[R] Two envelopes problem

[markleeds at verizon.net]

  Duncan: Just one more thing which Heinz alerted me to. Suppose that we 
changed the game so that instead of being double or half of X,
we said that one envelope will contain X + 5 and the other contains X-5. 
So someone opens it and sees 10 dollars. Now, their remaining choices
are 5 and 15 so the expectation of switching is the same = 10.  So, in 
this case, we don't know the distribution of X and yet the game is fair. 
This is
why , although I like your example, I still think the issue has 
something to do with percentages versus additions.

In finance, the cumulative return is ( in non continuous time )  over 
some horizon  is a productive of the individual returns over whatever 
intervals
you want to break the horizon into. In order to make things nicer 
statistically ( and for other reasons too like making the assumption of 
normaility somkewhat more plausible ) , finance people take the log of 
this product in order to to transform the cumulative return  into an 
additive measure.
So, I think there's still something going on with units as far as adding 
versus multiplying ? but I'm not sure what and I do still see what 
you're saying in
your example. Thanks.

 
Mark



On Tue, Aug 26, 2008 at 11:44 AM, Mark Leeds wrote:

> Hi Duncan: I think I get you. Once one takes expectations, there is an
> underlying assumption about the distribution of X and , in this 
> problem, we
> don't have one so taking expectations has no meaning.
>
> If the log utility "fixing" the problem is purely just a coincidence, 
> then
> it's surely an odd one because log(utility) is often used in economics 
> for
> expressing how investors view the notion of accumulating capital 
> versus the
> risk of losing it. I'm not a economist but it's common for them  to
> use log utility to prove theorems about optimal consumption etc.
> Thanks because I think I see it now by your example below.
>
>                                            Mark
>
>
>
>
>
> -----Original Message-----
> From: Duncan Murdoch [mailto:murdoch at stats.uwo.ca] Sent: Tuesday, 
> August 26, 2008 11:26 AM
> To: Mark Leeds
> Cc: r-help at r-project.org
> Subject: Re: [R] Two envelopes problem
>
> 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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