[R] Two envelopes problem

[Mark Difford]

...

To pick up on what Mark has said: it strikes me that this is related to the
simplex, where the bounded nature of the vector space means that normal
arithmetical operations (i.e. Euclidean) don't work---that is, they can be
used, but the results are wrong. Covariances and correlations for instance,
are wrong, something that Pearson noted more than a century ago.

Taking logs of ratios of numbers (say a number divdided by geometric mean of
the other numbers, as in Aitchison's set of transformations) in this space
transfers everything to Euclidean space, so "squaring" the problem. This is
why taking logs fixes things ??

Mark.



statquant 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. 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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> ______________________________________________
> R-help at r-project.org mailing list
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> PLEASE do read the posting guide
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> 
> 

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