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:
“We are the creatures of imagination, passion, and self- will, more than of reason or even of self-interest.... Even in the common transactions and daily intercourse of life, we are governed by whim, caprice, prejudice, or accident. The falling of a teacup puts us out of temper for the day; and a quarrel that commenced about the pattern of a gown may end only with our lives.”
—William Hazlitt (1778–1830)
“Compared to football, baseball is almost an Oriental game, minimizing individual stardom, requiring a wide range of aggressive and defensive skills, and filled with long periods of inaction and irresolution. It has no time limitations. Football, on the other hand, has immediate goals, resolution on every single play, and a lot of violence—itself a highlight. It has clearly distinguishable hierarchies: heroes and drones.”
—Jerry Mander, U.S. advertising executive, author. Four Arguments for the Elimination of Television, ch. 15, Morrow (1978)