[PairTheBoard]

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For your first point, these are two different ways of saying the same thing, but with an essential difference of order. I say the common argument is "if you always switch, that distribution must have infinite expectation." You say "if the assumed prior distribution has finite expectation, then always switching is EV neutral."


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And I insist that my statement makes sense while yours does not. You say, "if you always switch, that distribution must have infinite expectation". That's simply not true. The prior distribution can have finite expectation AND you can always switch. You have mispoken. Your statement is incomplete. I suppose I should guess that what you mean to say is, "If a prior distribution is assumed and it's assumed that always switching is not EV neutral then the prior distribution cannot have finite expectation". That statement is implied by mine. I agree the form is different. I don't think the difference is essential.

I agree this is one approach to resolving the paradox - whether the person taking it adopts your statement of the fact or mine. It speaks to one of the hidden assumptions people mistakenly make when they think that envelope amounts can be chosen from a uniform distribution on an infinite scale of money. It's really an elaboration on that point and thus succeeds in breaking down the paradox at that false premise.

However, I don't think it gets to the heart of the paradox. One of the best explanations I've seen for this was made by BBB when he said,

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BBB -
If we have some basis of information on which to determine the probability that N is $100 versus the probability that it is $50, for example using what we know in general about how people such as our benefactor might be willing to put in envelopes and give away, it turns out that we should clearly switch if we determine that the probability that N is $100 is more than half the probability that N is $50, and we should clearly not switch if N is more that twice as likely to be $50 as it is to be $100. But to simply guess that N is just as likely to be $100 as it is to be $50 and going from there is totally baseless and meaningless. That would be like if I told you I had a coin in my pocket that was not necessarily fair and I asked you what were the chances that it would come up heads if I flipped it. Unless you had some kind of information on which to determine what the coin might be like, it would be totally meaningless for you to gess 1/2 just because that would be the answer if the coin were fair.

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You said,
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This gets to your second point. We agree (I think) that if you have consistent prior beliefs about everything, as required for Bayesian statistical calculations, that either switching is EV neutral or you have infinite prior expectation. It's not clear that infinite prior expectation is irrational, despite there being only a finite amount of money in the world. For one thing, maybe there's an infinite amount in the universe. For another, you don't an infnite realization to support an infinite expectation.


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I still don't know what you mean by "consistent prior beliefs about everything". You have yet to explain this. It may be that you are not dealing in the realm of mathematics. If that's the case you should be more clear about where you depart from mathematics and explain more precisely just what you are saying in whatever realm you are dealing. I believe there is some controversy about certain so called "Baysian" approaches which lack sound mathematical foundations. I suppose if this is where you're at there's not much basis for argument with you. We will have to take your pronouncements on authority.


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But that's theory stuff, my concern is that people don't have consistent prior beliefs. Bayesian statistics achieves consistency by ignoring things that are important to real decision-making. What is a young person's prior about his lifetime income or amount of marital happiness? Isn't the choice of going to college or dumping a girlfriend somewhat like the envelope choice? Do you think understanding the Bayesian "resolution" of the paradox will help him decide?


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"Isn't the choice of going to college or dumping a girlfriend somewhat like the envelope choice?"

I see no relationship between the two.

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Sometimes our best information about possible outcomes comes from the first choice we are offered. We have no idea what things cost until someone offers us something. We instinctively turn it down, because we realize the chance of the first offer being the best is pretty low. We get a few more offers, until someone offers us something that appears to be a low price relative to what we've been seeing from the others. Isn't this a pretty good description of how you make some decisions? Would you be surprised if I proved that it shows you have an infinite prior expectation?


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You would have to explain your model more precisely for us to see its implications. I don't see how it relates to the envelope paradox.


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Yes, the USD/GBP example involves two people measuring expectation in different units. I think this is very common in real decisions. Even if we are both betting money, if our credit ratings are different, then we are not computing in the same units. And since the money is presumably a means to some end, if there is price uncertainty about what we plan to spend it on, there's also a difference.


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Maybe people do this. But I don't see how the observation that people often do bad math relates to the envelope paradox.

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I don't say (a) and (b) imply each other, I say that if you assume both, you get the envelope paradox.

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Let's be clear here. Your statement of (a) and (b) are

(a) when you pick one of two envelopes with different amounts at random, there is a 50% chance that you'll get the one with the higher amount

(b) if you have a 50/50 chance to win $2 or lose $1 you have positive expected EV.

Now you say, "if you assume both, you get the envelope paradox"

There's no need to assume (b). (b) is simply a true statement. Also, if by (a) you mean that if you chose one of the two closed envelopes there's a 50-50 chance it will have the larger amount then that is simply a true statement as well. It's you who are missing the heart of the paradox. It's right here. It's when someone opens an envelope, sees $2 in it and says that (a) implies the premise of (b) that they are making a mistake.

Instead, you remain unclear as to what you mean by (a) so that you can conclude,

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The Bayesian resolution rejects (a). It says that after seeing the amount in the envelope you pick you have to change your estimate of the probability it is the higher amount.

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There's no need to reject (a) unless you were unclear as to what it meant. If you are clear with (a) and mean that if you chose one of the two closed envelopes there's a 50-50 chance it will have the larger amount then there is a repeatable experiment you have in mind that can prove out this statement of probabilty. The experiment is for numerous people to pick one of the two envelopes at random. The more people who pick, the closer you will see half picking the higher amount and half the lower amount. There's a 50-50 chance that any one of them will pick the higher amount.

Consider this 2 envelope game. One envelope contains a dollar and the other envelope contains nothing. Again you have an
(a) when you pick one of two closed envelopes at random, there is a 50% chance that you'll get the one with the dollar

and a
(b) The expectation for a person picking an envelope at random is 50 cents.

Now a person picks an envelope, opens it sees it contains nothing and is given the option of switching. (a) remains true as clearly stated. (b) remains true. There's no deep philisophical conflict here between subjective Baysians and Objectivists. There's simply elementary conditional probabilty saying that once seeing the content of the first envelope the probablity for the contents of the second changes from the apriori probabilty. The exact same thing happens in the Original Two Envelope Problem. It's just not as obvious because people tend to percieve all envelope amounts as being equally likely - which we know they are not under any apriori conditions.


PairTheBoard