[extropy-chat] Possible Worlds Semantics

[Hal Finney]

Ian writes about Kripkean "possible worlds" semantics.  I've been trying
to learn a little about this because Robin has used it in some of his
papers, and it is a widely used framework for economic and game theory
analysis of certain issues.

I found a good and gentle online introduction to the formalism here.
It is a draft of a widely referenced survey by John Geanakoplos:

http://www.tark.org/proceedings/tark_mar22_92/p254-geanakoplos.pdf

The topic is "common knowledge", information for which we can say that
I know it, you know it, I know you know it, you know I know it, I know
you know I know it, and so on, indefinitely.  This seemingly simple topic
produces perhaps the most paradoxical results in economics, some of them
due to Robin's original work.  It is impossible for people to disagree
(or at least to disagree persistently, or to agree to disagree), it is
impossible for people to bet, it is impossible for people to trade on
futures markets, etc.  In other words, rational people would not engage
in a great many behaviors which generally seem to be perfectly reasonable,
which raises the question of what exactly it means to be rational.

Understanding these surprising results requiress understanding the
logic behind them, which is developed in a special formalism dealing
with Kripke's "possible worlds".  Geanakoplos' paper starts off with an
introduction to the concept and the formalism, and is the best source
I have found for understanding how it works.

To whet your appetite I will quote one of the puzzles he mentions, which
deals with issues of what we know, what we know of what others know,
what we know of what they know of what we know, and so on:

"A generous but mischievous father tells his sons that he has placed
10^n dollars in one envelope and 10^(n+1) dollars in the other envelope,
where n is chosen with equal probability among the integers from from 1
to 6.  Since the father's wealth is well known to be 11 million dollars,
the sons completely believe their father.  He randomly hands each son
an envelope.  the first son looks inside his envelope and finds $10,000.
Disappointed at the meager amount, he calculates that the odds are
fifty-fifty that he has the smaller amount in his envelope.  Since the
other envelope contains either $1,000 or $100,000 with equal probability,
the first son realizes that the expected amount in the other envelope
is $50,500.  Unbeknownst to him the second son has seen that there is
only $1,000 in his envelope.  Based on his information, he expects to
find either $100 or $100,000 in the first son's envelope, which at equal
odds come to an expectation of $5,050.  The father privately asks each
son whether he would be willing to pay $1 to switch envelopes, in effect
betting that the other envelope has more money.  Both sons say yes.
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?
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?"

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.

Hal