[extropy-chat] Possible Worlds Semantics

[Chris Hibbert]

Spike responded:
> By the line of reasoning presented, one would trade.  But with the
> given information one might be tempted to trade even without looking
> in one's own envelope.  What have you actually learned by looking?

In my interpretation, you learned that you didn't have $10M.  In all but 
the extreme positions, two possibilities remain out of the original 6, 
and in each of those 4 middle cases, the expected value of trading (as 
long as the other party doesn't have to consent) is better than keeping.

> Even if there is a one dollar charge to trade, one would still want
> to trade. But after paying a buck and trading, the same line of
> reasoning would still apply. So one would pay another buck to trade
> back.

If you look at your envelope and then trade, and then look at the new 
envelope, there's no doubt about whether you improved your chances.

If you look, trade, and then don't look at the new envelope, then you're 
happier with the expected value of the current unviewed envelope than 
the one you've seen.

If you don't look before trading (in the original problem) then you 
don't know whether you have the $10M and so there's no reason to trade.

(Out of order):
> Modify the question only slightly by saying the envelopes contain 
> 10^x, the other contains 10^(X+1). X need not be an integer for this 
> to work: one envelope contains ten times the amount in the other, 
> with no hard limits. By the line of reasoning presented, one would

How do the edge conditions change in your reformulation with X rather 
than N?

I think a hard limit on the max value matters, but the continuity or 
lack thereof of the payouts doesn't.  If there's a max, and my envelope 
contains within a factor of 10 of the max, then I absolutely wouldn't 
trade.  If my envelope is between a factor of 10 and 100 of the max, 
then I would swap if my brother's consent isn't required.  If my brother 
has to agree, then I wouldn't swap, since I know he'd only do so if his 
envelope weren't within a factor of 10 of the max.

I don't know how to analyze it if there's no claimed maximum.  You have 
to convince me that I believe in an equal probability distribution up to 
and including unimaginable sums.  If there isn't an EPD, then the game 
is completely different.

Chris
-- 
   Pictures from my trip to the Four Corners area:
    http://discuss.foresight.org/~hibbert/Canyon02/canyon.html

Chris Hibbert
hibbert at mydruthers.com
Blog:   http://pancrit.org