The following response to the Wallet Game Parodox was sent to curiouser.co.uk by Robert M. Martin, professor of philosophy at Dalhousie University, Canada:
Let’s assume for the moment that each player has $1, $2, or $3 in his/her wallet. Let’s also ignore ties, pretending that the game is merely restarted whenever they happen, so in effect they never occur.
Straus’s mistake is reasoning that whatever is in his wallet, he has an equal chance of losing it vs. gaining more than it. This is right if he happens to have $2 in his wallet, but it’s not true for $1 or $3. If he has $1 in his wallet, there’s no chance of losing; if he has $3 there’s no chance of winning. Here’s a list of the possibilities:
| Straus Has | Morris Has | Outcome for Straus |
| $1 | $2 | + 2 |
| $1 | $3 | + 3 |
| $2 | $1 | - 2 |
| $2 | $3 | + 3 |
| $3 | $1 | - 3 |
| $3 | $2 | - 3 |
Notice that on this list that the number of +2 outcomes equals the number of –2, and the number of +3 equals the number of –3. So it evens out; because each outcome is equally probable, in the long run Straus will be no better or worse off.
The same thing is true for a larger number of possible sums in the wallets. Consider the outcomes when each player might have $4 in addition:
| Straus Has | Morris Has | Outcome for Straus |
| $1 | $2 | + 2 |
| $1 | $3 | + 3 |
| $1 | $4 | + 4 |
| $2 | $1 | _ 2 |
| $2 | $3 | + 3 |
| $2 | $4 | + 4 |
| $3 | $1 | - 3 |
| $3 | $2 | _ 3 |
| $3 | $4 | + 4 |
| $4 | $1 | - 4 |
| $4 | $2 | - 4 |
| $4 | $3 | - 4 |
Here there are three possibilities each of + 4 and – 4: two each of + 3 and – 3; one each of + 2 and – 2. It evens out again. It will even out no matter how what finite range of wallet contents we consider. It’s always the case that Straus’s probability of winning, and expected gain, both vary inversely with the sum in his wallet.
But the problem was posed without mentioning a finite range within which the amounts in both wallets must fall. The assumption made in setting up the problem seems to be that there is no upper or lower bound to the amount in either wallet. (How could Straus have less than zero? Well, if he had an IOU made out to Morris for $5!) Now, if there really is no upper or lower limit, then the differences we noticed above, which occur as the contents of the wallet get nearer to the upper or lower limit, do not obtain. In fact there is an equal probability for whatever amount Straus’s wallet contains that Straus will lose that amount or win more than it. So without a finite range, the paradox arises again.
But there must be a finite range of allowed possibilities, or else no probabilities make any sense. For example, if there are an infinite number of possibilities of what’s in Straus’s wallet, then the probability that he has any particular amount is zero. Similarly, the probability that Straus has less than $1 quadrillion in there is zero, as is the probability that he has more. And of he has, say, $5 in there, the probability that Morris has more is infinitely large. But so is the probability that Morris has less. What’s clear here is a phenomenon noticed in other areas of probability: infinities produce paradox, and turn what looks like sense into nonsense.
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