Re: [R] Two envelopes problem

From: Duncan Murdoch <murdoch_at_stats.uwo.ca>
Date: Tue, 26 Aug 2008 11:26:24 -0400

On 8/26/2008 9:51 AM, Mark Leeds wrote:
> Duncan: I think I see what you're saying but the strange thing is that if
> you use the utility function log(x) rather than x, then the expected values
> are equal.

I think that's more or less a coincidence. If I tell you that the two envelopes contain X and 2X, and I also tell you that X is 1,2,3,4, or 5, and you open one and observe 10, then you know that X=5 is the content of the other envelope. The expected utility of switching is negative using any increasing utility function.

On the other hand, if we know X is one of 6,7,8,9,10, and you observe a 10, then you know that you got X, so the other envelope contains 2X = 20, and the expected utility is positive.

As Heinz says, the problem does not give enough information to come to a decision. The decision *must* depend on the assumed distribution of X, and the problem statement gives no basis for choosing one. There are probably some subjective Bayesians who would assume a particular default prior in a situation like that, but I wouldn't.

Duncan Murdoch

Somehow, if you are correct and I think you are, then taking the
> log , "fixes" the distribution of x which is kind of odd to me. I'm sorry to
> belabor this non R related discussion and I won't say anything more about it
> but I worked/talked on this with someone for about a month a few years ago
> and we gave up so it's interesting for me to see this again.
>
> Mark
>
> -----Original Message-----
> From: r-help-bounces_at_r-project.org [mailto:r-help-bounces_at_r-project.org] On

> Behalf Of Duncan Murdoch
> Sent: Tuesday, August 26, 2008 8:15 AM
> To: Jim Lemon
> Cc: r-help_at_r-project.org; Mario
> Subject: Re: [R] Two envelopes problem
>
> On 26/08/2008 7:54 AM, Jim Lemon wrote:

>> Hi again,
>> Oops, I meant the expected value of the swap is:
>> 
>> 5*0.5 + 20*0.5 = 12.5
>> 
>> Too late, must get to bed.

>
> But that is still wrong. You want a conditional expectation,
> conditional on the observed value (10 in this case). The answer depends
> on the distribution of the amount X, where the envelopes contain X and
> 2X. For example, if you knew that X was at most 5, you would know you
> had just observed 2X, and switching would be a bad idea.
>
> The paradox arises because people want to put a nonsensical Unif(0,
> infinity) distribution on X. The Wikipedia article points out that it
> can also arise in cases where the distribution on X has infinite mean:
> a mathematically valid but still nonsensical possibility.
>
> Duncan Murdoch
>
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