From: Duncan Murdoch <murdoch_at_stats.uwo.ca>

Date: Tue, 26 Aug 2008 11:26:24 -0400

*>
*

> 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
*

*>
*

*> ______________________________________________
*

*> R-help_at_r-project.org mailing list
*

*> https://stat.ethz.ch/mailman/listinfo/r-help
*

*> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
*

*> and provide commented, minimal, self-contained, reproducible code.
*

R-help_at_r-project.org mailing list

https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. Received on Tue 26 Aug 2008 - 15:31:00 GMT

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,

R-help_at_r-project.org mailing list

https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. Received on Tue 26 Aug 2008 - 15:31:00 GMT

Archive maintained by Robert King, hosted by
the discipline of
statistics at the
University of Newcastle,
Australia.

Archive generated by hypermail 2.2.0, at Tue 26 Aug 2008 - 16:34:00 GMT.

*
Mailing list information is available at https://stat.ethz.ch/mailman/listinfo/r-help.
Please read the posting
guide before posting to the list.
*