One Observation behind Two-Envelope Puzzles

In two famous and popular puzzles a participant is required to compare two numbers of which she is shown only one. In the first one there are two envelopes with money in them. The sum of money in one of the envelopes is twice as large as the other sum. An envelope is selected at random and handed to you. If the sum in this envelope is x, then the sum in the other one is (1/2)(2x) + (1/2)(0.5x) = 1.25x. Hence, you are better off switching to the other envelope no matter what sum you see, which is paradoxical. In the second puzzle two distinct numbers are written on two slips of paper. One of them is selected at random and you observe it. How can you guess, with probability greater than 1/2 of being correct, whether this number is the larger or the smaller? We show that there is one principle behind the two puzzles: The ranking of n random variables X1, ... , Xn cannot be independent of each of them, unless the ranking is fixed. Thus, unless there is nothing to be learned about the ranking, there must be at least one variable the observation of which conveys information about it.two envelope paradox

Agreeing to agree

Aumann has shown that agents who have a common prior cannot have common knowledge of their posteriors for event $E$ if these posteriors do not coincide. But given an event $E$, can the agents have posteriors with a common prior such that it is common knowledge that the posteriors for $E$ \emph{do} coincide? We show that a necessary and sufficient condition for this is the existence of a nonempty \emph{finite} event $F$ with the following two properties. First, it is common knowledge at $F$ that the agents cannot tell whether or not $E$ occurred. Second, this still holds true at $F$, when $F$ itself becomes common knowledge.Agreeing theorem, common knowledge, common prior, no trade theorem

Anna Karenina and The Two Envelopes Problem

The Anna Karenina principle is named after the opening sentence in the eponymous novel: Happy families are all alike; every unhappy family is unhappy in its own way. The Two Envelopes Problem (TEP) is a much-studied paradox in probability theory, mathematical economics, logic, and philosophy. Time and again a new analysis is published in which an author claims finally to explain what actually goes wrong in this paradox. Each author (the present author included) emphasizes what is new in their approach and concludes that earlier approaches did not get to the root of the matter. We observe that though a logical argument is only correct if every step is correct, an apparently logical argument which goes astray can be thought of as going astray at different places. This leads to a comparison between the literature on TEP and a successful movie franchise: it generates a succession of sequels, and even prequels, each with a different director who approaches the same basic premise in a personal way. We survey resolutions in the literature with a view to synthesis, correct common errors, and give a new theorem on order properties of an exchangeable pair of random variables, at the heart of most TEP variants and interpretations. A theorem on asymptotic independence between the amount in your envelope and the question whether it is smaller or larger shows that the pathological situation of improper priors or infinite expectation values has consequences as we merely approach such a situation.Comment: Final corrections (fingers crossed

One Observation behind Two-Envelope Puzzles

and popular puzzles a participant is required to compare two numbers of which she is shown only one. Although the puzzles have been discussed and explained extensively, no connection between them has been established in the literature. We show here that there is one simple principle behind these puzzles. In particular, this principle sheds new light on the paradoxical nature of the first puzzle. According to this principle the ranking of several random variables must depend on at least one of them, except for the trivial case where the ranking is constant. Thus, in the nontrivial case there must be at least one variable the observation of which conveys information about the ranking. A variant of the first puzzle goes back to the mathematician Littlewood [7], who attributed it to the physicist Schrödinger. See [6], [3], [2]and[1] for more detail on the historical background and for further elaboration on this puzzle. Here is the common version of the puzzle, as first appeared in [5]: To switch or not to switch? There are two envelopes with money in them. The sum of money in one of the envelopes is twice as large as the othe