A Common Resolution
A common way to resolve the paradox, both in popular literature and in the academic literature in philosophy, is to observe that A stands for different things at different places in the expected value calculation, step 7 above. In the first term A is the smaller amount while in the second term A is the larger amount. To mix different instances of a variable in the same formula like this is said to be illegitimate, so step 7 is incorrect, and this is the cause of the paradox.
According to this analysis, a correct alternative argument would have run on the following lines. Assume that there are only two possible sums that might be in the envelope. Denoting the lower of the two amounts by X, we can rewrite the expected value calculation as
Here X stands for the same thing in every term of the equation. We learn that 1.5X is the average expected value in either of the envelopes, hence no reason to swap envelopes according to this calculation.
Read more about this topic: Two Envelopes Problem
Famous quotes containing the words common and/or resolution:
“Many women cut back what had to be done at home by redefining what the house, the marriage and, sometimes, what the child needs. One woman described a fairly common pattern: “I do my half. I do half of his half, and the rest doesn’t get done.””
—Arlie Hochschild (20th century)
“Breaking his oath and resolution like
A twist of rotten silk.”
—William Shakespeare (1564–1616)