I was trying to find some online papers on the
Two-Envelope Paradox, and didnt find much that was helpful beyond Chalmerss
paper. (Is there meant to be that last s
there? I can never tell what the rules are about forming possessives for names
ending in s.)
Actually, even Chalmers I think doesnt really get
to the heart of the problem. In Chalmerss resolution, the paradox arises
because we use dominance reasoning inappropriately. We infer from Exactly
one of these options is going to occur and Conditional
on any one of them occuring, Im better off switching than staying to Im
better off switching than staying. And thats a bad inference here,
and we can see its a bad inference here because the overall expected gain
from switching (B-A) can be represented as the sum of an infinite series in
which the sums of both the positive and the negative terms diverge.
I think this understates how much of a problem the
Two-Envelope Paradox really is. Or at least how much of a problem it is if,
like me, you dont take it to be immediately obvious that
rational decision making involves maximising expected utility. Ill spare you
the lesson on why thats something that shouldnt be taken for granted, and
just note that historically it wasnt taken
for granted. As much as it is orthodoxy amongst economists and philosophers
now, expected utility theory is a theory of
how to make decisions. Which is not to say there arent very good arguments for
that theory. There are. But as far as I know every single one of
them has as a premise something like If exactly one of
these options is going to occur and for any given
one, if it occurs, then I prefer A to B, then I prefer A to B. More
or less, what the Two-Envelope Paradox shows is that expected utility theory
undermines its own best arguments.
But only more or less. There are ways of
restricting the italicised premise in some of these support arguments there so
that (a) the resulting arguments are valid arguments to the conclusion that
expected utility theory is the one true decision theory and (b) the restricted
premise is still true according to expected utility theory. The problem is that
the restrictions end up looking rather ad hoc, so the philosophical force of
these arguments is questionable. This is all going by rather fast, but I think
the resulting dialectical situation is as follows. The italicised premise, which
has the form All Fs are Gs, is intuitively
true, but false according to expected utility theory. The restricted version of
it, All Fs that are not-Hs are Gs, is
strong enough to entail expected utility theory, and it is intuitive, but the
only reason it is intuitive is that the false generalisation All
Fs are Gs is intuitive.
Does this seriously undermine the case for expected
utility theory? Some days I think so, other days not.
None of this is to understate the value of Chalmerss
paper, which as I said is easily the
best thing on the web about the problem.
Anyway, what I really meant to do was draw your
attention to another problem, which for my current interests is even more
interesting. Its late, so Ill just state the problem and link to a long
discussion of it, but I highly recommend you try working it out before
following the link.