Tip of the Week #21 Tip Index

Go to the Prior Tip Why Flip a Coin?
Go to the Next Tip Analytica for Business Modeling and Monte Carlo Simulation
Return to MaxValue Home Page

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:

Adam and Bob open their envelopes and are delighted to receive the windfall. However, the person with the smaller amount feels less fortunate.
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.
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?:

A. If the envelopes remain sealed, it does not matter whether Adam and Bob switch.

B. Adam open his envelope and sees a check for $X. He calculates that it would be better for him to switch with Bob because the expected monetary value (EMV) is $1.25X = 0.5*$2X + 0.5*$X/2. Adam should ask Bob to exchange envelopes.

C. If Adam and Bob both open their envelopes but do not disclose their amounts to the other, then it does not matter whether they keep or exchange their envelopes.

I would like to test the calibration of your intuition. Please email me at 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.

Bonus question 1: Set up a Monte Carlo simulation to model the system and calculate the expected values and probabilities of the several strategies. This can be done in Excel or Lotus with the Rand function. Using a procedural language, such as QuickBasic, is somewhat easier.

Bonus question 2: Suppose the values are 1:1000, would this change your answer as to the best strategy (or whether it matters)? For example, suppose either person opens his envelope and finds a check for $1,000. The other person's envelope contains a check for either $1 or $1,000,000.

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