[PairTheBoard]

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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.

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I was not trying to ridicule you by asking you to address the 2E Proposition Bet Paradox. I think the 2E Proposition Bet Paradox is a better version of the 2E Paradox because it strips away a couple of irrelevant factors that make for psychological muddy waters. In the 2E paradox the fact that you are being given the amount in the envelope tends to hide the fact that you are simply making a bet when you switch. Also, in the 2E paradox, since switching is EV neutral, you tend to feel less psychologically compelled to make a decision about what is best.

The 2E Proposition Bet Paradox is right there in front of you. Even a frequentist will take the first bet.

1) Given 3-2 odds bet $10 your envelope contains the smaller amount.

This is clearly a +EV bet. Even after opening the envelope, assuming you don't know the envelope amounts. Although the frequentist will say the outcome has been determined he will still put his money on what amounts to a 50-50 proposition with 3-2 odds in his favor. He will do this in the same way he will bet that a randomly dealt card having been dealt face down is the Ace of Spades if you give him 80-1 odds.

But as soon as you change the bet to 2)

2) Given 3-2 odds bet the amount in the envelope that it is the smaller amount.

any gambler worth his salt knows the bet is no good. Yet the same Paradox arises. You are being given 3-2 odds on what you just said was a 50-50 bet. That's got to be +EV.

When confronted with the 2E Proposition Bet Paradox you respond,

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But it takes only logic, not probability theory, to see that the bet in (2) can be converted to an obviously poor bet of winning half the total amounts in the two envelopes if you have the smaller amount and paying two thirds of the total amount if you have the larger amount. That's not a mystery to me or anyone who thinks through it clearly.


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But with the original - But equivalent - Two Envelope Paradox you insist

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But there's the other side as well, the one that says the probability that you have the smaller amount is 50% so the expected value of switching is positive. You can't refute a paradox by strengthening one side, that just makes the paradox more puzzling. You have to show why one side is wrong.


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It's easy to see why that side is wrong in the 2E Proposition Bet Paradox. It's because you're being asked to bet the amount in the envelope. The amount you bet also determines whether you win or lose. Bet a little and win. Bet a lot and lose. Not +EV even with 3-2 odds.

The 2E Paradox is equivalent as a Paradox. In the 2E Paradox you are being asked to bet half the amount in the envelope and are given 2-1 odds. The amount you bet determines whether you win or lose. Bet a little and win. Bet a lot and lose.

In my opinion this Does give the reason "why one side is wrong." And I think it makes common sense.


PairTheBoard