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:
“The difference between people isn’t in their class, but in themselves. Only from the middle classes one gets ideas, and from the common people—life itself, warmth. You feel their hates and loves.”
—D.H. (David Herbert)
“Some hours seem not to be occasion for any deed, but for resolves to draw breath in. We do not directly go about the execution of the purpose that thrills us, but shut our doors behind us and ramble with prepared mind, as if the half were already done. Our resolution is taking root or hold on the earth then, as seeds first send a shoot downward which is fed by their own albumen, ere they send one upward to the light.”
—Henry David Thoreau (1817–1862)