Re: Observation selection effects

From: Jesse Mazer <lasermazer.domain.name.hidden>
Date: Wed, 06 Oct 2004 15:26:48 -0400

Stathis Papaioannou wrote:

>The problem is that you are reasoning as if the amount in each envelope can
>vary during the game, whereas in fact it is fixed. Suppose envelope A
>contains $100 and envelope B contains $50. You open A, see the $100, and
>then reason that B may contain either $50 or $200, each being equally
>likely. In fact, B cannot contain $200, even though you don't know this
>yet. It is easy enough for an external observer (who does know the contents
>of each envelope) to calculate the probabilities: if you keep the first
>envelope, your expected gain is 0.5*$100 + 0.5*$50 = $75. If you switch,
>your expected gain is 0.5*$100 (if you open B first) + 0.5*$50 (if you
>open A first) = $75, as before.
>
>Ignorance of the actual amounts may lead you to speculate that one of the
>envelopes may contain $200, but it won't make the money magically
>materialise! And even if you don't know the actual amounts, the above
>analysis should convince you that nothing is to be gained by switching
>envelopes.
>
>If the game changes so that, once you have opened the first envelope, the
>millionaire decides by flipping a coin whether he will put half or double
>that amount in the second envelope, then you are actually better off
>switching.

I don't think that's a good counterargument, because the whole concept of
probability is based on ignorance--if you were omniscient, for example, you
wouldn't have a need for probabilities at all. If someone puts $1000 in a
blue envelope and then flips a coin to decide whether to put $3000 in a red
envelope or to leave it empty, my expected gain from picking the red
envelope should be $1500 dollars--it doesn't make sense to say that from the
point of view of someone who saw the envelopes being stuffed, it is already
certain whether the red envelope contains the money, therefore *my* expected
gain from picking the red envelope should be either $3000 or zero. My
expected gain is based on my own ignorance of the outcome of the coin toss,
information that I don't have access to shouldn't play a role. Similarly, an
external observer who knows the content of both envelopes should play no
role in the calculation of my expected gain from switching in the
two-envelope problem. If I open the envelope and find $50, I don't know
whether the other envelope contains $25 or $100, so that information cannot
be used when calculating my expected gain from switching. But as I argued
before, if I know the probability distribution the envelope-stuffer used to
pick the amount in the envelope with less money, then seeing the amount in
the envelope I open will allow me to refine my estimate of the probability
it's the envelope with less money, there's no possible distribution the
envelope-stuffer could use that would insure that no matter how much I found
in the first envelope, the other envelope would have a 50% chance of
containing double that and a 50% chance of containing half that.

Jesse
Received on Wed Oct 06 2004 - 15:39:15 PDT