, July 1997, revised 15-Aug-97.

Copyright 1997 by John R. Schuyler. All rights reserved. Permission to copy with reproduction of this notice.