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:
“Thy fate is the common fate of all;
Into each life some rain must fall.”
—Henry Wadsworth Longfellow (1807–1882)
“Had I been less resolved to work, I would perhaps had made an effort to begin immediately. But since my resolution was formal and before twenty four hours, in the empty slots of the next day where everything fit so nicely because I was not yet there, it was better not to choose a night at which I was not well-disposed for a debut to which the following days proved, alas, no more propitious.... Unfortunately, the following day was not the exterior and vast day which I had feverishly awaited.”
—Marcel Proust (1871–1922)