The envelope paradox (2006)
Regarding the &quot;money in envelopes&quot; version in the second half of the post: When I first encountered this, I came up with the author&#x27;s answer (about how the &quot;always switch&quot; strategy implicitly assumes they come from an infinite distribution).<p>But over the years I&#x27;ve come to believe that it confuses things to arbitrarily claim that the opponent is choosing their numbers from a distribution, and then talking about what kind of a distribution it was. I think it&#x27;s much clearer to examine things in Bayesian terms, by noting that before opening an envelope the player expects it&#x27;s 50% likely that they chose low, and after opening it they can update their expectations. (The math works out the same either way, it just seems weird to me to impose rules on the opponent that weren&#x27;t part of the puzzle.)<p>(Also: the fact that people always seem to examine this puzzle by imagining what kind of distribution the opponent chooses from seems, to me, an artifact of just how accustomed we are to standard statistics, and how uncomfortable we are with probability puzzles that don&#x27;t involve values being chosen from predefined distributions.)