[AaronBrown]

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So there's no mystery with the proposition bet? It doesn't produce a true paradox with equally strong arguments on both sides for why it might be a good bet? All it takes is simple logic to see why it's not a good bet? Yet the Envelope Paradox is essentially the same situation. Only in the Envelope "Paradox" the Proposition asks if you want to bet half the amount in your envelope giving you 2-1 odds.

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I'm not sure why you keep belaboring this. I know the envelope switch is not a good bet, that's not the point. I don't believe that a friend and I can make each other infinitely wealthy by passing two envelopes back and forth all day (I tried, it doesn't work). You can, however, have two people get positive EV from opposite sides of a zero sum bet, if you tradie options with people who keep score in different currencies. But that's not the aspect of the paradox we've been discussing.

I think the hard part is to come up with a rigorous theory that spells out exactly when switching makes sense, that is also flexible enough to use for practical decision problems. You don't think that's hard, because you will accept things that I feel cannot be true and that are devilishly hard to apply. You're not alone in that, many people agree with you (although few are quite as broadminded as you, most limit themselves to one hard-to-swallow pill).

None of this makes me think you don't know basic logic or math, I don't know why you think that of me. If I were to get ad hominem, I'd bet you haven't done a lot of real world statistical analysis. Not because I think you've shown yourself incapable of it, but because the day-to-day experience of trying to apply your principles leads many practitioners to become agnostic like me.