[extropy-chat] Possible Worlds Semantics

[Chris Hibbert]

> Both sons say yes. 

The possible combinations are as follows:

1   10, 100
2   100, 1,000
3   1,000, 10,000
4   10,000, 100,000
5   100,000, 1,000,000
6   1,000,000  10,000,000

If you have 10M, you don't want to switch (you will only get worse.)
If you have 1M, you don't want to offer to switch (Your offer will only 
be accepted by someone with less than you.)
If you have 100K, it won't do you any good to switch. (Your clever 
brother would only trade if he had 10,000, not if he had 1M.)

The analysis can be run further, but it's starting to stretch thin.  If 
I had 10, 100, or 1000, I might be willing to switch.  I can't see why 
I'd switch with 10,000.  I wouldn't switch with 100K or 1M.

> The father then calls both of his sons in together and tells them
> that they have each offered $1 to switch envelopes, and asks them to
> shake hands on the deal, it being understood that if either son
> refuses the deal is off. The sons take a hard look at each other.
> What should they do?

If I'm face to face with my brother, and he's willing to switch, I give 
him credit for being as smart as me.

> Suppose instead that the sons were not permitted to look at each
> other, but instead they had to write their confirmation of the deal
> on separate pieces of paper and hand them to their father? What
> should they write?"

They should think this far through it, and the one with 10K shouldn't 
say yes.  The other one is stuck thinking that only someone with 10 or 
100 dollars would be willing to switch.

> I will add an additional question, to get the analysis started: is the
> first part of the reasoning correct?  That is, suppose you were the
> first son in the example above, you opened your envelope to see $10,000,
> are you correct to say that you would pay $1 to switch?  This is before
> you even know that your brother is going to be asked.

You want to know whether it's equally likely that the other envelopes 
has n+1^10 and n-1^10.  The answer is yes, in the absence of information 
about whether the other party would be willing to switch.  But someone 
with n+1^10 is much less likely to be willing to switch, especially if 
you have more than $100.  If you have to give an estimate of the value 
of the other envelope, (n+1^10 + n-1^10)/2 is the right number, but 
that's a different question of whether you'd go for a voluntary trade.

An involuntary trade?  Now I think you have to make the trade if you 
think the draws were random.  In any case but the 10M, you gain more 
than you lose in the long run by trading.  The asymmetry at the top 
makes it possible.

Chris
-- 
It is easy to turn an aquarium into fish soup, but not so
easy to turn fish soup back into an aquarium.
-- Lech Walesa on reverting to a market economy.

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