[ExI] Two envelope paradox

[Rafal Smigrodzki]

On Sun, Jul 27, 2014 at 10:31 PM, The Avantguardian <
avantguardian2020 at yahoo.com> wrote:

>
> So without further ado: Imagine that Ed McMahon, or somebody suitably
> cheesy, allows you  to choose between two sealed envelopes with money in
> them. Furthermore he informs that one of the envelopes contains double the
> money of the other. So you choose one but before you can open it you are
> asked if you would like to switch your envelope for the other. So you do
> the expected value calculations on keeping or switching envelopes-
>
> Your envelope contains x dollars. The expected value of the *other*
> envelope is (1/2)*2x + (1/2)*(x/2) = 1.25x therefore the *other* envelope
> has a higher expected value than the one you hold, so you should switch.
>

### Your envelope contains x or 2x dollars with equal probability, same as
the other envelope, so you should not switch.


>
> But wait a minute. If you switch envelopes, you could do the calculation
> again and realize the the expected value of the original envelope is one
> and a quarter of the one you now hold. You could use this reasoning any
> number of times to switch envelopes. You would be trading envelopes with Ed
> McMahon indefinitely, the other envelope always being more valuable than
> the one you hold. Thereby the paradox.
>
> So to help you make up your mind, Ed McMahon allows you to open the
> envelope you now hold. You open it up, and it contains $20. Then Ed Mahaon
> gives you one last chance to switch envelopes, should you?
>

### It depends on your expectation of the distribution of rewards in
envelopes in this game, as well as your valuation of money. If you think
that the median of this distribution is closer to $40 than to $20, you
should switch. If you think the median is closer to $10, you should not.
The exact boundaries will depend on whether your valuation of rewards is
linear and what kind of shape is the distribution of rewards.

The reasoning becomes more intuitive if you are told that the multiplier
between envelopes is higher, e.g. 1000000. If the envelope shown to you
contains 1 cent, you should switch, since it's very unlikely they would
bother with putting 0.00000001$ in the other, and you can only gain a cool
10k lot while losing at most a cent. If you see $1000, you should not,
since it's very unlikely they put 1 billion in the other envelope, and you
can lose 1k, which is mildly painful.

Rafal
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