Vsauce

Should You Switch? NO!

Not a real offer, for educational purposes only. I don’t have two envelopes filled with money. I don’t have any envelopes OR any money.

In The Two Envelopes Paradox — also called the Exchange Paradox — you know what the right answer is almost immediately. Until you don’t. And then you do. And then you’re not sure.

The problem isn’t so much the problem itself; it’s figuring out why the setup is wrong. Like several of Zeno’s Paradoxes, we’re lured into thinking about the scenario in a way that leads us down the wrong path… and by the time we realize it, we’re so deep in convincing (or troubling) math that we’ve lost sight of the real issue.

Just like with the Monty Hall Problem, even top academics have trouble elucidating clear, meaningful reasoning for why switching in the Two Envelopes Paradox is or isn’t valuable. That’s why Martin Gardner and others struggled with it for years, and why decades after the paradox (and its variants like the necktie and wallet-switching problems) debuted, it was still of academic interest in math journals and popular recreational mathematics publications.

But the Two Envelopes Paradox is an exercise in logic and probability that continues to be valuable, and probably more so than ever, with implications on how we approach math, science, and the world around us.

*** SOURCES ***

Barry Nalebuff, “The Other Person’s Envelope is Always Greener,” Journal of Economic Perspectives, Winter 1989: https://pubs.aeaweb.org/doi/pdfplus/10.1257/jep.3.1.171

Miles Mathis, “The Two Envelopes Paradox”: http://milesmathis.com/twoen.pdf

Eric Bliss, “A Concise Resolution to the Two Envelopes Paradox”: https://arxiv.org/pdf/1202.4669.pdf

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