Comment by jonath_laurent | Hacker News Reader

Comment by jonath_laurent | Hacker News Reader

I would argue that this Wikipedia article is misleading and that it confuses more than it clarifies when it comes to resolving the paradox.

In particular, I am disputing the fact that "no proposed solution is widely accepted as definitive" (exact quote). Indeed, the switching argument (or at least the argument as it is presented in the Wikipedia article) makes a clear and precise mistake that I claim any trained mathematician will immediately agree on once pointed to it.

This mistake is explained in the following comments: https://news.ycombinator.com/item?id=31566226, https://news.ycombinator.com/item?id=31567251 and https://news.ycombinator.com/item?id=31569991.

Once you have identified it, this mistake is all but subtle. It is not about infinite series or Bayesian reasoning. It does not require a 30 minutes rebuttal talk. It is simply about misusing the very definition of an expected value in a way that could be qualified as a typing error (see linked comments). This error can only go undetected because of the ambiguity of the English language. Most of the fancy mathematical discussions I've been reading are distractions. This paradox is about language and not about any deep mathematical fact of probability theory.

This is not the issue. When expected value is well-defined, it is perfectly fair to compute E[X] = E[E[X | A]], where E[X | A] is a function of random variable A ("conditional expectation"). If E[X | A] > A for every particular value of A, then E[X] > E[A], whenever the expected values E[X], E[A], and E[X - A] are all well-defined.

The issue in this case is that E[X - A] is not well-defined. The expected profit from switching is given by an infinite series whose positive and negative components are each of infinite total magnitude, thus yielding different sums when bracketed differently ("conditional convergence"; just a coincidence that the word "conditional" comes up here again).

This is a good comment and I believe your justification is equally valid.

The reason for the apparent disagreement is that in order to point out a logical flaw in an argument, one must make the argument formal and explicit enough first. There are several plausible ways that the Wikipedia argument can be translated into a machine-checkable formal proof (and this ambiguity is at the core of the paradox). When I make such an attempt in my head, I am not interpreting the expectation as a conditional one and the problematic step is about conflating a random variable with a fixed constant. In your attempt to formalize the Wikipedia argument, you try and use the iterated expectation theorem and the problematic step becomes different. Seeing from the comments, several people agree with my interpretation but clearly there is an ambiguity here that is part of the paradox.

Thanks for prompting me to add nuance to my comment.

The claim is that no solution is accepted as definitive, which is an empirically true fact about the academic literature. To respond to this with "no, my favored solution is definitive" is uninteresting.

I am in fact disputing this empirical fact, or at least the interpretation of it that is suggested in the article.

There is no such thing in mathematics such as an unresolved dispute over whether or not a five line proof with elementary concepts is flawed or not. The only possible situations are: 1) the proof can be checked to be correct at the level of axioms, 2) an incorrect reasoning step can be pointed to or 3) the proof is ambiguous and/or people cannot agree on what is being proved.

There isn't really a mathematical literature about the two envelopes paradox because the paradox is not really interesting from a mathematical standpoint. Or at least, making it interesting from a mathematical standpoint would require presenting a different argument than the one presented in the Wikipedia article. There may be a scholarly debate about different philosophical or linguistic aspects of this paradox. However, there is certainly no debate about where the flaw is in the presented switching argument once you make it precise enough.

(And the answer should not involve infinite series unless you are looking at a different, strong-arm version of the argument.)

The second simple answer from the article is basically this: pointing out that A is conditionally defined in step 1, but both definitions are used in later steps as if they are the same value. It then goes on to over-explain that a bit, but this answer is in there.

I agree with you. What I am taking issue with is the article presenting a multitude of (unnecessarily) complex refutations (as if one wasn't enough) and suggesting that there is no consensus on which one should be believed. This is not how people do mathematics!

A better article would first introduce a more precise version of the switching argument and then precisely identify the flaw in it. The more advanced mathematical and philosophical discussion that explores variations of the basic argument should be separated and contextualized clearly.

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