Newcomb's Problem

Additional material

RESULTS OF THE NEWCOMB PARADOX RESPONSES FROM CUGgers
                        (From Ron Barnette, Philosophy)

[ Editor: Ron Barnette circulated the following version of the problem
to a mailing list ("CUGgers"), asked for answers, and compiled the
responses; his comments on the responses are in parentheses, signed
"Barnette". ]

                           NEWCOMB'S PARADOX

A philosophy professor at your school has a reputation for being
brilliant as well as possessed of an enormous fortune he has dedicated
to his research. One day you get a request to report to his office at a
certain hour. On a table are two boxes. One of them, labeled A, is
transparent; in it you can see an enormous pile of $100 bills. The
other, labeled B, is opaque. He tells you that there is $10,000 in
transparent box A and that in box B there is either $1,000,000 or
nothing. The philosophy professor then tells you that he is going to
give you a choice between:

(1) Taking just what is in box B.

(2) Taking what is in both boxes.

(Think about what you would choose given this much information.)

Then, he tells you that this is part of an experiment. During
registration at the beginning of the quarter, you walked under a
peculiar device that reminded you of the machines at airports that are
used to prevent hijacking. You didn't think much about it at the time.
But, he now informs you that this machine was something he designed and
that it recorded an instant profile of your basic personality and
character traits. On the basis of your profile, he made a prediction
about what choice you would make, and he decided what to put in box B
on the basis of this prediction:

(1) If he predicted you would take both, he put nothing in box B.
(2) If he predicted you would take only box B, he would put $1,000,000
    in it.

(Now, think about what you would do given this much information.)

At this point you ask the professor how accurate his predictions have
been. He tells you, correctly, that 1,000 students have been given the
choice, and he has only been wrong once. In all other cases, students
who chose both boxes got only $10,000, whereas those who chose only box
B got $1,000,000. Then he tells you to choose.

What would *you* do?

------------------------------------------------------------------------
There were 24 responses to the paradox; 23 specified their rational
choice, and one provided some fun, which I include in the excerpts
below.  Given the nature of the responses, methinks we have another
paradox in the works! Apparently, what seems to some an *obvious*
preferred choice, appears to others as not obvious at all, but, in
fact, the lesser of the rational options. Let's have a look.
------------------------------------------------------------------------
Number who chose to open *both* Box A and Box B: 14

Number who chose to open Box B only: 9

WOW!

Sample responses, edited anonymously:
---
I'd have to honestly take both boxes and see what happened
---
I would have to say that I would take box B, on the basis that if it
did indeed hold $1,000,000 I would be much better off, and if it didn't
I still wouldn't have lost anything.  That is assuming that this
required no fee.  If it did include a fee, then I would go for box A,
on the basis that I would know that I wouldn't lose anything, and in
fact gain some.  If the fee were higher than the $10,000 I would choose
to keep my fee, and go without any of the boxes.  Kinda confusing, but
that is the way it goes.  *grin*
---
I must be missing something. But, let me say that Newcomb's Paraboxes
is at least fun to read.

Always take both. Get what is in A and then anything in B, as long as B
is not something that will reduce A's value, is a freebie. I have been
told the worst it can be is nothing so I take both and lose nothing.

[ Editor: on the prediction rules: ]

I am not sure if the conditional "would" in choice (2) means he will
put the $1 mil in after the choice is made. I suppose it doesn't really
matter. If it is to be read as a past tense action " He would have put
the $1 mil in prior to the test based on his prediction." then no
decision could now affect his decision. In this case, take both boxes
and lose nothing.  If it is to be a future action then still take both
boxes. His prediction has already been made. Since the issuing of the
money is determined by his prediction, not by the decision, taking both
boxes assures the most. The decision only has an effect on receiving
whatever is in, or will be put in, the boxes. If only B is taken, even
if it has the $1 mil based on the prediction, the $10,000 in box A
would be forfeited for no reason.

[ Editor: on the record of 999 correct predictions out of 1000: ]

Always take both boxes. At this point in time no decision will change
the actions which were based on his prediction.
---
Since I am not by nature a gambling person, I would have to presume
that if the device was accurate in it's assessment of me, then the
professor would know that I would not chose only box B. Since I only
have two choices, (B) or (A & B), I would have to chose A & B.  $10,000
in the hand is worth $1,000,000 in the BOX. (sorry for the pun) :-)
-----
I would take both boxes.  The decision made while in this room has no
effect on what the professor (using his machine) has already decided
about me and what he has already put in the boxes.
---
I would pick the box that had the million in it.  If he has been wrong
only once then the chances are that I'll be a million dollars richer!!

(Interesting to note that this response is the first to claim that the
*most likely* choice for maximizing payoff is to go with B only..
Barnette)
---
I'm not sure how much analysis you were asking for in your question.  I
was reminded of the logic class I had nearly 30 years ago, and had a
vague memory of "if p than q" statements.  I'm not sure, though, that I
could put this paradox in a truth table.  My guess is:  both.  Hope
this is what you wanted.
---
I would choose Box B.  (...he had only been wrong once, but, "In all
other cases, students who chose Box A got $10,000 and students who
chose Box B got $1,000,000.")  She who risks nothing, gains nothing....
---
I would take both because from a logical position I would only have
taken Box B.
---
[ Editor: after the first part of the problem: ]

At this point, I would take both boxes.  Being a student, and due to
the fact that we are all notoriously broke at all times, I'd have to
take what I could get.

[ Editor: after the rest: ]

The logical choice would now be box B only.  1.) The machine must have
factored in the personal response to the pre-choice information given
by the professor.  The accuracy of the predictions was too accurate to
have left that information out.   2.) And even more pertinent....
Based on the information given, if you choose two boxes, you are
guaranteed only $10,000.  If you choose box B, you are virtually
guaranteed to receive $1,000,000, unless you fell into the .01 percent
error group.

(Aha! "virtually guaranteed ...$1,000,000"---could this be true?...
Barnette)

[ Editor: one wrong prediction in 1000 is a .1 percent error rate. ]
---
(Given my penchant for self-referential statements, I loved the tone of
this one!..Barnette)

What is this, "Price is Right" or "Let's Make a Deal?" I'm game, so
I'll play.  Now, your professor knows that I feel that life is little
more than a never-ending series of choices.  The choice is always
between playing is safe or taking a risk.  Since I am by nature a risk
taker, your professor knows that I will take B.  But, I know he knows
that.  So, I should take both.  But, I know he knows that I know.  So,
I should take B.  But, I know, he knows that I know that he knows that
I know.  So, I should take both.  But I know, he knows that I know that
he knows that I know that he knows.  So, I should take B.  But, you
know, I don't give a damn what he knows.  And since we got little or no
raises and I have to pay Robby's tuition, I'll take both.  And the hell
with the fact that he knows what I know what he knows that I know what
he knows, etc. etc. etc. etc. etc.  etc.  And you're better for it
since now that I am $10,000 richer, I'll buy you and Candy dinner!  No
paradox.  Just the facts!
---
(Okay, you who judged the most reasonable choice to be *both* boxes,
how do you react to this line of *argument*---not just a claim---to the
contrary? A very detailed response...Barnette)

From a pure game theory point of view, one should take only box B.

If the predictor is accurate 99.9% of the time, then the two options
are:

You take both:  You expect .999(10,000) + .001(1,010,000) = $11,000
                            ^^^^^^^^^^^    ^^^^^^^^^^^^^^
                   predictor  correct          incorrect

You take only B:  You expect  .999(1,000,000) + .001(0) = $999,000
                              ^^^^^^^^^^^^^^^    ^^^^^^
                   predictor      correct       incorrect

The utility of choosing only B is over 90 times that of choosing both.

This does assume a benevolent, and not sadistic, predictor.

Of course, you, as a philosopher, know this paradox is used to try to
poke holes in both predestination and free will.

(Now we even see some theology creeping into the discussion...Barnette)
---
I would choose box B only, but given the "paradox" that may be the
wrong answer.

My reasoning: I would shoot for the million - an extra $10,000 isn't
much if I get the million.  At the second question, it is clear that
the rational thing to do is to take box B only.  After you walk through
the character detector, it presumably detects that you are rational.
Since it is right 99.9% of the time, you have a 99.9% chance of getting
$1,000,000.  This is what I would actually do.

That is what I would do, even not knowing about the results of the
previous tests.  Given the results of the other tests, I would still
take B only.  Of course, what is in box B was really determined when I
walked through the detector at the first of the quarter.  That would
imply that you could "change your mind" from what it would have been at
that time, and safely take both boxes now.  (Also, you could be that 1
in 1,000 where the machine is wrong.)  Nevertheless, if this actually
happened to me, I'd take B only.

I suspect, though, that since this is proposed as a paradox, perhaps
the "solution" is to take both boxes at this point.  If it thinks you
were rational, then you have a 99.9% chance of getting $1,010,000, so
your expected gain is slightly larger.  But then this gets into a sort
of logical loop, I think.
---
(This response provides a nice moment of fun...Barnette)

I have four answers to the paradox (depending on the personality of the
respondent).  I assume that the personality machine is a fake!

Person 1:  Having a keen sense of touch, she picks up box B, and feels
its weight.  If it is heavy, she picks box B.  (Victim of Empiricist
Bias)

Person 2:  Finding a coupon in the box(es) for the money and realizing
he has been set up, he tries to leave the office only to be confronted
by a man offering him a great deal on a condo on Lake Erie.  (Victim of
Capitalism)

Person 3:  Violating all the rules, she takes only box A. (Victim of a
psychopathic Personality)

Person 4:  Becoming totally disgusted, he leaves both box and quits the
experiment.  (Victim of Laboratory Burnout)

Person 5:  Having watched this kind of thing on TV, she looks in vain
for a wheel to spin.  (Victim of Game Show Addiction)

(Sounds like a social-science type to me!...Barnette)
-----
The more I think about it, if I was _actually_ in that situation, I
would take box B only.  Although taking both boxes arguably (sp?) gives
a higher expected gain, I think it leads to logical problems (I know
that you know that I know that you know ... ).

Therefore, I think it really shows that the situation is impossible to
begin with.  Such a person/being/machine can't exist.  This has two
nice benefits:

1. Proves that we have free will

2. Proves that an all-knowing being (such as God) can't exist.

(Now we've got *free will* issues to contend with!..Barnette)
----
My predicted behavior (as determined by the professor) has
predetermined the contents of box B. My actual behavior will have no
impact on what box B contains. I choose both boxes.

NOTE: In developing the criterion validity for his predictor (the
"personality detector"), the professor could not have "correctly" told
the 1000 subjects that his predictor had a .999 probability of being
correct in prediction (he would not know this probability until the
completion of his study). As such, that extra bit of information given
to me prior to my selection could alter the degree of accuracy of the
predictor.
----
if there is no penalty for taking both, take both. you end up with
either 10,000 or 1,010,000. is this all there is to paradox? why's it a
paradox?

(Read the responses to get an operational definition of what a paradox
prompts...Barnette)
-----
But he doesn't say he's GOING TO put money in the box; it is already
there. No action of yours will alter what is in the boxes.  The only
thing that was ever at issue was whether his assessment of you was
correct.  But whether it was correct or not, it has already been made,
and the money is already in the boxes.  So why not take both?  I'm with
[name deleted]: why is this a paradox?  Does paradox mean something
other than what I think it does?  Excuse me while I go get a reference
book!
----
[Name deleted]'s right, it's a no lose situation.  Are you sure that
there isn't more to the paradox than meets the eye.
---
I see the following paradox. The 'machine' has supposedly determined
prior to the professor's experiment how I would react. All hinges on
whether or not I have faith in the professor's machine.

1. Assuming the machine is infallible, I should choose B and walk away
with $1,000,000.

2. However, if I do not believe the machine has predicted my behavior,
I should choose both A and B and walk away with $10,000.

This is an interesting problem, with overtones of Calvinism. The
machine is comparable with an omniscient and judgemental god of
predestiny. A good Calvinist, believing in such a god, accepts the
doctrine (analogous with box B) and a reward of election ($1,000,000).
However, the non-believer could never receive the greater reward.

What is the lesson here, that non-believers like me are never going to
receive more than "$10,000"?

(More theology!...Barnette)
---
I feel that initially I would have chosen 2, which was both boxes.
However, after the professor told me that he had only been wrong ONCE
out of a 1000, the obvious selection for me was to choose just box B
and go for the million.  The reason for this is because no matter what
I choose, he is bound to be right.  So the best bet is to go for the
million, which is box B....
-----
I'm replying to your Newcomb's paradox posting.  I'd be interested to
see the results.

I would choose only box B.  The professor's device should predict this
choice, since I decided on it last fall!
----
I would take both boxes of course, because if the personality
measurement device had correctly predicted what I might do, I would be
10,000 richer.  However, since there is a 1/1000 chance that the
machine is wrong, I might receive $ 1,010,000 for my effort.
Interesting.

-- compiled by Ron Barnette, Philosophy Department

[ Editor: from a 1994 thread on whether a being that always predicts
  the future correctly is paradoxical. ]

Here is the situation that in my opinion shows a paradox in assuming
that an Oracle can always predict the future:

I ask the Oracle the following question about the future:
     Is it true that you will express a negative statement before you
     express an affirmative statement?
Then, no matter whether the Oracle answers affirmatively or negatively,
his prediction would be incorrect.

Of course, this is only intended to show that the Oracle cannot ALWAYS
be correct.  He can still be 100% correct when predicting people's
choices about the 2 boxes.  So, if put in a Newcomb-paradox situation, I
would choose the box B and pocket $1,000,000.  Even if I had believed in
free will before, given a new information (i.e. previous performance of
the Oracle) I would have to re-evaluate that belief.

-- Paul Leyderman <pleydrmn@math.ohio-state.edu>

In article <94045.004946RVESTERM@vma.cc.nd.edu>, <RVESTERM@vma.cc.nd.edu>
(Bob Vesterman) wrote:
>a couple of problems here:
>
>  (1) you don't use the assumption of future prediction at all.

"The oracle expressing a negative statement before expressing a positive
statement" is the event that might or might not happen in the future.
I am asking the Oracle to predict whether or not this event will happen.
This, of course, is clear.

>      this question can be asked of anyone, and will lead to the same
>      conclusion.

Precisely.

>      it says nothing at all about whether the ability to
>      predict the future is paradoxical or not.

It says precisely that the ability to predict the future (for anyone) is
paradoxical because it contradicts the rules of (naive) logic.  Here, by
"the ability to predict the future" I (with David Karr) mean "the
ability to correctly and unambiguously determine whether or not ANY
imaginable event in the future will happen."
Here is an analogy.  The assumption that God is omnipotent (or, that the
Oracle can predict the future) leads to a paradox because God cannot
create the stone so heavy that he himself cannot lift it (or, the Oracle
cannot answer my question correctly).  This, of course, does not imply
that the assumption of God creating the Universe in six days leads to a
paradox (or, that the assumption of the Oracle being always correct
regarding people's decisions about 2 boxes leads to a paradox, as I
already said in the previous post).
>
>  (2) the being can answer "maybe".

This answer is ambiguous, so it is not allowed under my definition of
predicting the future.  If this answer was allowed, then anybody can
predict the future by answering every question with "Maybe", and the
meaning of the word "prediction" is lost.
So, unless you give another reasonable definition of the word
"prediction", the answer to your question is "Yes, there is a paradox."

Now, try to answer my question:
     Where in the world is the city called Joe?

Also, so far I received only 2 answers to Martin Gardner's (less
serious) question about the bull, so I'll post it again:
     A bull had just swallowed a bomb which is about to explode.  What
     one word best describes this situation?
          Same as above, but the bomb had exploded.  What one word best
          describes the situation?

-- Paul Leyderman

[ Editor: no answers to these two side questions appear in the
  collected postings. ]

As I have pointed out before, paradoxes are a formal concept.
To "generate" a paradox involving a perfect oracle it is
"simply" a matter of creating a statement in whatever kind
of temporal logic you like of the form:
        oracle predicts situation X
        AND
        situation X is possible
        AND
        (IF oracle predicts X
        THEN situation X is NOT possible).

In its limited way the "add 1" example is of this form.  Hence
it IS the case that a system can be found in which the
existence of a perfect oracle leads to a paradox.

It is no good claiming "hey -- if we add these extra bits
to your statement then there isn't a paradox".  This doesn't
change the claim above; it's just an attempted straw man.

This doesn't say, of course, that a system can NOT be found
in which no such inconsistency exists; nor does it say
that, within a given system, certain situations are NOT
inconsistent with the existence of an oracle.  But these were,
as far as I can see, not the original question.

Now, if you are trying to maintain that the "real world"
may permit existence of oracles then you will first have
to write down what logical system obtains.

Quite a tall order.

-- Kym Horsell <khorsell@ee.latrobe.edu.au>

ok, "maybe" wasn't a good answer.  i'll try again.

"is it true that you will express a negative statement before you
express a positive statement" does not necessarily lead to a
contradiction.

answer number one: "it is not true that i will not."  this is a
negative statement, and says that the being will first express a
negative statement.

answer number two: "that's a nice try at getting a paradox.  no,
though, it's not true."  here the being first expresses a positive
statement, and says that a positive statement will come first.

you might argue that number two isn't really a prediction of the
future; instead it's a prediction of the past.  but whatever, if you
feel that way, concentrate on the first.

you might argue that number one is a prediction of the present, not the
future.  however, i'd argue right back at you that the answers "yes" or
"no" would be, too (just not CORRECT predictions of the present).

next "paradox"?

-- Bob Vesterman <RVESTERM@vma.cc.nd.edu>

Let me add something to my original question:

[ Editor: Leyderman wrote the addition into his quotation of
  Vesterman's reply: ]

>"is it true that, WHEN ANSWERING MY QUESTION, you will express a
>negative statement before you express a positive statement" does not
>necessarily lead to a contradiction.

This eliminates your 2-d answer, since "that's a nice try at getting a
paradox" is not the part of the answer to the question, so it can be
ignored.

The essence of the question is this:
     Here's a simple yes-no question, so answer it only with "yes" or
     "no".  Will your answer to this question be "no"?
You would argue (wouldn't you?) that the Oracle is not required to
answer every yes-no question with either "yes" or "no", that's why I had
to make the question more complicated.

Before asking my original question I should have explained what I mean
by "negative" and "affirmative".  I'm talking about meaning, not about
syntax.  By a negative answer I mean an answer whose meaning can be best
expressed by "no" or "it will not"; the meaning of an affirmative answer
is best expressed by "yes" or "it will".  In this context, the answer
"it is not true that i will not" is, in fact, an affirmative answer,
since its best analogue is "yes, I will".  So, your first answer still
leads to a paradox.

-- Paul Leyderman

Bob Vesterman (RVESTERM@vma.cc.nd.edu) wrote
> the problem ASSUMES that the being can foretell the future.  it does
> not assume that you have free will, no matter how much you want to
> think you do.

[ Editor: and Ben (of stsci.edu) replied: ]
>I just don't see that, not at all.  A truly omniscient being would
>(by hypothesis) know exactly what you are going to do *of your own
>free will*.  I mean, it's possible for me to know what you did
>yesterday of your own free will; does that mean that yesterday your
>will wasn't free?

A truly omniscient being cannot have any free will himself!  If he knows
everything about the future, he also knows everything about his own
future.  Then he has to act in accordance to what he already knows about
his actions, otherwise he didn't make the right prediction to start with
and he is not truly omniscient.  The being knows the "script" which
describes his entire life, and, in order to be omniscient, he has to
follow that script which would deny him any kind of free will.

-- Paul Leyderman

    I'd like to make a few observations about paradoxes and future
seeing oracles.  This is kind of rough, and may not be very
enlightening, but I had fun writing it, and I predict you will enjoy
reading it. :-)

    An oracle which doesn't tell anybody what it sees in the future
(or give some sort of indication) is no different from someone who
can't see the future and says nothing about what he doesn't see.
An oracle which selectively tells the future is no different, it can
avoid all paradox by only answering those questions which do not
lead to paradox.  The only interesting case I see is an oracle which
truthfully answers all questions put to it.

    I know of two "classic" time travel paradoxes:
    1.) You do the opposite of what is predicted.
    2.) Information can appear from nowhere.
    There were several attempts to apply the "do the opposite"
paradox, which led to a whole discussion about free will, and someone
brought up the billiard ball problem where knowledge of the future
changed the question asked!  I imagine the following scenario:
    You are at the base of a massive staircase leading to the oracle's
inner chamber.  You plan to confound him in paradox by asking "Are you
about to say no?"  Half way to the oracle you see a plaque which reads
"The oracle does not have to answer yes or no questions 'yes' or 'no'"
You pause.  You decide to ask a different question.  You will ask,
"What number will I say next" and then say a number 1 higher than what
the oracle says.  Half way to the oracle you see a plaque which reads
"What is aleph null plus one?"  You pause.  You decide to avoid
questions about aleph null's numerical status and instead you will now
say "1" if the oracle answers "2", or say "2" if the oracle says any
other number.  Resolved, you continue on.  Half way to the oracle you
see another plaque which reads "Wouldn't you like some Ice cream?"
You pause.  Ice cream was what you were just thinking about, and yes
you do want some, but wait!  The oracle is trying to prevent you from
asking your question.  That means that the oracle must not be able to
answer your question!  Filled with determination (and ignoring all the
remaining plaques) you burst into the oracle's chamber.  Before you
can ask your question, the oracle calmly informs you that everyone who
asked him a question they thought would lead to paradox died before
finishing their question.  You pause.  You are determined, the threat
of death is not enough to stop you.  Finally you ask "What number will
I say next?"  The oracle seems to ponder your question for a while then
says "Are you sure you wouldn't rather have some ice cream?"  "No, just
answer the question!" you scream.  The oracle looks tired and says "you
will not have an opportunity to say any number at all, as you are about
to die from gun shot wounds."  Several machine guns open fire.  "I knew
this would happen" sighs the oracle.
    Somehow I think the oracle of this story defeats the intent of an
oracle which truthfully answers all questions put to it, even if he
does avoid the "do the opposite" paradox.

    What about the information from nowhere problem?  I.e. suppose I was
writing a story, and couldn't think of an ending.  So I go to the oracle
and ask "How will my story end?"  The oracle responds "your story will
end with the oracle sighing 'I knew this would happen.'"  Of course, I
think, why didn't I think of that?, and proceed to write the ending in
exactly that way.
    This one is as easy to avoid as the last, just prevent anyone from
asking these tough questions!  "You'll never finish your story, as you
will die from . . ."  (That ice cream sure sounds good to me, maybe I
will have some after all.)

-- Scott Nelson <snelson@asd.sgi.com>

[ Editor: rec.puzzles discussion of Newcomb's Problem, 1993-1998.  The
early postings respond to Chris Cole's nomination of Newcomb's Problem
as the hardest puzzle of all time, and quote an earlier version of the
solution above, which called the expected-gain argument fallacious and
said that specifying that P(predict X | do X) is high would be
equivalent to specifying that the being could use magic (or reverse
causality) to fill the boxes. ]

In article <1993Dec1.051906.18927@questrel.com> chris@questrel.com (Chris
Cole) writes:
> While you are given that P(do X | predict X) is high, it is not given
> that P(predict X | do X) is high.

We are not even given that P(do X | predict X) is high!  We could have
P(do X | predict X) = 0, if the being has never made a prediction about
your actions (check this out with the problem statement).  It's trivial to
change the problem statement to fix this, but it's reasonable to
expect "the hardest problem of all time" to be correctly stated.

Assume that in fact the being has made many predictions about your
actions, and they have always been right.  Although it is true that
we do not know that P(predict X | do X) is high, there is another
relevant probability that is very high.  Suppose for simplicity that
all the being's predictions are - like this one - predictions of your
choice of just two options, X or the opposite of X.  What we have observed
is that

P( predict X | do X AND (predict X OR predict NOT X))  = 100%.

This is relevant as we know the being has either predicted X or NOT X
(picking option (1) or (2)), and in the past, every time the being
predicted either X or NOT X, and we did X, then - darn it - it had
predicted X!

Of course, this does not solve the problem, as we do not know what the
true value of this probability is.  For example, maybe all those
predictions were easy: "You will take a breath within the next two weeks."

> Indeed, specifying that P(predict X | do X)
> is high would be equivalent to specifying that the being could
> use magic (or reverse causality) to fill the boxes.

Nonsense!  Statements about conditional probabilities need have no
connection with causality at all.  Apart from this basic probabilistic
howler, the argument is irrelevant as nowhere are we given that the
being cannot use magic or reverse causality.  Even without magic,
if the being just knows you very well, that probability will be high.
It says to itself "Aha, you've read Chris's solution on rec.puzzles,
so I predict you'll take both, so I'll leave box B empty!"

> Therefore, the
> expected gain from either action cannot be determined from the
> information given.

This at least is true, and therefore, I fail to see how the problem
is "possible to solve".

-- Ian Gent <ipg@aisb.ed.ac.uk>

The problem lies with the predictor. How is it doing its work? Magic?
Detailed analysis of your brain? It seems that the idea of an entity
that can predict something faster than letting the outcome come out in
the normal course of events is impossible. (How is it supposed to
gather and process all this information? How does it represent itself
and predictions about its future behaviour?) Accordingly, if I were to
actually find myself in a Newcomb-style situation I'd have to conclude
that the predictor did not in fact exist and evidence that it did was
false for whatever reason. Forced to make a choice, I would probably
take both boxes and my $1000.

The reason this problem appears so intractable is because it is not
well-posed.  People struggle with it because they do not realize that
one of their initial assumptions leads to paradox.

-- Tony Lezard <tony@mantis.co.uk>

How can this be difficult? The being *put* the money in the boxes, that is
not 'puts' or 'is about to put'. Ie. when I am about to make my choice,
it has already made its prediction, and behaved accordingly.

Say P(1) is the probability it predicts 1, and P(2) the probability that
it predicts 2, given the entire previous history of the world at the
time of prediction.
Both P(1) and P(2) is >=0, and P(1) + P(2) = 1

Now my estimated win E(x) from action x is
E(1) = P(1) * 1000000
E(2) = P(1) * 1000000 + 1000

Thus E(2) is always better, though when P(1) is large, only marginally
better.

The only way you could influence P(1) is by trying to convince the
being that you would choose 1 if you were given such a choice, this
could include deciding beforehand to accept option 1.
It would still be advantageous to change to option 2 when you were
presented with your choice though.

The best strategy in a repeated experiment is still open for discussion,
I think, but this would rely too heavily on the being's psychology for
us to determine anything for certain.

Note to beings: This discussion is purely hypothetical, and does not
        apply to *real* situations, in which I would *always* choose
        option 1, I would! I would!

-- Stein Kulseth <stein.kulseth@nta.no>

As a matter of fact, I heard this problem in a lecture by Robert
Nozick about 12 years ago.  (Although if he invented it, why is it
called "Newcomb's Problem"?)

My impression was that this is very, *very* much a philosophical
problem, having to do with free will among other things.  For one thing,
the people who advise you to take both boxes have a particular concept
of free will that insists that once the alien has decided what to put
in box B, you are not constrained by any past events to open box A or
not to open box A.

As for the quality of the alien's predictions, my recollection from
the lecture is that you should assume a history that convinces you
that P(predict(X) | do(X) AND (predict(X) OR predict(NOT X))) is
indeed very high, for example several thousand people have tried
this same game against the alien before, and each option was tried
by a large percentage of them, and the alien has never been wrong.

Now if the "your choice is independent" crowd is right, then the fact
that in all the a few thousand people declined to open box A and
walked away millionaires, while thousands of other people looked in
box A and came away with only a thousand dollars, is just a
demonstration that stupidity has been rewarded.  If this is correct,
I would take my chance at becoming a stupid millionaire.

To put this another way, the problem setup is that everyone who
believed the "open both boxes" argument got $1000, while everyone who
was "duped" by some other argument to open only one box got a million
dollars.  So sometimes it pays to be a dupe.

Just to put the monetary prizes into proper perspective, I personally
know someone who suddenly became the recipient of an unexpected
million dollars.  It changed his life forever, and when all is
balanced was a great benefit.  Similarly, most people, including
myself, would be elevated several notches in financial security and
freedom by a million dollars.  In contrast, a thousand dollars would
not erase my current financial worries for even a moment, and I think
most people would soon have to worry where the thousand *after* that
was going to come from.

I'm curious as to what Nozick's current approach is to this question.

-- David A. Karr <karr@cs.cornell.edu>

Here's something about this paradox that I've thought of and which may be
original.

Consider the following setup: let S be the set of Turing machines which
halt on every input and always output zero or one.  (The Turing machine
is supposed to simulate the person making the choice, the input is the
process of explaining to the person what the game is, and the output is
the person's choice.)  Fix a string I.  (This is supposed to represent
the standard instructions that are given to every person.)  Let a
"perfect predictor" be something that, given any Turing machine M in S,
can predict the output of M when M is fed I as input.  Clearly, such a
perfect predictor exists, and is given by some Turing machine---all it
has to do is simulate M.  (It won't necessarily halt on every input, but
it will halt if fed any machine in S.)  This seems to suggest that free
will or something like it plays an essential role in the Newcomb
[Editor: spelled "Newcombe" here and below in the original] paradox
because no paradox arises with simple deterministic machines in place of
human beings.  For although all these Turing machines "should" take both
boxes, they have no ability to act as they "should."

So now let's see what happens if we try to put some simulation of free
will into the picture.  For example, take Chris Cole's benevolent friend
who can see what's in the boxes.  How can we factor this into the picture?
My suggestion: what this amounts to is allowing the Turing machine to take
the output of the perfect predictor as part of its input.  All the ways of
generating the Newcomb paradox seem to rely on the idea that the boxes
have already been filled---the money is already there, and it will not
vanish mysteriously in the process of your opening the box or anything like
that.  It seems to me that this is akin to having the perfect predictor
complete its run ahead of time and for the result to be available, sitting
around, and not subject to change.

We can already begin to see the outlines of a paradox now.  Instead of a
fixed input I, we have an input that depends on the output of the perfect
predictor.  We now want to know if there exists a Turing machine P which
satisfies the following conditions:

1. It always halts when given any M in S as input.
2. For all M in S, the output O(M) of P when given M as input is the same
   as the output of M when given O(M) as input.  (In other words, P must
   be able to predict what M will do even if M is allowed to act based on
   P's prediction.)

But if P exists then for each M in S there exists some input O(M) such
that M will output O(M) when fed O(M).  This is absurd, since the
machine which simply reverses its input lies in S, so P does not
exist.  (Alternative descriptions of P are possible, but in each case a
similar argument will show that no such thing exists.)  Note that this
argument shows something stronger---that even if we allow P to be a
more powerful machine than a Turing machine the same difficulty arises,
because at no stage in the argument do we need to construct an M in S
containing an embedded copy of P, as is sometimes the case with these
arguments.

What this seems to indicate is that it does not make sense to assume both
that an entity's behavior can be predicted perfectly *and* that the entity
can use predictions of its own behavior to alter its behavior.  (For
example, to take another well-known paradox---can God tell you which arm
you are going to raise ten seconds from now?)  The Newcomb paradox then
seems to derive its force from our intuition that both assumptions are
true in the case of human behavior.

There are different ways to break the paradox: one is to attack my argument
above and show that both assumptions are in fact compatible; another is to
claim that human behavior cannot be predicted perfectly; another is to say
that human behavior can be predicted perfectly but there is some reason that
such predictions cannot be made available to us (incidentally, this is often
the way Christian theologians who believe in predestination try to motivate
evangelistic fervor---some are destined to be saved but you don't know who);
finally one can deny that we have the power to act opposite to what we have
been predicted to do, even if the prediction is made available to us (this
is the basis of many amusing and somewhat eerie stories where someone is
told about something that will happen in the future and although they do
everything they can to forestall the event they inevitably fail).

-- Timothy Y. Chow <tycchow@math.mit.edu>

[ Editor: David Karr, in mail to Chris Cole and Stein Kulseth: ]

I think I've managed to give quite a number of different sets of premises
in Newcomb, and as a result they seem to have gotten a bit mixed together.
So Chris's statement that I believe the alien is cheating is only
partially correct (in the sense that I merely suspect that this
*might* be the case).

There are a number of ways the alien could develop a track record in
predicting human behavior without error, and I don't know for sure
which one it is using.  A few of the possibilities follow:

(1) Dumb luck, perhaps assisted by a good understanding of human nature
    and observation of the subjects.

(2) Determinism and advanced technology; the alien can somehow detect
    human mental states precisely enough to extrapolate them into the
    future with great accuracy.

(3) Sleight of hand, again perhaps aided by advanced technology (my
    example was a transporter to "beam" the money into box B as
    needed).

(4) Mind control; if the alien makes a mistake, it simply alters our
    memories so we believe it made the correct prediction.

(5) Anti-temporal causality (not "non-causality" as Stein terms it,
    but causal chains that don't always run forward in real time).
    This could be accomplished by a limited form of time-travel,
    for example, that somehow avoided the Grandfather Paradox.

Now my personal view of free will is such that (2) is a possibility
but not the most likely explanation---not impossible, just
phenomenally difficult.  I often can't make up my mind between one
thing and another, or may change my mind repeatedly before coming to a
decision.  So it seems to me (relying on the rather unscientific
method of self-analysis) that human thought processes are often highly
chaotic, in particular when two very attractive options are offered
and there is no clear rational basis for choosing between them.  And
it seems to me that there are very attractive arguments on both sides
if you believe (2).

In fact my guess is that any being that could do (2) could more easily
do (4), in analogy to the fact that it is often much easier to make it
rain than to predict whether it will rain, given that there are clouds
in the sky (and that you have all the necessary equipment at hand).

Similarly, the mechanism to accomplish (2) is magical enough (I think
this is Clarke's Law: any sufficiently advanced technology is magic to
those who don't have it) that the being could just as easily, perhaps
more easily, cheat by using some magic of type (3) (e.g., using a
matter transporter on the million dollars, or perhaps something a bit
less fantastic).

Possibility (5) (strange causality) I threw in just as a "what if," to
point out that we can't be absolutely certain this is *not* what is
going on.

Now as to the answers to the problem:

If you believe (1) (luck) then I think the answer is to take both
boxes.  That's why I posed Newcomb-Karr, where you get to play the
game 1000+9000 times for low stakes: the premise that the alien
correctly guesses the first 1000 binary choices by a human who is
trying not to have a pattern, is somewhere in the realm of 1/googol.
(Actually, guessing 1000 truly random choices would have probability
2^{-1000} which is about 10^{-300}, but humans are not perfectly
random so I knocked off "a few" orders of magnitude.)  Such an event
is highly unlikely to occur in the entire history of the universe, so
I wouldn't believe it in Newcomb-Karr.

If you have rejected (1) (due to the amazing true predictions of the
alien) but believe (2), then the puzzle is not entirely meaningless,
because the word "should" still has some good use in such a world.
But the puzzle statement becomes unfortunately a bit self-referential,
because the subject's reasoning process might proceed somewhat as
follows:

   Suppose my reasoning process leads me to choose B only.  Then (with
   high likelihood) the alien knows this, and has put the million
   dollars in B.  But since I know this, simple logic for evaluating
   choices leads me to pick A+B.  But again (with high probability)
   the alien knows this, so it will place nothing in box B.  By
   logical deduction, then, (with high probability) there is only
   $1000 in the boxes and that is the most I can get, which I obtain
   by choosing A+B.

In other words, this is a self-referential argument for rejecting B
and choosing A+B; but in Newcomb-Karr it is contradicted by the
empirical evidence that one can very frequently collect the prize from
box B.

Usually we aren't faced with such problems of self-referentiality, and
we use the question "what should I do?" to develop techniques for
optimizing our interaction with the world.  We can do this even if we
are totally deterministic creatures, because then we still have the
(deterministic) abilities of learning from experience, of reasoning
from beliefs, and of self-reflection.  So we take a set of
circumstances that are not dependent (as far as we can tell) on our
own reasoning about them, and apply that reasoning to extrapolate
the results of different actions we might take.  Empirically, this
works quite well in real life.  But real life has never yet
included a Newcomb situation.

If we interpret "should" as specifying which deterministic thought
process brings us the most benefit, and if the alien really is
infallible, then the thought process that leads to box B is the
one we "should" take.

If the alien's predictions are not quite perfect, it still might be
that the expected value of choice B is higher than that of choice A+B,
so we "should" choose B in that sense; on the other hand, in a
non-probabilistic game-theoretic sense I think choice A+B is favored
because it maximizes both the minimum and maximum outcomes.
I just happen to prefer a probabilistic analysis in this case.

If our thought processes are deterministic enough for the alien to
make correct predictions almost all the time, but there is a small
element of non-determinism that could cause us to have a sudden change
of mind at any time, then obviously the best outcome is for us to
enter into a deterministic process leading to choice B, and then
*after* the alien places the money, leap non-deterministically to
choice A+B.  The drawback to this plan is that I do not know how to
make myself perform such a leap, even if it is possible.

Another interpretation of the problem, which corresponds to the
non-deterministic free will position, is that the alien makes its
prediction based on the mental state of someone, but at the very last
second before the choice is to be made, you are allowed to take that
person's place.  Obviously you are not bound by the thought processes
of the person who came before you, there is no meaningful causal
relation (even common causality by a third event) to link your choice
with the alien's, and you should take both boxes, getting $1000 more
than you could have gotten from box B alone.  I think this is roughly
Stein's argument; non-causal free will allows you to perform a mental
reset and complete do-over after the alien makes its prediction, which
is as good for you as if you got to stand in for an entirely different
person.

The only reason I haven't tried to develop the last line of reasoning
further (or to refine away its rough spots) is that it doesn't seem to
agree too well with any scenario that allows me to reject possibility
(1) (luck).  I have been interpreting Newcomb to require enough
evidence to reject (1) in favor of some other possibility, since (1)
leads rather trivially to the conclusion that you should take both
boxes---essentially, it asserts that the fact that "the being has
never been wrong before" is irrelevant to your analysis, as someone
just claimed in rec.puzzles.

Of course, if you believe (3), (4), or (5), then the obvious
choice is box B since empirically that works out much better for you.

I think the original purpose of Newcomb's Problem is to illustrate the
difficulties of position (2) (determinism).  I think part of the
apparent paradox stems from the fact that our heuristic use of the
word "should" is to invoke a perfectly rational agent who comes into
existence at the moment a decision is to be made, and who can make the
decision based on all available information but unconstrained by any
past events.  Unfortunately we don't know how to invoke such agents in
real life; all we know about is our own imperfect thought processes,
and we make a guess about what a perfect agent would do.  That's where
the contradiction comes in, I think: classical theory gives the answer
A+B, but the mechanism used by this theory (rational agents) is
rejected by the assumptions of the problem (mechanistic predictability
of human action).

In the case of Newcomb-Karr, where you have the experience of playing
1000 times already with the alien, Chris correctly uses classical
analysis to conclude that you should choose A+B in the remaining 9000
games (even if you were foolish enough to choose the wrong choice in
400 of the first 1000 games, thereby collecting nearly $40,000 more
than you did in all 600 correct choices).  As a practical matter in
such an unlikely situation, I personally would "learn" to take box B
only and happily collect my hundred-dollar bills as long as they kept
coming, particularly after all 600 attempts to gain an extra ten cents
resulted in much worse outcomes.  You might say I'm relying purely on
inductive reasoning from the prior cases of this one phenomenon,
throwing all other reasoning to the wind.

-- David A. Karr <karr@cs.cornell.edu>

In article <1993Dec4.220235.29659@questrel.com> chris@questrel.com (Chris
Cole) writes:
> Since Dave has now presented a VERY GOOD ARGUMENT for taking one box,
> I feel duty bound to present a VERY GOOD ARGUMENT for taking two boxes.
>
> Suppose the back of the boxes are transparent, so that you cannot see into
> the boxes but someone else can.  In fact, a friend of yours (someone who
> you trust and who you know to have only your best interests at heart) is
> at this very moment looking into the boxes.  You know he or she can see
> whether there is money in the boxes.  The rules of the game do not allow
> you to ask him or her about what is in the boxes, so instead you try to
> imagine what he or she is thinking.  If there is $1,000,000 in the first
> box, he or she will of course want you to open that box.  Since there is
> $1000 in the second box, he or she will also want you to open that box.
> In either case, whether the $1000000 is there or not, he or she will be
> mentally pleading with you to open both boxes.
>
> Now, your friend -- who knows what is in the boxes, who has your best
> interests at heart, who wants you to get the most money you can, and
> who is right now actually looking at the money -- this best friend is
> mentally pleading with you to open both boxes.  What are you going to
> do?

You mean like the very good friend of the prisoner who already had heard
the response of the other prisoner?

-- J. A. Durieux <xerox@cs.vu.nl>

In article <1993Dec8.173535.13336@cs.cornell.edu> karr@cs.cornell.edu
(David Karr) writes:
> In article <1993Dec8.154932.13015@nynexst.com> gene@nynnexst.com (Gene
> Miller) writes:
>> Probability arguments just do not apply here. There is no reason to
>> assume that all previous and future game instances are random
>> observations from the same universe.
>
> If this logic applies to Newcomb then it applies to Monty Hall as well,
> I think.  Which it doesn't.
[...]
> The perfect chooser would begin in a state of mind that led clearly,
> directly, and above all very predictably to the coward's choice of box
> B only.  But at the last instant (after the predictor placed the money
> but before the chooser made the choice), the chooser would suddenly
> make a completely unpredictable leap of thought, and would choose both
> boxes.  If you know how to do all this, let me know.

Many people will conclude from David's excellent arguments that the
expected value of choosing one box is higher than the expected value of
choosing two boxes.  This would lead to a crisis in decision theory,
since it would be the only known case where the principle of dominance
contradicts the principle of maximizing expected gain.  The principle
of dominance states that between two choices, if under all possible
states of nature the outcome of choice A is better than the outcome of
choice B (choice A "dominates" choice B), then choose A.  In the
Newcomb problem, the principle of dominance recommends choosing both
boxes, since whatever is in the boxes, you get more by taking both than
by taking one.

My purpose here is to demonstrate that there is no conflict between the
principle of dominance and the principle of maximizing expected gain.
This is because there is not enough information to calculate the
expected gain in Newcomb's Problem.  The rec.puzzles archive entry
demonstrates this using a formula; experience dictates that many people
suffer MEGO ("My Eyes Glaze Over") when presented with formulas, so I
will use the old standby: enumeration of cases.

I follow here Isaac Levi, "Newcomb's Many Problems," _Theory and Decision_,
6 (1975), 161-175.  Let's consider three possible histories:

Case 1:
Predicts choice of one box     Predicts choice of both boxes
Chooses one box    900,000                                10
Chooses both boxes 100,000                                90

Case 2:
Predicts choice of one box     Predicts choice of both boxes
Chooses one box    495,045                            55,005
Chooses both boxes  55,005                           495,005

Case 3:
Predicts choice of one box     Predicts choice of both boxes
Chooses one box         90                           100,000
Chooses both boxes      10                           900,000

Let "A1" signify "pick one box," "A2" signify "pick both boxes," "H1"
signify "predict pick one box," and "H2" signify "predict pick both
boxes."  Assuming for the sake of argument that probabilities can be
assigned based on the relative frequencies presented in these tables,
we have:

        P(A1|H1)  P(A2|H2)  P(H1|A1)  P(H2|A2)  P(H2|A1)  P(H1|A2)
Case 1:       .9        .9  .9999888  .0008991  .0000111  .9991008
Case 2:       .9        .9  .9000000  .9000000  .1000000  .1000000
Case 3:       .9        .9  .0008991  .9999888  .9991008  .0000111

The accuracy of the demon is, since P(H1) = 1 - P(H2):

P(A1 & H1 or A2 & H2) = P(A1|H1) * P(H1) + P(A2|H2) * P(H2) = .9

and is high in all cases, as it should be.  The expected gains are
given by the formulas:

E(A1) = $1,000,000 * P(H1|A1) + $0 * P(H2|A1)
E(A2) = $1,001,000 * P(H1|A2) + $1,000 * P(H2|A2)

The expected gains, by case:

            E(A1)          E(A2)
Case 1:   $999,988.8    $1,000,100.7
Case 2:   $900,000.0      $101,000.0
Case 3:       $899.1        $1,011.1

In cases 1 and 3, the expected gain is greater when two boxes are picked.
In case 2, the expected gain is greater when one box is picked.  Now,
the crux of the difficulty is that we do not know which of the three
cases obtains.  Therefore, we cannot use expected gain to decide what
to do.

David has presented an iterated version of the Newcomb Problem.  It is
not legitimate to argue that in the Newcomb-Karr Problem, we know the
probabilities and they are like case 2.  The Newcomb-Karr Problem is not
all that different (except for rhetorical effect) from the original
Newcomb Problem.  While we are given relative frequencies in both the
original and the -Karr version, we are not given probabilities.  In
order to deduce probabilities from relative frequencies, you have to
know something about the experimental set up.  In Bayesian terms, you
have to know the "prior probabilities."  We are not given these here.

In correspondence, David has stated that he would be willing to assume
that the probabilities are given by the relative frequencies.  I would
not.  If faced with making that assumption versus going with the
dominance principle, I would let dominance be my guide.  But I admit
that this choice may seem arbitrary to others; I also admit that I was
a one-boxer after first hearing about Newcomb's Problem.  So, I do not
claim that I am right and David is wrong.  Remember how this discussion
got started: I claim that Newcomb's Problem is The World's Hardest
Puzzle, and is unsolved at present.

-- Chris Cole <chris@questrel.com>

[ Editor: a correspondent, not named in the collected material, replied
to the argument above; Chris Cole's answer follows. ]

> My argument is this.
>
> I quite agree that we cannot know what the prior probabilities are when
> it comes to our particular run up against the being.  Nevertheless,
> these conditional probabilities do exist.  And since they exist,
> if we knew the probabilities, our best course of action would be given
> by the expectation calculation.  Obviously, if the conditional
> probabilities have certain values, we should take one box, while if they
> have other values we should take both.  The problem remains that we
> do not know these values.  Now, for the sake of argument, let's assume
> we are in case 2, and a large amount of past history suggests that
> the probabilities point to taking one box.  Well, as you quite correctly
> say, we cannot prove that this is the correct answer.  To say "pick
> one box" is to say that we believe that the probabilities are in
> a certain range, and this is unjustified.  However, it is equally true
> that to say "pick both boxes" is to say we believe the probabilities
> are in another certain range.  We have no justification for making
> either assumption.  In other words, even in agreeing with you that
> we cannot know the probabilities, I believe that the dominance argument
> is still irrelevant, because to say "pick both boxes" is not based
> on the dominance argument, but on the implicit belief that the
> probabilities have certain values.
>
> To summarise.  In any given situation, there _is_ a best thing to do to
> maximise our expectation, which is different on different occasions, but
> we may be (indeed probably will be) _unable_ to know which it is.  I see
> no reason for deducing from that fact that we should always pick both
> boxes.
>
> I think the fundamental problem is that we are stretching common sense
> too far - probably why philosophers like the problem!  In any everyday
> situation, we have two common sense arguments which point both ways in
> this case.  One is: it's reasonable to extrapolate probabilities from
> observations on many thousands of apparently identical cases.  In
> certain situations, this would lead to picking just one box.  The other
> is: nobody can be a (nearly) infallible predictor of human beings.
> This suggests that we try to fool the being that we will pick one box,
> then pick both, because in any real situation, we could probably do
> this well enough to weight the probabilities in our favour.  Both
> arguments are good in general for our ordinary lives, but they break
> down in this case precisely because the problem has set up an entirely
> extra-ordinary situation.

These three paragraphs contain a very clear presentation of a position,
which I HOPE is your central position, because I at least understand
it.  It also explains why we have been miscommunicating previously
because I failed to understand that this was your argument.  This is,
I suspect, entirely my fault, for reasons I will get into below.

In the Newcomb problem, there are two opposing arguments: dominance and
expected value.  Dominance seems to argue for taking both boxes;
expected value seems to argue for taking one.  Both arguments are
very good.  It is not enough, says Nozick (and I agree), to provide
more good reasons to believe in either argument; you must undermine
the opposing argument AND explain why it seems so good, but is not.

I have tried to do both of these things.  I have shown that the expected
value argument is unsound without the assumption of unknown prior
probabilities (or the assumption of reverse causality, magic,
infallibility, etc.).  There are other examples of this type of
fallacious reasoning in the archive.  Here is one:

[ Editor: Chris Cole quoted the entry probability/apriori in full
here. ]

Having shown that the expected value argument is unsound, I still have
to explain why it so appealing.  Here I think the answer is that
Newcomb's Problem is fiendishly designed to trick our intuition.  The
whole point of the Problem is to try to get us to believe in a kind of
reverse causality.  This reminds me of the question, "what do you
conclude after flipping a fair coin 1000 times and seeing it come up
heads every time?"  The right answer is, "I conclude that a very
unlikely event has occurred."  But it is very hard not to answer, "I
conclude that the coin is unfair," or since this contradicts the
assumptions of the question, "I conclude there must be magic at work."
(Actually, the answer I like best is "I'll worry about it when it
happens.")

OK, now to your argument, and why I didn't get it.  You are saying that
the dominance argument is tantamount to specifying certain prior
probabilities (or ranges, rather).  This is not true.  Dominance
operates when there are no known probabilities, and does not need to
worry about the fine points of the philosophy and interpretation of
probability assignments.  Dominance says, simply, if one choice is
always better than or equal to the other, take it.  You can use
dominance when you have no idea of the probabilities (objectivist
language) or when the probabilities do not exist (subjectivist
language).

Your argument assumes that all decisions are based on probabilities.
If I ask you which would you rather have, $1 or $10, you do not need
to invoke probabilities to make the decision.  Dominance suffices.

Now, it is a trivial theorem of decision theory that if outcome A
dominates outcome B, the expected value of A must be greater than or
equal to the expected value of B.  Newcomb's Problem seems to violate
this theorem.  This would be a real crisis in decision theory.  It
would entail major foundational rethinking along the lines of the
rethinking Tarski did about truth theory when presented with Richard's
[Editor: "Richards'" in the original] Paradox.  But, fortunately, the
crisis abates when it is realized that the expected value cannot be
computed.

The reason I was having trouble understanding your position, I think,
is that it was new to me.  The position that all decisions are implicit
probability assumptions is not one with which I am familiar, so I did
not recognize your arguments.  When someone has argued Newcomb's
Problem as many times as I have, it is easy to fall into the habit of
quickly assuming you understand someone's position from the first few
things they say.  I had trouble placing you, and my replies probably
seemed strange.  I apologize.

-- Chris Cole <chris@questrel.com>

The difficulty with Newcomb's Paradox is that it's a decision problem (DP)
which doesn't have all the preconditions we expect from a DP.  Let's say
your friend Bill calls you up on the telephone and tells you that he has
to know how to pick the light or heavy coin out of a dozen coins using
a balance and the minimum number of weighings---what should he do?  You
think it over for a while and then tell Bill what to do.  You will probably
assume, without thinking about it, that how you analyze the problem won't
affect Bill; only the final answer you give Bill will matter to him, and
even here, only direct effects of what you tell Bill will be relevant.

A DP is an idealized form of this situation.  The separation between
Bill and ourselves, his counselors, is even more absolute (especially
since Bill is imaginary and we are real) so the assumption that our
analysis process cannot affect Bill is taken for granted.  However, in
Newcomb's paradox, this is no longer true.  If Bill calls us up and says
``I'm in this room with an Omniscient Being and two boxes...'' then the
OB must have predicted Bill's phone call, and our own thoughts
afterwards---as well as the postings to rec.puzzles, if we take the
question to Usenet.  This is a violation of our expectations.  The
temptation is to deny that the OB can really do this, which leads to
telling Bill he can take both boxes.

Another way of thinking about a DP involving Bill is to say ``Suppose
Bill were perfectly rational; what would he do then?''  We may suppose
that Bill has a black box, the Perfectly Rational Decision Engine (PRDE)
embedded within him, which he consults in all situations.  But now
the contents of the PRDE will affect what the Omniscient Being
will do, which is again an unexpected case of feedback, since the
outcome of our debate on Usenet might decide what circuitry should
be in the PRDE.

In a variant of Newcomb's Paradox, it's even clearer that these
implicit preconditions for a DP are not met.  An analysis of Fred's
family history reveals that, when a certain genetic defect is present,
Fred's ancestors almost always become flutists and die young of
a horribly painful disease, and when the genetic defect is not
present, Fred's ancestors almost always do not become flutists,
and live a normal lifespan.  (This variant is due to R. Nozick.)
Fred does not know whether he has the defect or not.  In this case,
if Fred calls you up on the phone, wanting to know whether to take
up the flute or not, the problem conditions imply that what you
say to him will make no difference.  You might as well say
nothing.  And if Fred and his ancestors had PRDEs, this strange
correlation between being a flutist and dying young would
presumably never have arisen.

BTW, if I were in Bill's situation, I would take only box B,
for the million.  But this isn't `rational' in the sense alluded
to above.

-- David Moews <moews@math.berkeley.edu>

The FAQ contains a solution for Newcomb's Problem. This is surprising,
since there is no generally accepted solution to the problem.

I won't explain the problem, since it's in the FAQ. But here is something
the FAQ says.

****
The following argument might be made: 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. However, this
argument is fallacious. In order to compute the expected gain, one would
use the formulas:
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)
While you are given that P(do X | predict X) is high, it is not given that
P(predict X | do X) is high. Indeed, specifying that P(predict X | do X)
is high would be equivalent to specifying that the being could use magic
(or reverse causality) to fill the boxes. Therefore, the expected gain
from either action cannot be determined from the information given.
****

But this is incorrect. The main error is that specifying probabilities,
including conditional probabilities like P(predict X | do X), does not
entail anything at all about causal connections. So reverse causality or
magic are not in question at all. For example, see R. Jeffrey _The Logic
of Decision_ for a theory of probability (and decision!) that eschews all
mention of causality.

Another error is this sentence:

     While you are given that P(do X | predict X) is high,
     it is not given that P(predict X | do X) is high.

Well, strictly speaking, as stated in the FAQ you are not given any
probabilities, only frequencies. But if you allow your probability
judgments to conform to frequencies, then you will judge that the
predictor is very likely to be correct. And that does mean that P(predict
X | do X) is high. And in particular, suppose that you think that the
probability that the predictor will predict X is nearly the same as the
probability that you will do X (for each X, of course). Then these two
expressions *must* also be approximately equal: P(do X | predict X) and
P(predict X | do X) (by Bayes' Law). So if one of them is high, then the
other is too.

So the argument said to be fallacious in the FAQ is not fallacious at all.
The expected value of choosing only the one box is much higher than the
expected value of choosing both. Yet most people think it is foolish to
take only one. This is precisely the paradox. Decision theory seems to
tell you to do something stupid. I do not, myself, think that it is stupid
to take one box. That is what I would do. I am happy to follow the
standard advice of decision theory in this situation. Sadly, I doubt that
this will ever help me in real life.

-- Jamie Dreier <pl436000@brownvm.brown.edu>

> >Nozick's most recent book, _The Nature of Rationality_, has his most
> >recent thoughts on the problem. His new view will surprise many who read
> >it.
>
> Aw... Don't tantalize us. Please summarize.
> --
> Dave Ring

Oh, jeez. Well, I'll try. I don't have a copy of the book.

Nozick points out that classical decision theory, which is non-causal (in
the sense of making no reference at all to the concept of causation and
treating it as a bit of metaphysics irrelevant to decision-making) gives a
different answer to the Newcomb problem from a newer version of decision
theory, called simply 'causal decision theory'. The latter instructs you
to disregard probabilistic dependence that doesn't arise from the proper
causal relations. He points out that each kind of theory is perfectly
consistent, and that there are intuitive arguments on both sides without
any decisive victor. He speculates that the most sensible approach is to
take each kind of theory as having some chance of being correct! So, if
the prizes in the Newcomb problem are very disparate, as in the original
problem, and the probability of the predictor being correct is very large,
then if you think that the classical sort of theory has even a small
chance of being right, you should take one box. But if the prizes were
much closer, or the predictor's success only a small amount larger than
random, then you should take two boxes if you think the causal version of
the theory has a pretty good chance of being right.

Personally, I find this sort of odd. It is hard for me to think of one
theory having some *chance* of being right. It isn't exactly an empirical
speculation, is it? It's more like thinking that Goldbach's conjecture has
a high chance of being true.

Still, it's an interesting twist.

-- Jamie Dreier <pl436000@brownvm.brown.edu>

Jamie Dreier says:
> This depends on how the counterfactual is evaluated.  If you had taken
> just the first box, *holding constant what the predictor predicted*, then
> you would have gotten nothing. But the counterfactual can also be
> evaluated thus: if you had taken just the first box, what would the
> predictor have predicted?

This must be close to the heart of the problem. Can my actual choice
influence the prediction or not? I for one do not find the problem very
interesting if we allow this (it then boils down to estimating the
probabilities for this, and weighing these against the extra $1,000
value).

So I like to assume:
  1) My choice is made *after* the prediction is made and the money placed.
  2) No 'reverse causality', time travel or sleight-of-hand trickery is
     used to fit the prediction to my actual choice
  3) I have at least some tiny amount of free will (otherwise definitely
     not interesting :-)

With the above assumptions it is obvious (to me at least :) that the
best *choice* is to take both boxes ($1000 more for you), yet ironically
people who don't understand this are better off than the people who do.

Here is a Newcomb variant to think about for people who still feel that
with the above assumptions, one box is still the best choice:

  "The Newcomb Being" is generously handing out money. Based on whether it
  predicts people will say "yes" or "no" to accepting $1,000 it first hands
  out $0 or $1,000,000 respectively. Then people have the option of
  accepting the $1,000. Lucky you get $1,000,000, then do you want another
  $1,000?

  ...just before a loud-and-clear "yes" emerges you realize that of all
  the many people you have seen received money, none of the $1,000,000
  receivers have accepted the other $1,000, whereas all the $0 receivers
  naturally did, and you take a little time to ponder this. Now do you
  want another $1,000?

-- Stein Kulseth <stein.kulseth@nta.no>

David Karr (karr@cs.cornell.edu) wrote:
> The thing is, Newcomb's is not an ordinary physics question, it's a
> philosophical paradox.  I don't believe I will ever end up in such a
> situation; nobody will ever reliably be able to predict my last-minute
> choices at restaurants, for example.  But I have no proof that this is
> absolutely the way things are and must be.  The paradox then asks an
> apparently counter-factual question: what *would* my best choice be if
> there *were* a predictor with such demonstrated ability.  This
> question is not answered by the statement, "There is no such
> predictor."

Actually, it may be answered by the statement, "There is no such thing
as a 'best choice' in this situation."  I don't have any problem
with believing there could be such a predictor.  After all, look
at all these people who have written here and elsewhere vehemently
pledging that they will take the two boxes!!  It's not stretching
the imagination too much to say that you can predict that they will
do as they promised.

In essence, what is happening is this: the predictor has the ability
to classify people into a "one-box" group and a "two-box" group.
Then the predictor decides to reward the "one-box" people by giving
them a million dollars and punish the "two-box" people by giving
them only a thousand dollars.  Unfortunately for us, traits like
mysticism and bad understanding of causality are associated with
being a "one-box" person, while traits like a good understanding
of logic and causality are associated with being a "two-box" person.

Therefore, from the predictor's standpoint, there IS no choice.
The apparent 'choice' is merely a by-product of the victim's
membership in one group or another, which is the product of his or
her background, training, intelligence, and experience.  To talk
about what the victim "SHOULD" do at this point is beside the
point.

To make this clear, let me propose a nasty variation of the Newcomb
problem:

Kimball's Nasty Variation of the Newcomb Problem

This problem starts with the ordinary Prisoners' Dilemma.  Two prisoners
are kept separate; if they both stand firm they will get light
sentences; if they both squeal they both get heavy sentences; if one
squeals and the other does not, the squealer gets off and the
other one gets a -really- heavy sentence.

The prisoners sit down and think about what to do, but are then
told:  IN ADDITION, the king of this land, an absolute tyrant, -hates-
logical thought.  Both prisoners are being carefully watched and
their thought processes can be pretty much accurately inferred.
When both have made their choices, the one who made the -most logical-
choice, taking into account all the conditions of the problem including
this one, will be tortured to death.  So now, go ahead, figure out what
to do :)

Oh, by the way, if it turns out that the most logical thing to do is
to flip a coin, of course doing that will get you tortured to death
as easily as anything else.  Other means of deliberately trying to
be illogical won't help either... the more creative you are, the worse
off you will probably end up.  Unless you thought of that too..

Of course if you can make a strong argument why following strategy X
is a good idea, you are probably just proving that you should get killed
for doing it.  In these conditions, does it even make any sense to
ask what the "best choice" is?  The tyrant hates logicians.  If you
are a good logician, you probably get killed.  Where is the choice?

-- P. Kimball <achepak@interaccess.com>

The Universe, it turns out, is symmetrical. It has a plane of symmetry.
One day, cruising around in your spaceship, you unexpectedly arrive at the
Plane of Symmetry of the Universe. To your surprise (though upon
reflection, so to speak, it isn't at all surprising), so does your
doppelganger. As you approach the plane, it *appears* to be a mirror. But
it isn't a mirror, there is a real (half of the) universe on the other
side, and it is an exact mirror image of our side, in all respects. Your
doppelganger is not a reflection, but a real person just like you.

Cautiously, you move right up to the plane and try to cross. But of
course, you can't, because your doppelganger moves right up too and blocks
you. Not his fault! He thinks you're blocking him. You wave. So does he,
naturally. The laws and initial conditions of physics are just the same on
either side. Whenever something happens on our side, the mirror image must
happen on the other. Your doppelganger is *just* like you. His inner
workings are as yours.

And now, Newcomb's Demon arrives.  Fiendishly, he offers you and your
doppelganger a chance to play Prisoners' Dilemma. Each of you gets a
button, one red, one green. Your red button is next to your right hand,
and of course your dopple's red button is next to his left hand. If you
both press the red buttons, you each get a million dollars. If you press
red and your dopple presses green, you get nothing and your dopple gets
$1,001,000. Likewise, if he presses red and you press green, you get
$1,001,000 and he gets nothing. If you both press green, you each get a
thousand dollars. (The payoffs have the structure of the Prisoners'
Dilemma, with RED being the cooperative move.)

Reasoning as a good two-boxer, you notice that RED dominates GREEN. No
matter what your doppelganger does, you do better pushing the RED button.
And you cannot possibly *cause* your doppelganger to push the RED button,
of course. But you also realize that the conditional probability that he
will push RED given that you do is very high--extremely close to one, you
reckon. And also the probability that he will push GREEN given that you
do. So the expected value of pushing RED is very much higher than the
expected value of pushing GREEN.

[ Editor: by the payoffs above GREEN dominates RED ($1,001,000 vs.
$1,000,000 if he presses RED, $1,000 vs. $0 if he presses GREEN), as
"play the dominant strategy ... You reach toward the GREEN button"
below assumes. ]

But, you recall, following the dictates of expected value and decision
theory is a sucker's game in circumstances like this one. You have been
assured by paragons of rationality on rec.puzzles that if you cannot
*raise* the probability of the favorable event, then the fact that its
conditional probability given an action of yours is high, is an irrelevant
fact. For example, you have repeatedly chosen two boxes when zany martians
offered you choices, and though you are always disappointed that the
martians are intent on rewarding irrationality and punishing smart
behavior like yours, you are at least satisfied that you did the smart
thing every time. So you intend to do the smart thing again. You intend to
play the dominant strategy.

You reach toward the GREEN button.... and so does your doppelganger.
Suddenly he looks alarmed (as do you, of course). "Hm." you each murmur.
You smile at each other. You slowly each reach for the RED button....

Which button will you push? Come on, two-boxers, push that GREEN button.
You aren't getting all confused about causality, now, are you???

(The variant just offered is due to John Leslie.)

-- Jamie Dreier <pl436000@brownvm.brown.edu>

In article <3i0e81$dhs@nntp.interaccess.com> achepak@interaccess.com
(P. Kimball) writes:
> In essence, what is happening is this: the predictor has the ability
> to classify people into a "one-box" group and a "two-box" group.
> Then the predictor decides to reward the "one-box" people by giving
> them a million dollars and punish the "two-box" people by giving
> them only a thousand dollars.

This is making assumptions about how the predictor works.  I happen to
like this set of assumptions, because I would rather believe that
human actions are in principle predictable by extrapolating our
current state, and that the predictor has sufficient access to
detectors and computers to make this extrapolation in this case, than
believe in reverse causality or magic.

> Unfortunately for us, traits like
> mysticism and bad understanding of causality are associated with
> being a "one-box" person, while traits like a good understanding
> of logic and causality are associated with being a "two-box" person.

I disagree strongly with this part of the analysis.  I think some
(but not all) of the "two-box" people are simply stating that they
will discard any experimental evidence, no matter how overwhelming,
that disagrees with their (arbitrary and unproven) notions of free
will and causality.  (Some of them also believe in mystical concepts
such as mind-body duality.)

If someone were to come up to me right now and say, "I've predicted
what you would do in a Newcomb situation, here are the two boxes,
which do you choose," then I expect I'd take both boxes.  I have no
reason to disbelieve in my current set of assumptions about the real
world, in which the chance is phenomenally low that I'll ever meet
someone who can *really* predict my decisions.  (In this respect I
agree with Dr. Holt.  I don't know if it's possible in principle for
someone to do that, but I can't believe I'll ever see it happen.)

But consider the Newcomb setup I described, in which the predictor has
correctly predicted the actions of thousands of people, with large
percentages both of the two-box and one-box type.  This is an outcome
that according to my current world view is phenomenally unlikely.
Consequently, it would lead me to doubt my current world view.
This is a perfectly rational response; in fact, I claim that the
opposite response, persisting in a world view despite the fact that
one's observations contradict that view, is highly irrational.

I'm not sure if the foregoing was clear enough, so to summarize:

  (A) I don't believe I will ever encounter a Newcomb predictor.
  (B) If someone just came to me claiming to be a Newcomb predictor
      and gave me the choice of boxes, I'd most probably take both.
  (C) If someone *demonstrated* their Newcomb predictor ability
      (this would involve a series of correct predictions that I
      currently believe to be impossible, see point (A)), and *then*
      put me in the Newcomb situation, I'd be strongly persuaded to
      take both boxes.

[ Editor: "take both boxes" in (C) contradicts the next paragraph
("Given a sufficiently strong setup ... I'm a one-box person"); "take
one box" seems to be meant. ]

In other words, lacking sufficiently strong setup for the Newcomb
paradox, I'm a two-box person.  Given a sufficiently strong setup
(one that would teach me a new view of the world), I'm a one-box
person.

> Therefore, from the predictor's standpoint, there IS no choice.
> The apparent 'choice' is merely a by-product of the victim's
> membership in one group or another, which is the product of his or
> her background, training, intelligence, and experience.  To talk
> about what the victim "SHOULD" do at this point is beside the
> point.

This is true in the sense that the question, "What should the chooser
do?" is usually asked in such a way as to completely discount
background, training, intelligence, and experience.  Typically, we
ask, "What would a perfectly rational person do in this circumstance?"
Of course no such person exists.

I would even argue that a "perfectly rational person" is a
mathematical impossibility.  A very informal argument is that such a
person would have to embody a set of rules and a process for reasoning
on them that make the "correct" decision in all cases where a choice
is presented.  But the universe of possible choosing situations is as
rich (I argue) as arithmetic, hence the rules and reasoning process
would have to be as powerful as a system of arithmetic.  But there is
no system of arithmetic that can correctly answer all arithmetic
questions that can be posed to it, and similarly there is no system
for a perfectly rational person to follow that will yield correct
decisions to all the choices that can be posed to it.

Fortunately, when certain kinds of self-reference are ruled out, the
known problems with arithmetic systems don't arise.  Unfortunately,
Newcomb's Paradox contains self-reference, since it raises the
possibility that the predictor uses knowledge of the chooser's system
of reasoning, and that the chooser's system therefore has to reason
about a system that refers to itself.

> To make this clear, let me propose a nasty variation of the Newcomb
> problem:
> [Prisoner's dilemma, but ...]

This has little to do with the Newcomb problem, because *logic*
involves drawing deductive conclusions from assumed premises, whereas
the whole trick to Newcomb is in deciding what premises to assume.

The other difference is that in Newcomb, there is a clear dichotomy
(on which all observers would be able to agree) between the people who
actually *do* choose one box and the people who actually *do* choose
both.  So everyone can see (after the closed box is opened) whether
the predictor predicted correctly.

In the nasty Prisoner's dilemma, it would not generally be the case
that all observers could agree on whether the prisoner's thoughts were
logical, and so there is really not a good way for them to see whether
the king judged correctly (by executing or not executing the
prisoner).  In fact, if you assume everyone *could* agree on whether
the prisoner acted rationally, then "rational" doesn't even seem to be
well-defined.  After all, if it is rational to flip a coin, then
you'll be executed.  But it is irrational to do something that gets
you executed when you could have done something else, so it is
irrational to flip the coin.  But then flipping a coin will prevent
you from being executed, so it's rational to do so.  But then ...

-- David A. Karr <karr@cs.cornell.edu>

In article <pl436000-1702951038290001@bootp-122.college.brown.edu>,
pl436000@brownvm.brown.edu (Jamie Dreier) writes:
> stein@hal.nta.no (Stein Kulseth) wrote:
> > [Variant with no hidden information]
>
> The reason is that in your game, the player *knows* that
> he has a million dollars.  ... . The conditional probability that he has
> the million, then, is one, whether he additionally conditionalizes on
> *taking* the extra thousand, or on *not* taking it.

Yep, but even in the variant, the player knows that he has X dollars
where X is either $0 or $1,000,000. You can assign a probability to
each, and while the probability that he has the million may not
be one (unless the predictor is itself predictable), the conditional
probability that he has the million does *not change*, whether he
additionally conditionalizes on *taking* the extra thousand, or on
*not* taking it.

-- Stein Kulseth <stein.kulseth@nta.no>

Stein:
Yes, it does!

The conditional probability will depend on how reliable you take the
predictor to be. That is: pick some conditional probability, the
probability that the predictor guessed you'd pick both, given that you did
pick both. That will be the probability that you have the million, given
that you picked both. And that is the predictor's 'accuracy' level, so to
speak.

By the way,

 "the player knows that he has X dollars where X is either $0 or $1,000,000."

This is very odd! He knows that he has either $0 or $1,000,000, that much
is clear. He knows he has X, where X is either $0 or $1,000,000, sounds as
though he either knows that he has nothing, or knows that he has a million
dollars, and THAT's not true. ("Knows that" does *not* distribute across
disjunction!)

-- Jamie Dreier <pl436000@brownvm.brown.edu>

[ Editor: in Dreier's second paragraph, a predictor who guessed you'd
pick both leaves you $0, not the million, so "the million" there
should apparently read "nothing". ]

In article <3iiaa7$obl@crocus.csv.warwick.ac.uk> mareg@csv.warwick.ac.uk
(Dr D F Holt) writes:
> In article <pl436000-2002951146390001@bootp-122.college.brown.edu>,
> pl436000@brownvm.brown.edu (Jamie Dreier) writes:
>> mareg@csv.warwick.ac.uk (Dr D F Holt) wrote:
>>> Suppose the predicter makes his/her prediction at 3.00, and I make my
>>> choice at 3.15. Are you saying that the choice that I make at 3.15 is
>>> some kind of theoretically computable function of the state of the
>>> physical universe at 3.00 (either with certainty or with high
>>> probability)? That, I really cannot believe.
>>
>> Well, actually, yes, I was saying so. Why is it so difficult to believe?
>>
>> Suppose, first, that the universe is deterministic. (This is not contrary
>> to known facts.)
>
> Well it depends what you mean by known facts. It is completely contrary
> to current scientific opinion. Of course it is not impossible for
> this to be wrong, but I think it is very foolish to adopt personal
> viewpoints which fly in the face of informed research without
> good reason.  [...]
>
> Remember that (again according to current scientific belief), there are two
> known sources or reasons for the inherent unpredictability of the
> universe. One is quantum mechanical, and the other results from
> chaotic processes.  I would think it much more likely that the
> unpredictability of someone's decisions results from chaos rather than
> quantum mechanics - but I may well be wrong about that.

This is basically the argument why it is extremely unlikely, given
what we know of the universe, that we will encounter a Newcomb
predictor.  I happen to agree strongly with this argument.

I note, however, that we are talking here about *predictability*, not
determinism.  In particular, chaos theory assumes a deterministic
model of some process, then shows that your inexact measurement of the
initial conditions, not randomness in the process itself, is
responsible for your inability to predict the outcome of the process.

Of course it is precisely predictability, not determinism, that is the
heart of the question.  So the objections to the existence of a Newcomb
predictor are valid.

So we could perhaps agree that no Newcomb predictor will ever be
observed: we will never see anyone make a large number of successful
Newcomb-type predictions in a row.  But that says that the problem
situation will never arise; and *any* strategy is equally good for
dealing with decisions that never arise.  This makes the question
rather boring.

It's more interesting to consider Newcomb's paradox as a
counterfactual question: if such a situation *were* to arise, what
would you do?  The problem with relying on current scientific opinion
in that case is that I think if a Newcomb predictor *were* to be
demonstrated, it would seriously challenge, perhaps change, current
scientific opinion.  (Not that I think people would abandon quantum
mechanics or chaos, but they might reassess their application to the
human brain.  I don't know enough of the relevant sciences to say what
the implications would be.)

> Yes, but so what? What I am saying is that IF it was somehow determined
> at 3.00 what I was to decide at 3.15 (I am assuming there is no
> compulsion involved in the crude sense of someone holding a gun to my
> head), then it would be impossible for me to be made aware of what I was
> to decide at 3.10.- because if I was, then I could do something
> different.

What you've pointed out is that if I could somehow predict your
actions during the next 15 minutes, there are certain things I
couldn't do with that prediction without invalidating it.  This is no
surprise, and it's not even a challenge to the notion of determinism.
Nor does it have any application to the Newcomb paradox, since the
predictor does *not* tell you what you're going to do---it makes the
prediction and keeps it a secret until after it's too late for you to
change your mind.

> This is really the same argument as the one which proves that
> it is impossible to write a computer program which determines
> how another program will behave.

Now this is something I *do* have the background to address, and I can
state categorically that the assertion is misapplied to this
discussion.  A computer program *can* predict how another program will
behave ... within the next specified, finite amount of time.  I think
what you're referring to is the "halting problem," a result of which
states that there's always some program out there that we can't know
whether it will ever stop running (assuming we don't somehow abort it,
of course).  But there are many programs whose behavior we *can*
predict completely, and we can look at *any* program and predict what
it will do for some time in the future---and the more work our
predicting program does, the farther into the future it can look.

> This is diverging a bit from Newcomb's paradox - but my
> whole point was that I think the paradox is a psychological problem,
> rather than one to do with predictability in the physical universe.

I think the problem is that "rational decision making" isn't quite
what most of us would like it to be.  (In fact, it's possible we're
*all* wrong about what's "rational" in general.)  The problems in our
flawed premises only come up in bizarre counterfactual cases like
Newcomb's paradox, however, so there's no practical reason to fix
them.

-- David A. Karr <karr@cs.cornell.edu>

The literature on this paradox is comparable in volume to the literature
on the unexpected hanging paradox.  I have not, however, systematically
collected all the articles on Newcomb the way I have for the unexpected
hanging.  (Fact is, I don't have *any* articles on it in my possession.)
Does someone have at least a partial bibliography?

Thinking about it briefly the other day, I convinced myself that Newcomb
really reduces to the following simpler paradox: there is a being who has
never been wrong about predicting your actions.  You're annoyed at this
and you ask the being to predict which hand you're going to raise next.
You fully intend to raise the opposite hand from what the being predicts.
Can you successfully carry out your intention?

The reason I believe Newcomb reduces to this is as follows.  The being
must presumably use some sort of procedure to determine what your choice
will be.  Furthermore, the procedure must be capable of delivering its
verdict *long prior to* your actual decision.  Presumably, if such a
procedure exists, there is no reason why you cannot employ the procedure
yourself to predict what your own action is going to be.  You can then
be perverse and make the opposite choice from what the prediction
procedure predicts you're going to do.

I've flagged what I think are the weakest links in this argument with
the word "presumably."  But even these weak links seem quite strong to
me.  If the argument is correct, then observe that the simpler paradox
is no paradox at all unless the being is supposed to be a *perfect*
predictor.  Thus we may analyze Newcomb by breaking into two cases:
(1) the being is a perfect predictor and (2) the being is not a perfect
predictor.  Case 1 reduces to the simpler paradox and case 2 should
(although I have not worked out all the details) be relatively easy
to analyze compared to the original Newcomb paradox.

-- Timothy Chow <tchow@umich.edu>, December 1996

Although it does seem to reduce to the "free will/determinism" paradox,
I find that paradox interesting to think about but not really useful.

On the other hand, I've always thought that Newcomb's paradox bears
an uncanny resemblance to the Prisoner's dilemma.

In both paradoxes, there are four results that can happen, and the
results must be subjectively ranked by you in a certain way for the
paradox to appear.

(A) Prisoner's dilemma:
 (1) Worst case: You cooperate, but your opponent does not.
 (2) Slightly better, but still bad: Both of you do not cooperate.
 (3) Much better than case 2: Both of you cooperate.
 (4) Slightly better than case 3: You screw your opponent.

(B) Newcomb's paradox:
 (1) Worst case: You pick one box, and the predictor is wrong ($0).
 (2) Slightly better, but still bad: You pick both boxes and get a bit
     ($1000).
 (3) Much better than case 2: You pick one box and get $1000000.
 (4) Slightly better than case 3: You pick both, and get $1001000.

Note that in both paradoxes, your choice gets to restrict the possibilities
to either the set {(1),(3)} or the set {(2),(4)}.

The paradox gives you an assumption that your "opponent" gets to restrict
the possibilities into {(1),(2)} or {(3),(4)}.  (In the Prisoner's, this
is done by "you may not communicate with your opponent"; and in Newcomb's,
this is done by "The being may not change his decision once it is made.")

This leads to the "rational" argument:  "Choose {(2),(4)}, since (2) is
better than (1), and (4) is better than (3).  Since your opponent must
choose either the first two or the last two, this is the best strategy."

However, the assumptions of the paradox are also so that choosing {(1),(3)}
makes (3) likely, and that choosing {(2),(4)} makes (2) likely.  (In
the Prisoner's, this is done by "assume that the other prisoner thinks in
the same way you do"; and in Newcomb's, this is done by "assume that the
predictor is always right.")

This leads to the "likelihood" argument:  "Choose {(1),(3)}.  Why?  Since
(1) and (4) are unlikely, your choice reduces to choosing between (2) and
(3).  Since (3) is better than (2), you should choose {(1),(3)}."

The main difference between the two is that most renditions of Prisoner's
try to favor the "rational" argument, while most renditions of Newcomb's
try to favor the "likelihood" argument.  This is done by changing the
ratio [(3)-(2)]/[(2)-(1)].  (Note that [(2)-(1)] is usually about equal
to [(4)-(3)], to simplify calculations.)  If the ratio is very large,
the paradox resembles Newcomb's, and if the ratio is small (around
the order of 1), the paradox resembles Prisoner's.

-- Wei-Hwa Huang <whuang@ugcs.caltech.edu>

rhoads@hyla.rutgers.edu (Glenn Rhoads) writes:
> But you still have not refuted my argument nor resolved the paradox.
> The impossibility of the superpredictor logically follows from the
> infallibility of the expected-utility and dominance arguments. To
> refute this, you have to show that either expected-utility or dominance
> does not apply.  What you have shown so far tends to support the
> correctness of the expected-utility argument, you have not yet
> attempted to show that one of the two principles is incorrect.

True, but there appears to be a similar flaw in your logic. I agree
that both the expected-utility and dominance arguments are compelling,
so arguing that one is right is not sufficient to show that the other
is wrong -- one must actually show why the other is wrong. But aren't
you doing the exact same thing? There seem to be compelling arguments
that your actions can be predicted with at least slightly over 50%
accuracy. So we have three compelling statements, not just two. And
extending your statements above, you cannot refute the existence of a
superpredictor (i.e. someone with >50% prediction rate) by arguing the
correctness of the other two compelling arguments -- you must actually
show that the third compelling argument is incorrect. In other words,
everyone, regardless of their position, should be attacking one of these
three points, not arguing in favor of the other two.

In addition, Martin Gardner has pointed out that it's really necessary
not just to show why one of the compelling arguments is incorrect, but to
explain why it's convincing despite being incorrect.

> If you believe that you should take both boxes. Then what could possibly
> be wrong with the expected utility argument?  If you take only the left
> box, then the superbeing will very likely have predicted it and you
> will very likely have a million dollars.  If you take both boxes, then
> again the superbeing will very likely have predicted it and you will
> get only $1000.  Thus, you will very likely get a lot more money by
> taking only the left box.

Here's what I see as wrong with the expected utility argument. I assume
here that the prediction rate is >50%, but not 100%. The expected utility
argument observes that people who make choice A on average do better than
people who make choice B, and concludes that if I make choice A, I will do
better than choice B. But the expected utility argument is only valid
when the people in situation A do better on average than those in
situation B, because of some causal chain of events between their
choice and their situation. If there is only a correlation, the
expected utility argument does not hold. For example, I might sample
the incomes and voting patterns of people throughout the country, and
note that richer people are more likely to vote Republican. It would be
incorrect to argue that since the expected income of those voting
Republican is higher than of those voting Democrat, I should vote
Republican to maximize my income (assume I don't particularly care about
politics). Similarly, in Newcomb's paradox, those choosing one box
(Republican voters) on average do better than those choosing both boxes
(Democrat voters), but it does not follow that you will get more money
if you choose one box (vote Republican).

As I see it, what's happening here is that taking one box is highly
correlated with looking like the sort of person who will take only one
box. The predictor has decided to punish people who look like they will
take both boxes, and so on average, people who take both boxes do worse,
because of the correlation. But they are punished not for taking both
boxes, but for looking like the sort of person who will take both boxes.

OK, I know that everyone and their dog has some sort of analogy to
Newcomb's paradox that they think proves their point, but I think this
is a good one: Imagine that I hold a chess tournament, and invite people
with all kinds of different ratings, ranging from Kasparov to utter
novices. I know roughly how good people are based on looking at their
ratings. After everyone gets here, I announce that to make the matches
more even, that 15 moves into the game, players will be given painful
electric shocks to hamper their concentration, and that the higher the
player's rating, the more painful the shocks. The level of shocks will
be independent of the actual play in the first 15 moves of the game.
After the first few rounds of the match, it becomes apparent that I
have designed the shocks to more than overcompensate for
the rating difference -- A person with a
rating of 1500 is over 50% likely to beat Kasparov, because while
Kasparov will have a better position at the end of the first 15 moves,
after that point Kasparov will be too busy writhing on the floor to
think clearly. You realize this after the first few matches. Now looking
at the first few games, you notice that people who come ahead in trades,
control the center, and so forth in the first half of the game, usually
lose. The expected utility (read chance of winning) is higher for those
who play the opening in a manner regarded as poor under normal
circumstances. Now if you buy the expected utility argument in Newcomb's
paradox, does it not follow that you should intentionally play a poor
chess opening at this tournament? But I think it is clearly silly to
play chess poorly in the opening in the example above. The people in
control have decided to punish players who appear to be good based on
their history (people who seem like they would pick both boxes),
and because this is correlated with good chess moves in the opening
(people who actually pick both boxes), good players (people who
correctly argue from dominance) on average do worse.

Anyway, that's why I think you should take both boxes. The other
question is, if I am correct in stating that the expected utility
argument fails in this case, because correlation is not the same as
causation, why do so many intelligent people who understand the
difference between correlation and causation nevertheless take one box?
I would suggest that the real reasons are based not on expected monetary
rewards, or beliefs about free will and determinism, but on two
psychological reasons.

First, the problem is usually described in terms of these
aliens with amazing powers. These aliens are pretty much gods, and with
a 99% prediction rate, it's easy to treat them not simply as good
predictors subject to the rules of logic, but gods, who have magical
extra-logical powers. I think the argument loses some of its force if
you turn them into your best friend, psychologist, parent, or some other
person who can predict you with 60% accuracy.

Second, in reality, you really aren't trying to maximize your
expected dollar reward, but your utility, which includes these weird
psychological factors. Now I'm looking at this box that might have a
million dollars, and I'm pretty excited about that, because with a
million dollars I could quit my job and go to the Bahamas. On the other
hand, $1,000 would be nice, but hey, I've got that much in the bank
right now. Now OK, the dominance argument still works, and the expected
utility still doesn't work. But am I 100% sure in my logic? And will I
still be 100% sure in the future? Because most likely I'm a tiny bit
uncertain, and without computing utilities or anything, I know I'm
going to feel like a chump if I pick both boxes and get $1,000, when all
my dummy friends take one box and get $1,000,000. Afterwards, my logic
may still be good, but that's not going to console me when my friends go
party with their $1,000,000. On the other hand, if I take one box, and
get nothing, I realize I could have had $1,000 more, so I feel a little
dumb, but not completely awful -- and besides, my unpleasant feelings
about having made the wrong decision get outweighed by my anger at the
stupid aliens, who apparently can't predict me so well. So the real box
is:

                Aliens predict one          Aliens predict both

Take one        $1,000,000=happy            $0=angry

Take both       $1,001,000=happy+epsilon    $1,000=feeling like the
                                            biggest chump
                                            in the world

If I use this table, it is not clear that dominance still holds, because
while happy+epsilon > happy, I might prefer to be angry at the aliens
with $0, than feel like the biggest chump in the world with $1,000

Anyway, there's my solution to Newcomb's paradox.

-- Monwhea Jeng <momo@possum.physics.ucsb.edu>

It's not that hard.  At the heart of Newcomb's paradox is a
contradiction.  There are two assumptions here, one of which is
unspoken:
1) The superpredictor knows (with virtual certainty) which way
   we'll pick
        and the unspoken assumption
2) That we actually make a meaningful decision when we pick a box.

The two statements are incompatible.

Let's slip into analogyville again.  Imagine we're not people, imagine
we're a bunch of AI programs, and the superpredictor has gained
his amazing predictive abilities because he has access to our source
code.  Now, which of us get the million dollars?  The ones which
pick 1 box.  Dominance argument works fine.
        What's wrong with the utility argument?  Which "should" you
pick is a meaningless question.  You'll pick the way you're programmed
to pick.

        But we're not a bunch of AIs with published source code, you argue?
Well, maybe we are, maybe we're not.  If we are, and the
superpredictor ever shows up and starts playing this game,
the AIs that accept that they're AIs will get the million bucks and
the rest won't.  If we're not, then the superpredictor is an
impossibility.

-- George Weinberg <georgemw@slip.net>

In article <1993Dec4.220235.29659@questrel.com> chris@questrel.com
(Chris Cole) writes:
> Since Dave has now presented a VERY GOOD ARGUMENT for taking one box,
> I feel duty bound to present a VERY GOOD ARGUMENT for taking two boxes.
>
> Suppose the back of the boxes are transparent, so that you cannot see into
> the boxes but someone else can.  In fact, a friend of yours (someone who
> you trust and who you know to have only your best interests at heart) is
> at this very moment looking into the boxes.  You know he or she can see
> whether there is money in the boxes.  The rules of the game do not allow
> you to ask him or her about what is in the boxes, so instead you try to
> imagine what he or she is thinking.  If there is $1,000,000 in the first
> box, he or she will of course want you to open that box.  Since there is
> $1000 in the second box, he or she will also want you to open that box.
> In either case, whether the $1000000 is there or not, he or she will be
> mentally pleading with you to open both boxes.
>
> Now, your friend -- who knows what is in the boxes, who has your best
> interests at heart, who wants you to get the most money you can, and
> who is right now actually looking at the money -- this best friend is
> mentally pleading with you to open both boxes.  What are you going to
> do?

If there is money in both boxes, unquestionably your friend will be
mentally pleading for you to take both boxes.

But if there is only money - $1000 - in one box, your friend will be thinking
"The idiot's going to take both boxes!"

-- Ian Gent <ipg@aifh.ed.ac.uk>

[ Editor: for a predictor-free variant due to Smullyan, see
decision/newcomb.smullyan. ]

[ Editor: Stein Kulseth's March 1995 proposal for a new archive entry,
with the replies, and four postings from a February 1998 thread
"Paradoxes".  Kulseth's proposal began by quoting the problem and the
archive solution of the time (the two-boxer argument that the
expected-gain formula is fallacious); that quotation is omitted
here. ]

This is intended as a first step towards a new archive entry giving
full explanations of both major "solutions".  (The existing archive
entry is a 2-box view, and yet even 2-boxer me don't agree with all of
it).

I feel it would be appropriate to first try to reach some consensus on
some issues.  As a start I have tried to identify a set of relevant
assumptions and deductions, in the optimistic hope that we can agree on
these.  To work this into an archive entry feedback is necessary,
especially from prominent "1-boxers".

Now follows a set of assumptions which I think should be accepted by
both one-boxers *and* two-boxers, please tell me if you disagree with
any of these, or if there are any relevant assumptions missing.

Assumptions to make:

A1) Box content should *not* change

       There are numerous techniques that could be used to change the
       box contents afterwards to fit the actual choice made, fi.
       magic, time travel or ordinary sleight-of-hand, and in a
       real-world situation one cannot completely rule out these.
       However if we allow for this to be possible in the puzzle, the
       implied massive evidence for the predictor's performance would
       indicate that the choice might directly affect the box content,
       and option (1) would trivially become the best strategy.  To
       keep the puzzle interesting we assume that box content does
       *not* change.

A2) We have some free will.

       If we do not have free will to the extent that either option is
       available to choose, then there is no puzzle.  Then we just act
       as fate has told.

A3) The predictor is not foolproof/we are not entirely predictable

       Though the questions of predictor accuracy / our predictability
       are quite different from the question whether we have free will
       or not, they behave equivalently when it comes to analysing
       this puzzle.  If we assume a foolproof prediction, then the
       unpredicted option is not available as a choice *within our
       analysis*.

A4) Massive evidence show that the predictor works very well

       The puzzle text doesn't say so explicitly, but it should be
       assumed that enough evidence has been given for anybody to
       accept that the predictor *will* predict correctly any of the
       two options chosen, in almost all cases - regardless of whether
       one would have believed this to be possible or not.

A5) "Fancy" schemes will not be considered

       In any Newcomb discussion, especially when the puzzle text is
       not carefully given, someone will introduce random answers,
       sidebets, or other fancy schemes.  The text as given
       specifically says that all such schemes yield $0 in box B.  So
       let's just forget about such schemes in our analysis.

A6) The goal is to maximize one's expected payoff

       It is possible to bring in other goal functions, one might
       assign weight to the pleasure of proving the predictor wrong, or
       one might state that "$1,000 is enough for me, I don't care
       about the additional $1,000,000".  Such goal formulations might
       make the puzzle either trivial or impossible to solve.  The
       interesting 1-box/2-box debate arises when the goal is to
       maximize expected payoff, so we will assume this.

A7) You yourself don't know the prediction until afterwards

       If you do then option 2 becomes the universally accepted best
       choice, so let's assume not.

A8) You only get one experiment (per person)

       If this is a repeated experiment, other elements will have to be
       taken into account, esp "how will my choice this turn affect the
       next turn's prediction".  While this is an interesting variant,
       it is just that, a variant.

A9) The problem doesn't change if it is repeated (with different
    persons)

       No reason why it should, it should only strengthen our belief in
       the predictor (as mentioned in A4).  I only state this
       assumption because some of the deduced statements below will
       consider results from repeated experiments.

Then comes the tricky bit, what statements can be deduced from these.
I hope we all can agree on the following four...  And please add to
the list if there are other relevant statements you think we might all
agree on.

S1) If many experiments are done, the mean payoff for 1-box-choosers
    will be $1,000,000 or insignificantly less.  The mean payoff for
    2-box-choosers will be $1,000 or insignificantly more.

S2) If you can be predicted to choose 1 box you will do better than if
    you can be predicted to choose 2 boxes

       [No problems with these I guess.]

S3) A predicted 1-box-chooser who manages to choose 2 boxes will net
    $1,000 more than if he had chosen in accordance with the
    prediction.  A predicted 2-box-chooser who manages to choose 1 box
    will net $1,000 less than if he had chosen in accordance with the
    prediction.

       [No problems with this either, I hope.  It follows directly
        from the puzzle text and A1.  And although this is half the
        heart of the 2-box argument, I suspect that 1-boxers also might
        agree.  However they will put less weight on it as these cases
        do happen very, very rarely (significantly rarer than 1 in
        1000).  Tell me if I'm wrong, I admit that I don't understand
        the entire 1-box line of reasoning.]

S4) Consider a series of N experiments, where R1, R2, W1, and W2 are
    the number of right and wrong 1-box and 2-box predictions,
    N = R1 + R2 + W1 + W2, and W1 and W2 small.

    Now P(predict 1 box | 1 box chosen) can be estimated as
    R1 / (R1 + W2).  This is close to 1, and so the estimate of
    P(predict 2 boxes | 2 boxes chosen): R2 / (R2 + W1)

    Similarly an estimate of P(predict 1 box | 2 boxes chosen) is
    W1 / (R2 + W1).  This is close to 0, and so the estimate of
    P(predict 2 boxes | 1 box chosen): W2 / (R1 + W2)

       [Oh, boy, did I just lose all my fellow 2-boxers?  I hope not.
        Note that this is in disagreement with the archive solution
        that states that it is not given that P(predict X | do X) is
        high.  If I remember my statistics, these are straight-forward
        estimates derived from the available statistical data.  Also,
        I hope, that this is central to the 1-box line of reasoning
        (again corrections are welcome).  My two-box-view does however
        *not* find this statement central in determining the best
        choice.]

At last, the two views, where we will still disagree.  (And yet again I
apologize, if I have totally misunderstood the 1-box view, someone else
ought to write this, using other deduced statements if necessary)

V1) "S4 says it all, really.  Expected payoff *can* be estimated as
     E [ P(predict 1 box | 1 box chosen) ] * $1,000,000 =
     R1 / (R1 + W2) * $1,000,000 = $1,000,000 - smallchange
     S4 even accounts for the effect of S3 which merely reduces
     expected payoff just a little below $1,000,000."

V2) "S4 is not that important, it merely restates that your expected
     payoff is greater if you can be predicted to take 1 box *and* that
     you're then liable to do so (S2 and A4).  However both of these
     are outside of our control.  So S3 really is end of story, 2 boxes
     is the best choice netting you $1,000 more, regardless of
     prediction."

I'd really appreciate feedback.  I feel that this Newcomb Paradox is so
interesting it deserves a better, more extensive archive entry.

-- Stein Kulseth <stein.kulseth@nta.no> (7 March 1995)

In article <1995Mar7.144954.26928@nntp.nta.no> stein@hal.nta.no (Stein
Kulseth) writes:

>A1) Box content should *not* change

We all make this assumption, but in the end I think it is not
sufficiently well supported by the problem statement in the archive.
For the problem to be interesting (i.e., to cause a sufficient number
of people to attempt a rational argument in favor of one box), the
predictor has to be fairly convincing, by being impossibly successful
("impossible" relative to our current technology, that is).  But given
a being that is so much more technologically advanced than we are, it
seems foolish to rule out a priori any other "impossible" feats it
might attempt, such as causing money to appear or disappear from a
closed box.

I think the assumption can be better supported if you (the chooser in
the story) have had a role in the development of the prediction
technique such that you know the technique is a genuine case of
prediction by forward causality, and not some "cheating" method that
uses materialization at a distance, time travel, mind control, or any
other such nonsense.

This proposal has the advantage (or perhaps disadvantage) that it
forces us to agree not only on the predictions and prior outcomes, but
also on the model of causality and the mechanism by which the
predictions were produced.

>A2) We have some free will.

This begs the question of the definition of free will.  I think this
is central to the paradox.

>V1) "S4 says it all, really. Expected payoff *can* be estimated as
>     E [ P(predict 1 box | 1 box chosen) ] * $1,000,000 =

Minor error here:  The estimated expected payoff is

  P(predict 1 box | 1 box chosen) ] * $1,000,000
    + P(predict 2 boxes | 1 box chosen) ] * $1,000

(I prefer to simply use P for estimated probability, since we never
need to use the real probability in formulas, and to reserve the
symbol E to signify expected value.)

Admittedly, this is a trivial modification to the solution.

>V2) "S4 is not that important, it merely restates that your expected
>     payoff is greater if you can be predicted to take 1 box *and*
>     that you're then liable to do so (S2 and A4). However both of
>     these are outside of our control.
                          ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

This last is an interesting statement.  It seems to say that we cannot
control the current state and future progress of our own mental
processes.  I don't understand how to reconcile such a statement with
the notion that I have free will.

This touches on what is missing for me in the one-box presentation.  I
like my decisions to be based on a model of reality (causation and
mechanism) which is informed by observation (and statistical
inference).  The statistical inference alone is not the whole story.

My understanding of the notions of causality and mechanics behind the
two-box argument is that we assume that at the instant before your
final decision needs to be made, your mind is wiped totally blank of
all previous thoughts (a hard reset, if you will) and it then proceeds
to execute a perfectly rational decision process (which I think is an
unsound concept, but that's what's assumed) at blindingly fast speed.
Or at least that's how you're *supposed* to make a decision, and
governs the "correct" outcome, whether or not this is the actual
mechanism.  It seems to me that this model ironically wipes out both
free will *and* my favorite model for how the predictor works.

I have the experience of free will, but the above is not how it works.
My actions are free not because they are unconstrained by what I am
and what's in me, but because they are constrained by what *I* am and
what's in *me*, as opposed to being constrained by someone outside me
pushing buttons that say "raise index finger," "push with middle
finger," and so forth.

Moreover, I like to think that due to my actions being controlled by
*me* rather than by forces outside me, I have some control not only
over my immediate actions but over my future actions, because the
essential part of *me* has continuity.  Sometimes I might call this
"self-control."  If I didn't have such continuity and control of my
future, then who knows, the future me might suddenly start behaving
like a completely different person for no discernible reason---and with
high subjective likelihood for such a thing to happen.  So my marriage
vows, among other things, would be hollow and worthless.

Moreover, in a world in which people make decisions by spontaneously
(not deliberately) cutting loose from all previous mental states in
order to engage in a "rational decision" mental process, I don't see
how a predictor can perform accurate (and interesting) forward
predictions.  Either everyone comes up with the same decision, so
prediction is trivial---e.g., it's a given that box B will always be
empty anyway, so you know your choice is $1000 or nothing at all---or
the incidence of "rational thinking" is random, hence unpredictable,
hence the predictor will be practically unable to accumulate the
necessary record of correct predictions.

This is perhaps adding some assumptions that not everyone will agree
upon, but to me they're significant.

The whole one-box argument, of course, rests on the notion that the
prediction record of the predictor is indicative of an actual
underlying high probability of correctness, and not merely a
statistical mischance.  If we rule out all explanations of this
correlation other than detection of current mental state and forward
extrapolation, then the premise that the predictor exists requires that
this forward extrapolation actually works, i.e., that my concept of how
my "free will" functions is valid.  Hence you cannot simply make your
decision at the instant you reach out to indicate your selection.  You
must have made your decision, in effect, before the boxes were filled.
At least, most people's minds must work that way, else the predictor
could not have made so many successful predictions.

The problem with being a one-boxer, then, is not that you have to
abandon the chance at getting $1,001,000, but that it's hard to know if
you've really made the right decision in advance.  Which decision is
right is obvious; what's hard to tell is which of the two choices
you've "made."  Even I'm not completely sure what my own behavior would
be in such bizarre circumstances with such high stakes, which is why in
an earlier post I contemplated various tricks that I might use to
"steel myself" to choose one box in a real Newcomb situation, for
example drugs and self-delusion.

Of course the above is a mechanistic one-box rationale.  Jamie
Dreier's reasoning is much simpler and more direct: do what works.
That is the solution presented in argument (V1).  It's couched in
conditional expectations, but that's the essence.  It's a good rule in
general, and it obviates any need to discuss how the predictions are
made, causality, the nature of free will, or any of the other stuff.
In fact, it's really the only valid rule we have for behavior, when you
come down to it: all our models of causality, the state and function of
the world, etc., are justified solely because they help us figure out
how to do what works.  People like me, however, still need the crutch
of a mechanistic model.

-- David A. Karr <karr@cs.cornell.edu> (8 March 1995)

(Note: for those who couldn't care less about Newcomb and his elusive
million-dollars, there's an ObPuzzle at the end of the post)

David Karr writes:
|> >A1) Box content should *not* change
|>
|> We all make this assumption, but in the end I think it is not
|> sufficiently well supported by the problem statement in the archive.

Actually the main reason I make this assumption is to keep the problem
interesting.  Any competent illusionist or three-card-trickster could
show off a near-perfect prediction in a two-boxes setup like this.  So
without this assumption I feel that the one-box solution is trivial.

|> >A2) We have some free will.
|>
|> This begs the question of the definition of free will.  I think this
|> is central to the paradox.

Perhaps it is, again, I choose an assumption in order to keep the
problem interesting.  If there is no provision for choosing differently
than predicted then there is no problem.

|> Minor error here:  The estimated expected payoff is
|>
|>   P(predict 1 box | 1 box chosen) ] * $1,000,000
|>     + P(predict 2 boxes | 1 box chosen) ] * $1,000

Surely not?  If 2 boxes is predicted and you choose one you get $0 not
$1,000.  However, as you say this has little significance.

|> This last is an interesting statement.  It seems to say that we
|> cannot control the current state and future progress of our own
|> mental processes.  I don't understand how to reconcile such a
|> statement with the notion that I have free will.

Clumsy statement by me.  By saying that the being's prediction is
outside of my control, I mean: at the instant where I have to make my
choice I cannot further influence the prediction.  By saying that my
liability to choose is outside of my control, I don't mean that I have
no free will, only that my choice will be dominated by thought
processes that were predictable (otherwise the being would not be able
to keep up such a good record of correct predictions).  In other words,
I will not be able to deviate very much from my predicted behaviour,
but in a lucky (or unlucky :-) moment I might happen to come up with an
unexpected choice.

|> My understanding of the notions of causality and mechanics behind
|> the two-box argument is that we assume that at the instant before
|> your final decision needs to be made, your mind is wiped totally
|> blank of all previous thoughts (a hard reset, if you will) and it
|> then proceeds to execute a perfectly rational decision process
|> (which I think is an unsound concept, but that's what's assumed) at
|> blindingly fast speed.

Oh no.  Not by me at least.  If I should build a model I'd rather say:
A person's choice is primarily governed by thought processes that can
be predicted, say in 99.99% of cases these thought processes win out.
However every so seldom (0.01%) unpredictable thought processes will
govern our choice - which might be the same or the other option.

And BTW, even the 99.99% might also count as free will, only as
predictable free will, or as you say:

|> My actions are free not because they are unconstrained by what I am
|> and what's in me, but because they are constrained by what *I* am
|> and what's in *me*

... couldn't agree more.

|> Of course the above is a mechanistic one-box rationale.  Jamie
|> Dreier's reasoning is much simpler and more direct: do what works.
|> That is the solution presented in argument (V1).  It's couched in
|> conditional expectations, but that's the essence.  It's a good rule
|> in general, and it obviates any need to discuss how the predictions
|> are made, causality, the nature of free will, or any of the other
|> stuff.

Apropos this, here follows an ObPuzzle, for all of you who have grown
tired of the discussion (BTW maybe it's time to move this discussion to
email?)

ObPuzzle:
Out of my pocket I pick an ordinary coin.  I throw it 10 times, and get
7 heads and 3 tails.  If I throw the coin two more times, what are the
probability that both are heads?
a) 0.25
b) 0.25 + x (x small)
c) 0.49
d) 0.5090909...
e) 0.5625
Justify all the above calculations before making your choice.

-- Stein Kulseth <stein.kulseth@nta.no> (17 March 1995)

[ Editor: no answer to the ObPuzzle was found in the collected
postings. ]

[ Editor: the 1998 thread "Paradoxes".  Monwhea Jeng's chess-tournament
analogy, which these postings discuss, is given above. ]

Monwhea Jeng <momo@possum.physics.ucsb.edu> wrote:
>My claim is that all the cases above, and the chess tournament, are
>equivalent in all relevant aspects. If you take both boxes in
>Newcomb's formulation, you should also take both boxes in the Big
>Brother/60% Predictor, and you should play the chess opening poorly.
>And if you come to different conclusions in these cases, what are the
>relevant differences?

Perhaps the key difference between Newcomb and the chess tournament is
that your chess ranking only measures your ability to analyze chess
positions and respond to them.  It does not say anything about how you
will respond to the threat of an electric shock, which is the new
situation in which you've been put.  On the other hand, the Newcomb
predictor does purport to say something about how you'll respond to
the situation in which the Newcomb paradox puts you.

I think there is a particularly nasty kind of self-reference underfoot
in Newcomb.  This self-reference really is incompatible with the
concept of rationality that we tend to slide into when arguing which
decision is best.

>Given that the arguments that
>1) dominance works 2) expected utility works and 3) superpredictor can
>exist,

The application of dominance to Newcomb implicitly assumes that you
can make your decision, starting from an unbiased clean slate,
entirely after the situation has already been set up (in this case,
the predictor has already predicted).  In most decision questions, this
assumption makes no difference---i.e. there is absolutely no harm done
if I have secretly "precomputed" my decision before the setup and
simply remembered what I already decided.  In a mind-probe version of
Newcomb, however, precomputation clearly has a different effect from a
decision made completely after the setup.

Anything much less than a mind-probe predictor, and it isn't
sufficiently clear that it *will* demonstrate good enough powers to
trigger the expected utility argument.

So I choose dominance as the argument to suspect.

Perhaps the answer to Newcomb is there *is* no best strategy, at least
none that humans can apply, because of our limited mental abilities.
But is the notion that we might be predictable at some level really so
horrid?  I would think that total unpredictability would be worse:
imagine that just when you'd arranged your life entirely to your
liking, you wake up one morning to find that you hate everything about
yourself because your desires changed.

-- David A. Karr <karr@shore.net> (6 February 1998)

momo@sbphy.physics.ucsb.edu (Monwhea Jeng) wrote:

> What I mean is that if you can find a partition that satisfies the
> conditions you stated, AND the actions that you are deciding between
> have no causal influence on which member of the partition you are in,
> then the conclusion follows.

I see.  I'm not quite sure whether that's right, but it's close enough.
(I think you'd better restrict it to *finite* partitions, for one
thing, but we don't want to start worrying about that paradox now!)
It's close enough to Causal Decision Theory.

> I see that you've shown a different way to calculate utilities that
> leads to a different conclusion as to what you should do in the chess
> case. But I don't think it affects my argument that the chess example
> shows that the expected utility argument in Newcomb's paradox is
> wrong (have fun parsing that sentence!).

Hm.  Well, I think it's possible to cook up the chess example so that
it just *is* a Newcomb paradox, anyway.  So if you haven't completely
done that yet, let's just stipulate that you could.

My main point was that in ordinary sorts of cases, you are going to be
able to find some known information to conditionalize on, and this will
do away with the paradox.  And when you don't conditionalize on all
relevant information, well, we already knew that was a mistake.

In the chess example as I understood it, the known info was that you
can see that certain moves are better.  The moves you make are
evidence, but their (probabilistic) influence gets 'screened off' by
this other information.  In the original Newcomb, it is the choice
itself that's evidence, and nothing can screen it off.  (I know this is
not terribly lucid, I can give a clearer example.)

> I don't understand. I believe you earlier said that you thought that
> both the argument for taking one box and for taking both boxes were
> correct, and so you agreed with Glenn Rhodes that the "claim that such
> a predictor is impossible has a certain plausibility."

I did say those quoted words, but not for that reason.  For independent
reasons.

> Yet here I have given a (sort of) plausible situation in which the
> predictor gets over 60%, and you have raised no objection. Is there
> some qualitative change in the logic of the arguments when we ramp
> 60% up to 95%? And if you agree with my conclusion that both boxes
> should be taken in the example above, shouldn't you agree to take
> both boxes when the predictor works by psychology or physics, or some
> unknown combination of the two, rather than by a Big Brother network
> of spies?

No.  In the Network of Spies case, I will be able to conditionalize on
information I actually possess (I know my own history!), and then use
that updated probability function to compute expected utilities, and
the eu of two boxing will be higher.

If the predictor works by running a perfect computer simulation of me,
then I can't do anything like that.

Here is a bizarre thought experiment for you.

Suppose that our sadistic infallible predictor sets you up to play a
little Prisoners' Dilemma sort of game, and your 'opponent' is that
perfect computer simulation of yourself.  You have a green button and a
red button, and the simulation has a virtual red button and a virtual
green button.  To make things more intense, the simulation is presented
on a screen in front of you, and the image looks just like you, as
though you were looking in a mirror.

If you each press your green button, you will each get $1,000,000.
(The simulation is going to get virtual money, but that's what it
wants :-))  If you each press your red button, you will each get $1000.
If one of you presses red and the other green, then the green-presser
will get nothing, and the red-presser will get $1,001,000.

(Thus, pressing green is like taking one box; pressing red is like
taking both; simulation pressing green is like predictor predicting you
will take one; simulation pressing red is like predictor predicting you
will take both.)

Since you have no causal influence over the dopple-you, you feel you
ought to opt for the dominant strategy, namely, pushing the red button.
As you start to reach for it, you see the image on the screen reaching
for its red button too.  You hesitate.  It hesitates.  You feint toward
the green button.  It feints.  You decide to press the green button and
reach confidently toward it; so does dopple-you, but then at the last
moment you feel a little funny and stop... and so does the image....

Could you really bring yourself to press the red button?  I don't think
I could!

-- Jamie Dreier <pl436000@brownvm.brown.edu> (6 February 1998)

pl436000@brownvmDOTbrown.edu (Jamie Dreier) writes:

>In the chess example as I understood it, the known info was that you
>can see that certain moves are better. The moves you make are
>evidence, but their (probabilistic) influence gets 'screened off' by
>this other information. In the original Newcomb, it is the choice
>itself that's evidence, and nothing can screen it off.

I'm partially repeating myself here, which is perhaps a sign that it's
time to stop, but I don't quite understand what you mean by the
statement that "the known info was that you can see that certain moves
are better."  I know P-K4 is better than P-KR4 in a normal game, but it
doesn't seem legitimate to take as a starting point that it is
therefore better in the electric shock game.  That's what you're trying
to show.  After all, in the Newcomb case, I have an opaque box and a
thousand dollars sitting on a table.  I've picked up the opaque box and
put it in my pocket.  I'm about to walk out of the room, but then I say
to myself "should I also pick up the $1,000"?  Why can't I say that
"the known info is that I can see" that it's better to have $1,000 more
than I have now?

>No.
>In the Network of Spies case, I will be able to conditionalize on
>information I actually possess (I know my own history!), and then use
>that updated probability function to compute expected utilities, and
>the eu of two boxing will be higher.
>
>If the predictor works by running a perfect computer simulation of
>me, then I can't do anything like that.

But in the last section of the Network of Spies case, I postulated that
I hadn't discussed Newcomb's paradox in years, and had actually
forgotten what I said about it.  Given this, do you still agree that I
should take both boxes?  And if you agree with that, what's the
difference between that case, and a physicist who runs a simulation of
me, where I don't know the results of the simulation?

>Could you really bring yourself to press the red button? I don't
>think I could!

Ouch!  Nice example.  IF I could somehow be 100% sure that the
predictor was not cheating in any way, then I think it would be
rational to press the red button.  The trouble is that I think this
case, unlike the Newcomb case, is actually impossible.  I have been
considering the Newcomb case with a predictor with a success rate p,
with 50%<p<100%.  You are now giving me a case where p is actually
100%.  Everything I do is mirrored perfectly.  Because of chaos and
quantum mechanics, I do not think p can be 100%.  I see no reason chaos
would prevent over 50%, but there is some p_max that cannot be
surpassed.  So the example you give I think is actually impossible, and
if it were to actually occur to me, I would suspect the superpredictor
of cheating, and press the green button.

I suppose that there's some success rate in the original Newcomb case
-- say 99.9999% -- where I have to accuse the predictor of cheating and
take one box.  I've taken it as given that the evidence is such that I
believe the 95% (or 60%) was obtained legitimately.

-- Monwhea Jeng <momo@possum.physics.ucsb.edu> (6 February 1998)

I think discussion of the Newcomb problem cannot come to a conclusion
because the question being asked is the wrong one.  As stated, the
problem asks "What do you choose?", but people have little interest in
the answers.  The real question only comes after someone answers, and
it is "Why do you choose that option?".  This is where the paradox
arises because the answers are trying to give a logical argument for
choosing one option over the other, but if there were a valid logical
argument for choosing one option over the other, this would give the
predictor a reason to assume you would choose that option, which, in
turn, gives you a reason for not choosing that option.

That is where the causality problem arises.  Your choice can 'cause'
the earlier prediction if that choice is the result of an algorithm
which the predictor can test in advance.  This is equivalent to posing
the problem to an AI program - you can run a copy of the program and
use its answer as the prediction.

-- Charles Bryant <ch@chch.demon.co.uk> (8 February 1998)