[R] Two envelopes problem

[Duncan Murdoch]

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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