[PairTheBoard]

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I was a little dissatisfied with my Random Switchers, since I was supposed to be replying to your idea about calling off the fixed $100 bet. So here is an idea about Random Call-Offers.

There is a sequence of Groups. Group n decides whether or not to call off the bet in this way. They generate a random number U, uniform on (0,1), and call off their bet if

A > sqrt{n|ln(U)|}.

Their EV is

50(2e^{-y^2/n} - e^{-4y^2/n}),

where the values in the envelopes are y and 2y. For small n, Group n will not do very well, since they will be calling off their bet too often. But the EV will increase monotonically with n. At about n = 1.64y^2, the EV of Group n will be about the same as the Never Call-Offers, $50. But the EV will continue to increase, reaching a maximum at about n = (3y^2)/ln(2). This Group will have an EV of about $59. After that, the EV will decrease monotonically, with the limit being $50. In other words, all Groups with n sufficiently large will outperform the Never Call-Offers.

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That's remarkable and I think an interesting new addition to the archive of 2+2 Two Envelope Facts, Results, and Implications.

I'm a little suprised because I thought the 2-1 odds were so great as to put any improvement out of reach. I guess it should be noted that for any fixed envelope amounts y,2y picking n large enough improves on the Never-Call-Offers. However, no matter how large n is chosen, if the envelope amounts y,2y are large enough the Never-Call-Offers still do better.

But wait a minute. This is actually amazing isn't it? Why can't you just wait until you see the envelope amount before you choose which n to use for your random choice? You won't know whether you're looking at y or 2y. But just choose n large enough to work for either. That way you can always insure an improved decision over just Not Calling-Off. Are you sure this is right? If so I think its a significant advance.

Wouldn't this varying n method also work for asssumed prior distributions for Y (other than the point mass)?

PairTheBoard