Reference Material:
Newcomb’s paradox is a philosophical thought experiment involving
rationality, possibility, free will, and determinism. It’s been analyzed and
reanalyzed way too many times, but it’s a perfect playground for the ideas
involving possibility from the last post and major topics from still earlier
posts. Let’s go.
You need a million dollars, so you do what any self-respecting
rationalist would do – head to the casino to win it. In a less-travelled corner
of the casino floor, you find an open door into a darkened room. A sign above
the doorway reads “Win A Million Dollars!”. Perfect. Inside, a mad scientist
presents you with 2 boxes. Box A is clear and contains $1000 cash. Box B is
opaque, and the mad scientist claims it contains either $1M or nothing. She
then gives you with a seemingly bizarre choice – you’re allowed to choose
either box B or both boxes, and then leave carrying whatever box[es] you’ve
chosen. Strange, to be sure, but taking both boxes seems like an easy way to
walk away with a sure $1000 and potentially the $1M you set out for. But of
course, there’s a catch. As you walked through the door, a brain scanner imaged
your brain and created a detailed simulation of your cognition. It then
predicted whether you would choose box B or both boxes. If it predicted you
would choose only box B, the mad scientist went ahead and put the $1M inside
before presenting the boxes to you. If the cognition simulation predicted you’d
take both boxes, the mad scientist left box B empty. The mad scientist claims
the prediction is iron clad. After all, it’s never been wrong, and she’s been
doing this her whole life.
How does knowing about the brain scanner affect your decision? You might
reason as you briefly did before. The $1M is either in box B or it isn’t, and
there’s no changing that now. You’ve already walked through the scanner. If box
B has the money, taking both boxes nets you $1M + $1000, slightly better than
the $1M in box B alone. If box B is empty and you take both boxes, you wind up
with $1000 – obviously better than the 0$ box B would get you. It seems
whatever the status of the money in box B, taking both boxes leaves you better
off.
Except this seems to completely ignore the predictive accuracy of the
cognition simulation. If you choose both boxes, it would have predicted you
would choose both boxes and, if the mad scientist is to be trusted, box B would
be empty. If you believe the mad scientist’s claims about the scanning
technology and her manipulation of the $1M, it seems you should choose only box
B. Only then will you get the $1M.
How should you reconcile these rational sounding, but conflicting, ways
of thinking? One approach is to consider the possible outcomes of the scenario
using the language of possible worlds. Consider all the possible worlds where
the scenario described above plays out as described. What we’ll vary across
this set of possible worlds is anything the scenario doesn’t explicitly
mention, as long as what we vary doesn’t produce a contradiction. For example,
in one possible world, you visit the mad scientist only once, on a Tuesday at
age 44. In another, you go once a week on a random weeknight for your entire
life. In still another, you only go when raging drunk.
Now, consider each possible world in this set and note the choice you
made and the money you wound up with. If the brain scanner and cognition
simulation work as described, and we believe the mad scientist about how she
manipulates the $1M, then in the possible worlds in which you choose box B,
you’ll get $1M, and in those where you choose both boxes, you’ll get just
$1000. In none of these possible worlds do you choose box B and get nothing,
and perhaps more importantly, in none of these worlds do you choose both boxes
and get $1M. That this is true is simply a logical translation of the
description of the scanner, cognition simulation, and mad scientist in the original
scenario. The only possible outcomes of your choice are: take box B and get
$1M, or take both boxes and get $1000.
If this is right, taking both boxes seems quite foolish. But what are we
to make of the totally sane-sounding reasoning that leads many to choose both
boxes? “The $1M is either in box B or it isn’t, and there’s no changing that
now. You’ve already walked through the scanner.” So far, so good. “If box B has
the money, taking both boxes nets you $1M + $1000.” Wait, what? There are no
possible worlds in which you take both boxes and box B has $1M in it.
Considering this as a live possibility flatly contradicts the claimed efficacy
of the scanner and cognition simulation. It’s this move – considering possible
what is in fact impossible – that leads people astray here.
But why is the mistake so common? This is where free will and
determinism enter the picture. It seems like the reasoning above rests on the
following simple idea – I’m free to choose whichever box[es] I like. If this is
true, then the game theoretic reasoning works. I compare the result of choosing
both boxes to the result of choosing only box B in the cases with and without
the money in box B. Regardless of the status of the money in box B, choosing
both boxes leaves me better off. It’s the dominant strategy. The problem is
that the scanner, cognition simulation, and mad scientist combine to undermine
my freedom. It’s no longer possible for me to choose both boxes when the money
is in box B. This possibility is crucial to the strategic dominance of choosing
both boxes. Without it, choosing both boxes is not only no longer dominant, but
much worse. Of course, the idea that I’m not free to choose either box is very
difficult to accept. It seems to me that the resistance to this idea, this lack
of freedom, is what motivates the 2-box choice.
It could be said that your freedom is only undermined because of the
claim that the cognition simulation yields perfect predictions. Predicting
human behavior that accurately is impossible, you could argue. So make the
prediction 90% accurate. Now in 90% of the possible worlds in which you choose
both boxes, box B will be empty, and in 90% of the possible worlds where you
choose box B, you’ll get $1M. You don’t know which of these worlds you’re
actually in, but you know the probabilities over these possible worlds of
winning $1M conditional on your choices. If you agree that the rational
decision is to maximize the expected value of your winnings, it’s a no-brainer.
Your expected winnings choosing box B are $900k, whereas choosing both boxes
yields just $101k on average. So even if the cognition simulation is more
realistic (fallible), the game theoretic reasoning that suggests taking both
boxes fails. It fails, I believe, because a core assumption, that you’re free
to take either box, is, at best, misleading.
Fascinating, confusing stuff. More on this next time.
