<pre>
This is "Newcomb's Problem" or "Newcomb's Paradox."  It was first
written about by philosopher Robert Nozick, who heard it from physicist
William A. Newcomb.  It is called a paradox because two seemingly
reasonable lines of argument lead to opposite strategies.

The two-box strategy (the "dominance" argument):

You are presented with two boxes: one certainly contains $1000 and the
other might contain $1 million.  You can either take one box or both.
You cannot change what is in the boxes.  Therefore, to maximize your
gain you should take both boxes.

This is called the dominance argument because whichever state of nature
exists ($1 million in box B or $1 million not in box B), you gain more
by picking both boxes.  Thus the two-box strategy "dominates" the
one-box strategy.

The one-box strategy (the "expected gain" argument):

The expected gain of each strategy is:

	E(take one) =	$0 * P(predict take both || take one) +
				$1,000,000 * P(predict take one || take one)
	E(take both) =	$1,000 * P(predict take both || take both) +
				$1,001,000 * P(predict take one || take both)

Since P(predict X || do X) is near unity, your expected gain if you take
both boxes is nearly $1000, whereas your expected gain if you take
one box is nearly $1 million.  Therefore you should take one box.

Now, to resolve the paradox, it is not sufficient to provide better
arguments for any one point of view or to undermine the arguments for
any one point of view.  You have to bring out the hidden assumptions
that lead to these arguments.  In this case, the hidden assumption is
that P(predict X || do X) is near unity, given that P(do X || predict X)
is near unity.  If this is so, then the one-box strategy is best; if
not, then the two-box strategy is best.

Events which proceed from a common cause are correlated.  My mental
states lead to my choice and, very probably, to the state of box B.
Therefore my choice and the state of box B are highly correlated.  In
this sense, my choice changes the "probability" that the money is in
box B.  However, if you do not admit the possibility of reverse
causality, then since your choice cannot change the state of box B,
this correlation is irrelevant.

While you are given that P(do X || predict X) is high, unless reverse
causality is possible, it is not given that P(predict X || do X) is
high.  Thus without reverse causality, the expected gain from either
action cannot be determined from the information given.  Thus the
dominance argument holds and you should take both boxes.  With reverse
causality, the dominance argument cannot hold since your actions change
the states of nature.  Thus the expected gain argument holds and you
should take box B only.

See also, "Paradoxes," R. M. Sainsbury, Cambridge University Press,
Second Edition, 1995, chapter 3 and the references cited therein.
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