The famous Wallet Paradox invites two similar individuals to lay their wallets on the table, the one with the lesser amount of money to win both. Paradoxically, each might reason: “I have the advantage, because if I lose, I lose just what I have, but if I win, I win more than I have.” A follow-up analysis assumes that each has the same expected amount of money and asks for the best probability distribution or “best strategy” with that given mean. The following note is based on a senior colloquium talk.

The Wallet Paradox, initially put forth by Martin Gardner in his book Aha! Gotcha in 1981, was shown to have no optimal strategy for given mean by Carroll, Jones, and Rykken in a Mathematics Magazine article in December, 2001. Here we note that the strategies cannot be ordered. Indeed, consider the following three strategies with mean 10:

Strategy A: \$10 with probability 100%,

Strategy B: \$7.5 and \$12.5 with equal probabilities 50%,

Strategy C: \$7.5 with probability 2/3 and \$15 with probability 1/3.

One checks that A>B>C>A.