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In this post I described in some detail a certain paradox, which I will summarize:

$A$ writes two distinct numbers on slips of paper. $B$ selects one of the slips at random (equiprobably), examines its number, and then, without having seen the other number, predicts whether the number on her slip is the larger or smaller of the two. $B$ can obviously achieve success with probability $\frac12$ by flipping a coin, and it seems impossible that she could do better. However, there is a strategy $B$ can follow that is guaranteed to produce a correct prediction with probability strictly greater than $\frac12$.

The strategy, in short, is:

  • Prior to selecting the slip, $B$ should select some probability distribution $D$ on $\Bbb R$ that is everywhere positive. A normal distribution will suffice.
  • $B$ should generate a random number $y\in \Bbb R$ distributed according to $D$.
  • Let $x$ be the number on the slip selected by $B$. If $x>y$, then $B$ predicts that $x$ is the larger of the two numbers; if $x<y$ she predicts that $x$ is the smaller of the two numbers. ($y=x$ occurs with probability $0$ and can be disregarded.)

I omit the analysis that shows that this method predicts correctly with probability strictly greater than $\frac12$; the details are in the other post.

I ended the other post with “I have heard this paradox attributed to Feller, but I'm afraid I don't have a reference.”

I would like a reference.

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up vote 18 down vote accepted

Thanks to a helpful comment, since deleted, by user Stefanos, I was led to this (one-page) paper of Thomas M. CoverPick the largest numberOpen Problems in Communication and Computation Springer-Verlag, 1987, p152.

Stefanos pointed out that there is an extensive discussion of related paradoxes in the Wikipedia article on the ‘Two envelope problem’. Note that the paradox I described above does not appear until late in the article, in the section "randomized solutions".

Note also that the main subject of that article involves a paradox that arises from incorrect reasoning, whereas the variation I described above is astonishing but sound.

I would still be interested to learn if this paradox predates 1987; I will award the "accepted answer" check and its 15 points to whoever posts the earliest appearance.

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