The Newcomb Problem

Quote: jfalk

Thanks, MathExtremist. That article is quite interesting, but I'm afraid it doesn't quite resolve the issue for me. It parameterizes the game as a two person game and shows that the two ways of looking at the problem are not mutually consistent. The problem is that it gives a strategy to the Newcomb being. The consistenency requirement then imposes constraints on a joint probability distribution. But it's not clear to me this a two person game. If it's a one person game against a natural condition, in that case, all of the problems still follow through, and there is no guidance, or even potential guidance, as to which game you're playing. That, it seems to me is the problem. That said, the article is quite interesting and recommended to anybody well versed in game theory.

Well, the Newcomb being is said to have a strategy -- it's goal is to guess (in our formulation, with 90% accuracy) your future action. The time-reversal section of the paper, bottom of page 10, makes it clear that there are two interpretations of your relationship with the Newcomb being:
a) You believe that the Newcomb being *can* observe (or in the original, predict) your choice, and will act accordingly, or
b) You believe that you can hide your choice from the Newcomb being, and that the NB's accuracy is merely random chance.

I admit very freely that my interpretation of the problem is that there is a causality, and not an independence, to the question of the NB's accuracy, and that is why I interpret the problem as (a) rather than (b). But that wouldn't materially change if the NB was a natural condition, to use your phrase -- the only issue of interpretation is whether the predictions of the NB (or natural condition) are accurate (a) because of or (b) independently of the choices you make. Both (a) and (b) cannot be true simultaneously, and therein lies the "paradox".

To me, the problem statement screamed "causality", but I admit that's not necessarily the case. If you believe that the NB's choice (or the natural process) and your choice are truly independent, then picking both boxes is clearly the right choice.

BTW, I love this quote from Nozick:
"To almost everyone, it is perfectly clear and obvious what should be done. The diffculty is that these people seem to divide almost evenly on the problem, with large numbers thinking that the opposing half is just being silly."

Quote: MathExtremist

...BTW, I love this quote from Nozick:
"To almost everyone, it is perfectly clear and obvious what should be done. The diffculty is that these people seem to divide almost evenly on the problem, with large numbers thinking that the opposing half is just being silly."

ME, that quote reminds me of an exchange I had with mkl two days ago:

Quote: Doc

Quote: mkl654321

...If we accept the premise as given (fully realizing that premise's real-life implausibility), then the decision is trivial.

There just might actually be agreement on this statement. The problem remains that people disagree on which direction the "trivial" decision should go.

I have not read the paper that you reference, so I don't know what it says about time reversal. (I don't accept time reversal as real or as a rational part of making reality-based decisions.) I guess if the interpretations of the relationship with the being are strictly the (a) and (b) you have listed, I come closer to (b):

I view the prediction's accuracy as "random chance" to the extent that it might be either right or wrong with a probability of 90% right and is not a factual statement about what has been "seen" in the future. If the being actually can observe the future clearly, then there is no reason for a 10% error rate in the predictions.

On the opposite hand, I have allowed that I don't know the being's technique and would accept the premise that it has a "blurry", uncertain view of the future (or most any other technique, such as "reading" the players) that assists in it's achieving a high accuracy rate, though the outcome is still a random event. I don't know that this means a belief -- as stated in your option (b) -- that I can "hide" my choice from the being. I just think the being must not see completely clearly whether I hide or not. I don't view the ability or lack of ability to hide as necessary to making my decision -- since I don't believe in time reversal, I accept that whatever the money is, it is fixed in the boxes long before I make my choice and will not change because of my choice (though it is certainly decided by the being's much-earlier expectation of my choice). Of course, if it were indeed possible to deliberately deceive the being, the proper strategy would be to convince it that you would choose box A and then actually choose both, but that isn't required for me to make my decision.

I think this means I am closer to (b) but don't adhere strictly to (b). There must be other interpretations than the two you listed.

Quote: Doc

Absurd.

What is absurd is the wording of the entire question--it refers to an unreal (i.e., impossible) situation, and therefore the whole "paradox" is moot. This reminds me of the ancient, hoary "Zeno's Paradox" which, of course, is not a paradox at all--it's manufactured nonsense created by applying ambiguous wording to a real-life situation.

Obviously, the issue that never gets addressed is: is the Newcomb's 100% predictive ability immutable? In other words, are we ABLE to make a choice that the Newcomb failed to predict correctly? If so, then he never had 100% predictive power in the first place. If not, then the whole question of "choice" goes out the window--making the original question irrelevant.

So the real answer to the problem is that it CANNOT be answered given the stated conditions. Now, let's take a look at a lower stated predictive success rate--say, 90%. This DOES work, because we now have the ability to choose either option, without subverting either Newcomb's success rate (we could be part of either that 90% or of the other 10%) or our own free will (we are perfectly able to choose "wrong").

So I default to choosing Box A only in all those situations where the Newcomb's predictive ability is 50%< X < 100%, due to +EV.

Quote: Doc

Absurd. Your option (1) says that the million is definitely in box A which by definition says that the being predicted that you would choose only box A which (under your assumption in this particular post) assures there is 100% probability that you would choose box A. There is no free will involved -- your "choice" of box A is absolutely required of you. Similarly (but exactly the opposite) for your option (2). No free will available there either.

As I have mentioned before, the being makes the important decision as to whether to give you the $1 million or not, while (so long as there is neither a guaranteed accuracy of 100% or 0%) you have the choice of whether to take the extra $1,000. In those two extreme cases, you have no choice at all -- everything is determined by the being and by the rules. If you can actually make a free-will choice, what you choose will change the being's accuracy rate.

I don't understand how you can believe there is a free-will choice available when the being has previously made a 100%-accurate "prediction" of what you will do and definitely documented his advance prediction by placing a fixed amount of money in the boxes. How do you have the option to "choose" something other than what was predicted? The only answer I can see is if you believe that future actions change what happened in the past. If that is your genuine belief, we will never come to an agreement.

So success rate on predictions of about 90%, 95%, 99% and 99.99% etc..., is OK.
But success rate on predictions of about 100% is no.

OK then
If You believe, the Newcomb Being has a success rate on predictions of about 99.99%.(Based on your final(very last) choice, no the choice 24 hour ago).
Then the choice is:
(1) Choose box A only, get million dollars. 99.99% of the time.
(2) Choose both Box and get $1000. 99.99% of the time.

Is this sound better.

Doc, I am particularly interested in learning your answer to the question I asked earlier (look at my previous post, about an additional bet after the decision is made).

As for the causality reversal (future affecting past), I have given it some thought, and, I think, my argument is that for the predictions to be accurate with the high enough frequency in the absence of time travel and such, it has to be that the players really make their decisions way earlier than they realize. Take you or Wiz for example. You have already decided that you are taking both boxes even though the prediction has not yet been made. Likewise, my decision is to take one box.
I have suggested this problem to my wife and four of my friends, and got some different responses (I was able to accurately predict all four BTW), some people are of one opinion, and some are of the other, and all of them seem very unlikely to ever be convinced to change it. Their choice is essentially already made, way before the prediction is done. So, there is no paradox or causality reversal here.

MrCasinoGames: To cut it short, I have basically described my position as being:
(1) My own choice is not based on any calculation involving the being's percentage accuracy so long as the percentage is not a guaranteed 100% or 0%,
(2) I think that people who use the accuracy percentages to calculate expected values are doing so improperly, and
(3) If either 100% or 0% accuracy is certain, the player doesn't actually have any free-will choice at all.

I suppose it is a bit like saying discontinuities occur at 0% and 100% and you are asking me how close I can come to the cliff before I fall off.


weaselman: Please understand that I do not mean this comment as a criticism of you, for it likely reflects badly on me at least as much as on you.

I read your earlier post several times when I first saw it, and I now have re-read it several times. I still am not sure what you said. By adding a few words, I think I can interpret the first two lines, but I still get a bit lost in what the other two lines mean. I get jumbled every time I read them. Sorry.

Nevertheless, I made a guess at what they mean, and my answer is unlikely to satisfy you at all. I don't think I have any basis at all for guessing what the being predicted unless/until I know something about the technique he is using. And that is independent of the accuracy percentage (excluding a guaranteed 100% or 0%, which I have said leads to a different problem.)

As for your comments in the more recent post, that raises some additional possibilities. If the player has already made a decision well in advance, then the being does not necessarily need to see the future or be influenced by the future at all -- something like a skill at player reading could lead to the excellent success rate. But suppose the player wasn't even told he would be playing the game until after the being made the prediction about the future -- what would that do to the problem/game/strategy etc.?

My return three-part question is this:
(a) Are you confident that MrCasinoGames thoroughly believes he should choose box A only?
(b) If so, are you confident the being would be able to predict that choice and would place the million in box A for him?
(c) If so, do you think it would be a good idea for MrCasinoGames to change his plan at the very last moment and actually choose both boxes?

Quote: Doc

Nevertheless, I made a guess at what they mean, and my answer is unlikely to satisfy you at all. I don't think I have any basis at all for guessing what the being predicted unless/until I know something about the technique he is using.

I am afraid, I don't understand this line of reasoning at all. If you are are offered a 2-to-1 odds that the next card in the shoe is a ten or a face card, you can evaluate how good or bad that bet is without exact knowledge of the technique used to shuffle the shoe or how many decks it consists of. It is enough to know that the frequency of ten-value cards in the shoe is 4/13.
Likewise, if you know that some process yields a certain result 90% of the time, why do you need to be concerned about internal mechanics of that process.
Perhaps, you just find it hard to believe that the accuracy of the predictions can be that high? But then you are solving a different problem, because this one establishes the accuracy as a pre-condition, there is no wiggle room.

Quote:

But suppose the player wasn't even told he would be playing the game until after the being made the prediction about the future -- what would that do to the problem/game/strategy etc.?

I won't say that it does not matter at all if the player was told about the game or not, but I believe, it matters much less than you think.
The five people I mentioned earlier never heard about this problem before, yet, I was able to predict their decisions before I told them about it.
What I am saying is that, perhaps, the way you lean in this particular choice is reflective of some bigger thing - your life philosophy, if you will, and that makes it possible to deduce your future decision from simply knowing who you are, your position on other (maybe, seemingly unrelated) questions.

Quote:

(a) Are you confident that MrCasinoGames thoroughly believes he should choose box A only?

Well ... I don't know MrCasinoGames nearly as well as I know my wife, and I am no Newcomb being by any stretch of imagination, so, no, I am not confident, but yes, I do believe, that, if he is offered a choice now, he would choose box A.

Quote:

(b) If so, are you confident the being would be able to predict that choice and would place the million in box A for him?

Again, I can't claim to be able to predict the decisions of MrCasinoGames, leave alone the decisions of some Supernatural Being about his decisions :) So, I am afraid, I have to pass on this one. I don't know what it would predict.

Quote:

(c) If so, do you think it would be a good idea for MrCasinoGames to change his plan at the very last moment and actually choose both boxes?

Well ... yeah ... If he was confident of what was predicted, then it would be a good idea for him to change the plan.
But the point is, that if he was capable of that confidence (or courageous enough to change the plan without being confident) ... then there is a 90% chance that the being would have spotted that earlier, and the prediction would have been different.

Quote: weaselman

I am afraid, I don't understand this line of reasoning at all. If you are are offered a 2-to-1 odds that the next card in the shoe is a ten or a face card, you can evaluate how good or bad that bet is without exact knowledge of the technique used to shuffle the shoe or how many decks it consists of. It is enough to know that the frequency of ten-value cards in the shoe is 4/13.
Likewise, if you know that some process yields a certain result 90% of the time, why do you need to be concerned about internal mechanics of that process.

Sorry to give an incomplete answer -- I've got 5 minutes until I have to leave for a ball game.

I made an earlier post about random events and probabilities. The draw of the card from a shoe is a random event with a probability, and I can use that info to make my plans related to the next card. I do not consider the being's decision to be a random event with a probability upon which I can base some sort of calculated guess about what he predicted. It is the outcome of the prediction that is a random event with a probability, not the prediction itself. That is the basis of my comment that I believe people are using the percentages improperly to compute expected values.

Later.

Quote: Doc

It is the outcome of the prediction that is a random event with a probability, not the prediction itself. That is the basis of my comment that I believe people are using the percentages improperly to compute expected values.

Yes, I am talking about the outcome of the prediction.
After your decision to pick two boxes (or one) is fixed, and can no longer be changed, but before the boxes are opened, you are asked to guess the outcome of the beings prediction with a monetary prize attached to the right guess. Would you guess the prediction was right or not? What is the probability your guess is correct?

Quote: weaselman

... Would you guess the prediction was right or not? What is the probability your guess is correct?

The best information available to me is that the prediction has a 90% chance of being correct. That is true before I make my choice, after I make my choice, and even before the prediction is made. And it doesn't seem to depend upon what I choose. And it apparently is that same percentage even if I openly announced my commitment to a specific choice before the being made the prediction. All-in-all, it seems like a pretty good idea to bet on the being making a correct prediction.

I am afraid, though, that you are on the verge of suggesting that the being had a 90% probability of predicting the exact box(es) that I chose. I think that is likely the point this line of thought falls apart, because I don't believe it is appropriate to talk about a probability of what the being's prediction would be, was, or will be. It isn't a probabilistic random event. It is a choice/decision made by the being based on some unknown technique that I have no basis to guess about. I have no way to make that guess and no way to assess the probability that any such guess is correct.

You posed a question earlier that seemed to describe the situation of whether to take insurance in a blackjack game. Consider this card problem. I have four cards in my hand, an ace, 2, 3, and 4. I am going to show you one of those four cards. What is the probability that it will be an ace? I think it is obvious that you cannot name a percentage for that probability.

Alternative problem: when I predict my favorite basketball team will win a game, I am correct 90% of the time. They won a game last week; what is the probability I predicted that they would win that game? Is it obvious to you that you cannot name the probability of my prediction in this case either?

Similarly, I do not think you can apply a probability to what the being chooses to do/predict. It is not a random event. It is a deliberate choice.

Quote: Doc

I have no way to make that guess and no way to assess the probability that any such guess is correct.

But you don't need to assess it in the context of this problem - it's part of the problem's condition. Just like you know that the probability of the next card in the shoe being a ten or a face is 4/13, you also know that the probability of the prediction being correct is 90%.

It's an input parameter, you do not need to determine it, not if you are working on this problem.

Quote:

You posed a question earlier that seemed to describe the situation of whether to take insurance in a blackjack game. Consider this card problem. I have four cards in my hand, an ace, 2, 3, and 4. I am going to show you one of those four cards. What is the probability that it will be an ace? I think it is obvious that you cannot name a percentage for that probability.

Right, I can't name it. But if you tell me that you are going to show an ace about 90% of time, and I have reasons to believe you, then I know that the probability I'll see an ace is 90%.

Quote:

Alternative problem: when I predict my favorite basketball team will win a game, I am correct 90% of the time. They won a game last week; what is the probability I predicted that they would win that game? Is it obvious to you that you cannot name the probability of my prediction in this case either?

Hmm, no, it is not. I think, the probability is 90% (if you actually made a prediction).
If you offered me a better than 9-1 odds bet that you predicted a win, I would accept it.

Quote:

Similarly, I do not think you can apply a probability to what the being chooses to do/predict. It is not a random event. It is a deliberate choice.

If I offer you to pick one of two boxes, and tell you, that I, by my deliberate decision process, unknown to you, put a million dollars in one of them, and that I usually pick the first one 9 times out of ten. Would you argue that you don't see an advantage to you in picking the first box (supposing, you know me as an extremely honest person, that never lies)?

Regardless, you did say earlier that you considered the outcome of the prediction a random event, that can be assigned a probability. Are you taking it back now?

Quote: Doc

Similarly, I do not think you can apply a probability to what the being chooses to do/predict. It is not a random event. It is a deliberate choice.

Without getting back into the meat of the problem, this application of probability to non-random events is precisely what Bayesian probability is about. It's about quantifying and evaluating decisions under uncertainty. Choices made under randomness are a subset of uncertainty, but so are the deliberate but unknown actions of another rational agent. From Wikipedia, "In the Bayesian interpretation, probabilities are rationally coherent degrees of belief, or a degree of belief in a proposition given a body of well-specified information."

On the other hand, if you're a pure frequentist then your position holds. Probability to a frequentist is only about random events and their likelihoods.

Off-topic post:

Is anyone else having problems with performance of the WoV site? I am finding it very slow and unreliable, with multiple reports of server errors. Refreshing a page often requires multiple minutes. The MegaMillions web site was bogged down yesterday, but I understood that. I haven't had any problems with any other sites, so I don't think the problem is on my end. I don't see how this discussion group could be overloading any reasonable web site host. These problems have been giving me real pains in trying to respond to the posts above. My composed posts sometimes completely disappear when I click the "post" button.

I'll try again for an on-topic post.

Edit: weaselman and MathExtremist, I have tried multiple times to post my reply, but it just won't go through.

Wizard, this is the error message that appears all too often on my screen:

Quote: Error message encountered on Firefox browser

Internal Server Error

The server encountered an internal error or misconfiguration and was unable to complete your request.

Please contact the server administrator, webmaster@wizardofvegas.com and inform them of the time the error occurred, and anything you might have done that may have caused the error.

More information about this error may be available in the server error log.

Additionally, a 404 Not Found error was encountered while trying to use an ErrorDocument to handle the request.

Off topic: Yes, it's been very slow and unpredictable over the last few days. I use Firefox.

Quote: Switch

Off topic: Yes, it's been very slow and unpredictable over the last few days. I use Firefox.

Snap for me.

Very slow at times, indeed. Quite normal other times.
That usually happens when the web server is configured incorrectly, and it is discovered by hackers out there, that start using it as proxy. To rule this out, you can look for unusually looking GET requests (requesting urls from other servers, not WoV, using full url - like GET http://www.google.com) in apache access logs (assuming, you are using apache for web server).

Doc, I am afraid, Wizard is unlikely to see this exchange, because he had determined a while back that he had nothing more to add, and has likely stopped checking messages in this thread.

Well! Let's see whether the site will accept my replies today...

Quote: weaselman

But you don't need to assess it in the context of this problem - it's part of the problem's condition. Just like you know that the probability of the next card in the shoe being a ten or a face is 4/13, you also know that the probability of the prediction being correct is 90%.

It's an input parameter, you do not need to determine it, not if you are working on this problem.

Yes, it is a parameter of the problem that there is a 90% probability of the prediction being correct. But there is no parameter of the problem about the "probability" of the prediction being "box A". The prediction itself is not a random event. I have no basis to make a calculated guess about it.

Quote: weaselman

Quote:

Alternative problem: when I predict my favorite basketball team will win a game, I am correct 90% of the time. They won a game last week; what is the probability I predicted that they would win that game? Is it obvious to you that you cannot name the probability of my prediction in this case either?

Hmm, no, it is not. I think, the probability is 90% (if you actually made a prediction).
If you offered me a better than 9-1 odds bet that you predicted a win, I would accept it.

For purposes of full disclosure, my "alternative problem" is not real. I have several "favorite" teams and don't really make predictions about any of them. Consider these to be imaginary predictions about an imaginary favorite team, somewhat like the imaginary Newcomb being.

weaselman, the difficulty you may encounter in concluding there is a 90% probability that I predicted a win is that my (imaginary) prediction about my (imaginary) favorite team has always been that they will win their game, and they have a 90% win record. (They're very good!) The fact that I have a 90% accuracy rate in my predictions gives you no information about the "probability" of what I predicted. I don't even consider that a proper use of the word "probability". Even though I may have always predicted in the past that they will win, I don't consider it proper to say there is a 100% probability one of my specific predictions is for a win. It is something I choose deliberately, not a random event, and I could choose a different option any time I wanted to. Or maybe I have really been using a completely different, unknown prediction strategy. In any case, there is no basis for guessing the "probability" of my prediction, and it appears that viewing a deliberate choice as a random event misled you to an incorrect answer about a "probability".

As a side note, even if you were correct that there was a 90% probability that I had predicted a win, I don't see why you would expect me to offer you better than 9-1 odds that I predicted a win. I think you got something jumbled there (but I think it's not important).

Quote: weaselman

If I offer you to pick one of two boxes, and tell you, that I, by my deliberate decision process, unknown to you, put a million dollars in one of them, and that I usually pick the first one 9 times out of ten. Would you argue that you don't see an advantage to you in picking the first box (supposing, you know me as an extremely honest person, that never lies)?

No, I don't really think there is a definite advantage, and there might even be a disadvantage. Your undeclared "deliberate decision process" might just be that you place the money in box #1 the 9 out of 10 times when you play with someone who has consistently agreed with your viewpoint in this thread and place it in box #2 the 1 out of 10 times when playing with people who have posted dissenting views. That would not make you dishonest or a liar at all. It would just be an undeclared decision process about which I don't have any useful information.

Now on the other hand, if you had told me that you use some random process to decide which box to place the money in, that it has wound up in box #1 for 9 out of 10 times when you use that process, and that you plan to use the same random process when I play the game, then I do think it would be rational to expect box #1 to be more likely to have the money, and that is what I would choose. But that's a different situation.

Quote: weaselman

Regardless, you did say earlier that you considered the outcome of the prediction a random event, that can be assigned a probability. Are you taking it back now?

No, I don't think I have taken that back at all. The outcome of the prediction (correct or incorrect) has been stated as having a 90% probability of "correct". We just don't have any information about the "probability" of the prediction itself being "box A" vs. "both boxes", and (redundantly repeating myself over again once more for yet another time) I don't think that is proper use of "probability".

Apology to all for my always being so verbose.

Quote: MathExtremist

... this application of probability to non-random events is precisely what Bayesian probability is about. It's about quantifying and evaluating decisions under uncertainty. ...
On the other hand, if you're a pure frequentist then your position holds. Probability to a frequentist is only about random events and their likelihoods.

MathExtremist, I am a MathNeophyte, or perhaps a MathMinimalist. I have openly admitted that I have studied statistics and probability theory very little. It was not until reading a post of yours some time back that I had even heard of different camps among the statistics enthusiasts, and I certainly could not have described the difference between Bayesians and frequentists. I do recall Bayes's theorem about conditional probabilities (though I would butcher it if I tried to state it now), but I thought that when we studied it we were discussing random events, like the probability of having the queen of spades, given that you were dealt a black face card.

I am indeed aware of the use of the term "probability" for non-random events, such as, "There is an 80% probability that our super-star athlete will turn pro after his junior year." I don't take that as a claim that the athlete behaves in a random manner. I just thought it was sloppy terminology used to express "feelings" or "confidence" about some uncertain situation. I have likely used such expressions myself (I don't claim to always maintain consistency.) I just don't think I have viewed such a "probability" as a figure that is properly used to calculate a mathematically-optimal decision.

Based on your description, it appears that my current thinking falls in line with the frequentists. I know that in the two-envelope problem, I did not care for the logic that went something like, "There are only two possible amounts of money in the envelope, so we should base our decision on the figure half-way between the two."

Maybe next week my random walk will take me to join the camp of the Bayesians. Maybe I should apologize to weaselman because I didn't even know we had take up residence in long-established but opposing camps. I assume there is no agreement on whether the Bayesians or the frequentists are correct, is there?

Quote: Doc

Maybe next week my random walk will take me to join the camp of the Bayesians. Maybe I should apologize to weaselman because I didn't even know we had take up residence in long-established but opposing camps. I assume there is no agreement on whether the Bayesians or the frequentists are correct, is there?

It's not a question of correct/not so much when dealing with the sorts of topics we discuss here. It's primarily a distinction when dealing with true statistics - that is, estimation of probabilities based on small samples of the population rather than the whole population itself. Casino games aren't unknown distributions that need to be sampled, they're fully described populations (52 cards, etc.) with probabilities that can be calculated directly.

The Bayesian approach is growing in popularity in the statistical fields. I am not a practicing statistician as I have no call to collect (sample) statistics on anything and then analyze them. For those who do, the Bayesian approach presents perhaps a more flexible mechanism for analysis. The problem is that it's terribly non-intuitive; the idea that you should adjust your degree of beliefs in future events in a calculable way, based on your degree of belief and experiences in the past - that's just not something we're born understanding how to do. Of course, neither is the frequentist idea that probabilities are the asymptotic limit of a ratio as the number of samples goes to infinity. The closest thing we've got to something matching our intuition, if at all, is the 400-year-old classical idea of probability which says that whatever problem you're looking at can be divided into N equally-likely independent events, and the probability of one of them happening is 1/N. That concept is less useful when dealing with modern statistical tasks (clinical trials, etc.), but that turns out to be exactly how casino games work, and in fact is how the "classical" method was developed in the first place (Cardano, Fermat, Pascal, Laplace, etc.). It was developed by a bunch of guys trying to figure out how to get the edge in gambling games.

Quote: Doc

Yes, it is a parameter of the problem that there is a 90% probability of the prediction being correct. But there is no parameter of the problem about the "probability" of the prediction being "box A". The prediction itself is not a random event. I have no basis to make a calculated guess about it.

I am not asking you to. I am asking you to make a calculated guess as to weather or not the prediction was correct.

Quote:

weaselman, the difficulty you may encounter in concluding there is a 90% probability that I predicted a win is that my (imaginary) prediction about my (imaginary) favorite team has always been that they will win their game, and they have a 90% win record. (They're very good!) The fact that I have a 90% accuracy rate in my predictions gives you no information about the "probability" of what I predicted. I don't even consider that a proper use of the word "probability".

Well, yes, this is true. But lacking the information about the actual win rate of your team, I am making an estimate of the actual probability, unknown to me, which, I think, is reasonable.
There is nothing unusual about this. When thinking about a probability of the next card out of the shoe being a ten, there is always a possibility, that all the non-tens have been already drawn out, and the actual probability of getting a ten now is 100%. However, without that specific knowledge, I have to assume that the distribution of the past draws was about average, and thus getting a ten is still about 4/13 chance.

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Even though I may have always predicted in the past that they will win, I don't consider it proper to say there is a 100% probability one of my specific predictions is for a win.

Exactly. But if I am told that the accuracy of your prediction (not the "past success rate") is 90%, then I think it is perfectly reasonable to assume that the probability of you predicting a win in this particular situation is 90% even if I know that up until now you haev always been predicting a win (which I do not).

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As a side note, even if you were correct that there was a 90% probability that I had predicted a win, I don't see why you would expect me to offer you better than 9-1 odds that I predicted a win. I think you got something jumbled there (but I think it's not important).

Should it be 8-1? Or 1-8? I always get confused in these odds stuff.
What I mean is I would risk 90 bucks if you agreed to pay me more than 10 in case you have really predicted a win.

Quote:

No, I don't really think there is a definite advantage, and there might even be a disadvantage. Your undeclared "deliberate decision process" might just be that you place the money in box #1 the 9 out of 10 times when you play with someone who has consistently agreed with your viewpoint in this thread and place it in box #2 the 1 out of 10 times when playing with people who have posted dissenting views. That would not make you dishonest or a liar at all. It would just be an undeclared decision process about which I don't have any useful information.

How about if I also honestly told you that I don't have any bias either for you or against you? What if I don't even know who is going to be offered the box I am stuffing with money?
Still, it could be that I happen to never pick the left box on Thursday ... but just as well, it could be that all the tens are gone from that shoe. Without knowing the exact picture it is reasonable to replace the unknowns with averages.

I think, the key here is that I am not playing against you. If I was, then, certainly I could find an "honest" way to deceive you and deliberately make you pick the wrong box. But if I have no personal interest in misleading you, then knowing that I pick the left box more often then not should be good enough for you do conclude, that you have an advantage.

In fact, most (all) of the "randomness" you experience in a casino (and most other situations) is of this kind. If you could account for all the physics involved in a roll of dice, there would be nothing random about it. It is only "random" because you can't know the entire picture, and have to replace the missing pieces with what you believe to be a reasonable estimate. The casino could be (knowingly or unknowingly) using biased dice, and, in fact, since nothing is 100% perfect, it definitely is biased , and that roulette wheel is also skewed, and probability of some numbers hitting is a little bit higher than the others. But you don't know which numbers are preferred, so you just assume that all probabilities are equal, because it is a reasonable thing to assume.

Quote:

No, I don't think I have taken that back at all. The outcome of the prediction (correct or incorrect) has been stated as having a 90% probability of "correct". We just don't have any information about the "probability" of the prediction itself being "box A" vs. "both boxes", and (redundantly repeating myself over again once more for yet another time) I don't think that is proper use of "probability".

I'll repeat myself too, and say again, I never asked you about the probability of the prediction being box A. My question was about the probability of the prediction being correct, exactly the way you state it. If you agree, the probability of it being correct is 90%, what is the probability that the first box is empty, provided that you have chosen to take both boxes?

weaselman, MathExtremist probably was right in bringing up the subject of Bayesians and frequentists, even though I didn't know what they are and still don't really understand it. I think it is likely that you and I have different perspectives on the use of probabilities, and it is unlikely that either of us can build a thesis that will be convincing to the other. The more likely outcome of an extended discussion is that we would start trying to trick each other, then one or both of us would become annoyed, and instead of discussing the problem, we would be arguing about it. If we kept going down that path, it could turn into an on-line fight, and pretty soon we could start sounding like Jerry and mkl.

I have no interest in getting into a genuine argument with you or anyone else on this forum about any topic. Can we agree to disagree on this one? I'd rather maintain a friendly environment and move on to something else.

By the way, do you remember whether we agreed or disagreed on the two-envelope problem? Maybe in my capricious ways I have already moved to a different "camp." Give me a few more months, and I might be back on your side. Or you on mine.

Doc, did you and weaselman read the paper I posted about the resolution to the Newcomb problem? There's an especially illuminating section near the end where it discusses how you should behave if the NB acts after you make your choice, not before. Consider the problem restated, where you make your choice, and then afterwards the NB puts the money in the boxes accordingly.

If you believe the NB can tell what you picked, then the NB's choice is dependent on yours. If you believe the NB is guessing randomly (but accurately), then the NB's choice is independent from yours. Whether you view the NB's actions as dependent or independent is the entire crux of the paradox. You can't simultaneously hold both positions, and each position indicates a different choice on your part: if you think the NB's choice is dependent upon yours, you pick box A only. If you think the NB's choice is independent of yours, you pick both boxes.

Quote: Doc

By the way, do you remember whether we agreed or disagreed on the two-envelope problem?

I was wrong about that one. I thought that it was not possible to come up with an unbounded discreet distribution that would satisfy the required properties, but someone (don't remember who) had pointed out an example, that proved me wrong. After giving it some more thought, I concluded that the strategy of maximizing the expected value does not work in that situation because EV is infinite, and infinity does not behave like normal numbers (5/4*infinity is not greater than 1*infinity).

Quote: MathExtremist

You can't simultaneously hold both positions, and each position indicates a different choice on your part: if you think the NB's choice is dependent upon yours, you pick box A only. If you think the NB's choice is independent of yours, you pick both boxes.

I did read the article, but did not find it particularly enlightening. In particular, I think that it was established somewhere in the beginning of this thread, and was never disputed since, that the key to the problem is deciding whose choice is "primary", and whose isn't (the direction of causality). The rest of the discussion has been focused on which of the two options is the "right one".

Quote: Doc

I have no interest in getting into a genuine argument with you or anyone else on this forum about any topic. Can we agree to disagree on this one? I'd rather maintain a friendly environment and move on to something else.

I agree with you.

Quote: MathExtremist

Doc, did you and weaselman read the paper I posted about the resolution to the Newcomb problem? ...

What? You think I'm actually going to READ and accept input to my thought process?

No, I confess I did not read the article. I think I even said that once before. Maybe I will get around to it, but laziness may intervene. Just based on what you said in your second paragraph, I think the position "the NB's choice is dependent upon yours" probably aligns with the "future changes the past" scenario that I objected to in the original problem. Having the dependency occur in sequential time would be more acceptable, so I probably really should read about that.

Just became a member, so haven't read all the replies and this might already have been answered, but here's my answer:

If I chose both, I have a 90% chance of getting $ 1000 and 10% chance of getting $ 1,000,000, so my EV = 0.9*1000+0.1*1,000,000 = $100,900.
If I chose just box A, I have a 90% chance of getting $ 1,000,000 and a 10% chance of getting $ 1,000, so my EV = 0.9*1,000,000+0.1*0 = $ 900,000.
Clearly my EV is biggest if I chose just A.

Also, $1,000 wouldn't change my life, while $1,000,000 would, so maximizing my chances to get $1,000,000 would be the way to go for me.

It seems to be partially a Schrodinger's cat problem, only without cruelty to animals. I believe, in this scenario, an additional probability is required to resolve it.

The issue seems to revolve around the nature of prediction, or, from a practical viewpoint, whether the choice you are making now can affect what is to be in the box. If the being's predictions are based on mundane means, like psychological analysis, it can not (case P). If they involve means outside current scientific knowledge, it might (case Q).

If you don't know - as seems to be the case - then the right thing to do is to determine the probability of either.
If you have time to act, you could actually do it properly by collecting information on people who previously took the offer, to learn which choice did they make, whether the box A was full or empty, and, if you can interview them, how certain they were about their decision prior to making it.

The first thing you have to learn is the overall rate of winning the million. It is important in scenario P, because in that scenario your real probability is *not* 90%, even if you pick A. It is only equal to overall win percentage. Since it's still undetermined if you'll take A or A+B, the chance there is money in box A is only 90%*probability of you picking A, and since you don't know that probability yet, you have to take the average.
Compare it to the probability of flipping heads twice in a row, 1/4. After you flip it the first time and flip heads, it becomes 1/2, but right until then, it's 1/4. Same here, your picking box A is the first flip. Your 90% chance of getting the money comes from self-selection bias, putting yourself where most "money boxes" are. Since whether the money is there or not is determined before the first "flip" is made, the first probability applies in scenario P.
In scenario Q, however, the being actually knows what you are going to pick (presumably its 10% mistake rate comes from "prophetic noise"?), which means the money being in the box or not is determined, for purposes of cause-effect relationships, after the first "flip".

The second question is whether the being's 10% mistake rate is weighed towards either outcome. It could be that it short-changes some box A pickers, but never grants A+B their $1,001,000, or vice versa. If this rate is heavily weighed, corrected individual rates should be applied to the EV calculation.
The final thing you would do is study the pattern and determine if it correlates better with P-type or Q-type predictions. That will provide you with probabilities for either.

If you don't have any time or access to this information, you can just make a guess. This one can depend on personal beliefs, but it's always possible to quantify confidence in these beliefs. For instance, I believe it's 30% likely that FTL and temporal travel are not possible, 70% that one or both are possible, 40% that one or both will eventually be achieved by our species, and 4% that it will happen in my lifetime. I might then conclude it's 25% likely that this being has advanced predictive powers and 75% likely that it's just a good psychologist. That is if I haven't worked out these probabilities based on pattern correlation above.

The final adjustment is correcting Value for Utility. To someone in the third world, $1,000 may be a lot of money. To a compulsive gambler, $1,000,000 won is worth over 1000x the worth of a surefire $1,000. For some the utility of $1,001,000 is nearly identical to the utility of $1,000,000, for a millionaire it's equal to value, and to someone there is added pleasure in fooling the creature and getting the most possible.

The formula will then include the following parameters:
pA - being's probability of making a mistake against an A-picker
pB - being's probability of making a mistake in favor of an A+B-picker
pW - overall win percentage, known or estimated
pQ - probability that the being is using Q-type prediction
UA, UB, UAB - utility of box A's contents, box B's contents, and both, respectively

Then, for picking A and AB respectively, the EU is:
EU.A = (1-pQ)*pW*UA + pQ*(1-pA)*UA
EU.AB = (1-pQ)*pW*UAB + pQ*pB*UAB + (1-pQ)*(1-pW)*UB + pQ*(1-pB)*UB

Let's put some numbers in.
If we don't know pW, we can assume half. If separate pA and pB are not known, pA=pB=10%. I'll take linear utility curve here, i.e. EU=EV. For pQ, as above, 0.25. For linear utility, I'll cut the UB part and just add VB.
It arrives then at:
EV.A = .75*.5*1e6+.25*.9*1e6 = $600,000
EV.AB = .75*.5*1e6+.25*.1*1e6+1e3 = $401,000

As can be seen, for this scenario, EV.A>EV.AB, simply because VB is so small that it doesn't compensate for the probability of scenario Q, in which the being actually knows and not just guesses what you will take. But if UB was higher, then EV.AB could easily prevail. Assigning pQ without knowledge of prior pattern is pretty much arbitrary, but some would argue it's way lower, some higher, but there is a point everyone can assign and use in weighting these scenarios for the final decision.

It may look strange that EV.AB is more than 0.1*VA, but it's that way because we assumed it's 75% likely that taking both boxes doesn't change anything. This is an average EU that only applies to your most likely chances at a given moment and will not reflect overall statistics, because of uneven distribution: most people choosing AB have already lost, and most people choosing A already won, thus still providing a 90% overall prediction accuracy.

Is the being not by its nature forcing you to choose both boxes? - creating its own 90% success rate.

Its told you that you have money here, you many not get money here- also what if it wanted to hedge its own offer and has decided to keep the million, it would say that it predicted you would take both boxes so that you end up with nothing or just a grand. Also, Id imagine that humans look/seem pretty greedy to more superior beings and he would still say that we would take both boxes.

Either its because he wants to keep the million, we are greedy, or your playing it safe for a sure thing. I say he is predicting you to take both boxes, and thus box one is empty to begin with and your getting $1,000.

Choice A : A-B = $1,000,000 or $0
Choice B : A+B = $1,001,000 or $1,000

Choice B not only offers a guaranteed payoff, but also a higher 'jackpot'.

It says A and you go A+B = $1,001,000
It says A+B and you go A+B = $1,000
It says A and you go A= $1,000,000
It says A+B and you go A= $0

Mathmaticly your best choice for overall jackpot and guranteed return is to choose A+B which Im predicting is what the being would predict.

Quote: Peter

If I chose both, I have a 90% chance of getting $ 1000 and 10% chance of getting $ 1,000,000, so my EV = 0.9*1000+0.1*1,000,000 = $100,900.
If I chose just box A, I have a 90% chance of getting $ 1,000,000 and a 10% chance of getting $ 1,000, so my EV = 0.9*1,000,000+0.1*0 = $ 900,000.
Clearly my EV is biggest if I chose just A.

Also, $1,000 wouldn't change my life, while $1,000,000 would, so maximizing my chances to get $1,000,000 would be the way to go for me.

When I created this thread in 2010, I would have argued with you. However, nine years later, I can see your point and tend to agree.

Heh. I suppose the math is different but this reminds of video poker. Should I keep the sure thing high pair or should I go for the 4 to a royal flush? Let me consult the Wizard's basic strategy chart.

Quote: Gialmere

Heh. I suppose the math is different but this reminds of video poker. Should I keep the sure thing high pair or should I go for the 4 to a royal flush? Let me consult the Wizard's basic strategy chart.

I don't see the connection to video poker.

Ha, should have searched the site before creating the thread. Live and learn. Fun discussion here.

Maybe I’ll post the caterpillar game (assuming a search doesn’t turn up a thread on it first).

Quote: Peter

Also, $1,000 wouldn't change my life, while $1,000,000 would, so maximizing my chances to get $1,000,000 would be the way to go for me.

What if the numbers were bigger and it was $100 Million for Box A and $1 Million for Box B?

Just discovering this old thread.

I actually ‘’solved’’ the Newcomb paradox in my end paper back in 1986.
This kind of paradox is very similar in logical structure to reasoning ab absurdo . State an assumption, deduce a paradoxical result, conclude that the assumption must be false.

My conclusion was that such a being cannot logically exist. More precisely, the paradox exists because you assume at the same time that the being acts after knowing your action but you act after him. Time loop paradox.

For the record, I was working on Cournot equilibria in economics, and the possibility of anticipating rational action by the opponent (so-called Rational Conjectural Variations Equilibria). I destroyed the concept and confirmed the validity of the Cournot-Nash approach.

Maybe I'm not understanding the question properly, but it seems blindingly obvious to pick only box A.

The boxes are before you. Their content is fixed, nothing can change it. You know box B contains $1000. It is also blindingly obvious to pick box B in addition to A.

But you would have known I would be the type to pick box A, so you wouldn't possibly keep the $1,000 in there.

It's not out of science fiction to assume an ability to read other people. Successful poker players do it all the time. I would assume it is a useful skill in business and sales too. That said, I don't find the premise ridiculous. As I recall, the Newcomb Being gets to spend some time with the player before the player must make his choice. I find it an interesting question and one that I sometimes think about before falling asleep.

Quote: kubikulann

The boxes are before you. Their content is fixed, nothing can change it. You know box B contains $1000. It is also blindingly obvious to pick box B in addition to A.

This proves the absurdity of being rational in boundary condition cases. Similar to the Caterpillar Game.

Quote: unJon

This proves the absurdity of being rational in boundary condition cases. Similar to the Caterpillar Game.

’’Boundary’’?
This case is way out of boundaries. It is plain impossible.
In the Caterpillar or in the Selten paradoxes, we have a boundary case, in that we go to a limit. Not here.

Quote: Wizard

It's not out of science fiction to assume an ability to read other people. Successful poker players do it all the time. I would assume it is a useful skill in business and sales too. That said, I don't find the premise ridiculous. As I recall, the Newcomb Being gets to spend some time with the player before the player must make his choice. I find it an interesting question and one that I sometimes think about before falling asleep.

Who said anything was ridiculous. The thought experiment is interesting in the extreme. To prove the infinity of primes, Euclid posits the premise that their number is finite. It is by no means ridiculous, it is a pillar of mathematical thinking.

Simply, this is not a question of real-life ‘’reading people’’. That talent is exerted in the present, and is about estimating a course of action based on available info. We do that everyday, it’s part of being human (i.e. theory of the mind).
The specific of this model is in the interaction: a sort of game theory situation. The player is informed, not of what the Being predicts, but that this prediction IS what he will do. Feed-back loop means self reference means paradoxes.

This is precisely what I studied. Some authors were distorting the basic simunltaneity of decisions, without knowledge of the real action of the opponent, like in the typical Nash examples. I showed there was no way of justifying such anticipations rationally, that you ultimately came back to Nash.

In the Newcomb paradox, what happens is not cold reading. The game is definitely sequential. Whatever the Being anticipated, the player is faced with a physical situation which cannot be changed anymore. If he had to choose between boxes A or B, you are right, he would have to outguess which decision is best. it is what poker players, salesmen or teachers use. But the player here takes the ambiguous box anyway ; the question of what is best resumes to ‘’Should I take the other also?’’. Of course you should, it is $1000 for free: at that moment in the game your opponent cannot change the content of the first box anymore.
Whatever his reading skills in the previous stage, it is now too late.
The paradox arises only if you insist on his ability of acting in function of your decision. Or, because you are given an information : « the Being is never wrong (or some small percentage). » That does not appear in the real-life examples of poker or sales.
If it were simply cold reading, he acts in function of your personality at moment 1. NOT your decision at moment 2.

You might counter by saying: « If he reads you as a rational player, he’ll predict that you'll take both. If he reads you as a ‘delusional gambler’ or or other ‘magic thought’ type, he’ll predict your taking only the first box. » Alright, but after he closed the boxes and departed, what prevents you from taking both boxes anyway? The fact that some players take only one does not lift the paradox: they do it because, in some sense, they imagine their decision influences the content of the box, which cannot be. But for you or me, the question is: « whatever its reading skills, whatever its decision, it has no incidence on my choice. »

As an aside, the original Newcomb paradox states that the Being has perfect predictive ability. Your presentation introduces a percentage. Does it change the nature of the problem? I think not. I think we should know what that percentage means exactly. Is it, like I said, a percentage of correct assessment of personality, or must we admit it is a percentage of correct prediction? Also, how has it been established ? From a sample? OK but in what conditions ? Were all players informed the same way? But then how could the first ones in the sample be informed about it, when the statistical result did not yet exist? What were they told? How could they trust it?

Typical case of self-referencing percentages: « I know he is 90% correct, hence I take the decision X. » Either that leads to another percentage and you have another form of the paradox, Or it produces 90% and we have a tautology. Taking a random decision is provably suboptimal.
And now imagine: all players decide to randomize 50/50. The Being is said to predict that. OK, but then it can never achieve a 90% accuracy!
It reminds me of the case of the poll question: what is your probability of answering correctly this question? Self reference is the royal way to paradox.

The way I heard the problem, the Newcomb Being is not always correct.

I've changed my position, I think, to taking box A only, out of respect of the Newcomb Being's ability and wanting the $1,000,000.

Quote: Wizard

The way I heard the problem, the Newcomb Being is not always correct.

I've changed my position, I think, to taking box A only, out of respect of the Newcomb Being's ability and wanting the $1,000,000.

Let’s study a few scenarios.

Imagine everybody acts like you, choosing only A. How does Being achieve a 90% percentage? Simply by predicting AB 10% of the time. But that is not a predictive ability. Any frequentist or Bayesian being should quickly arrive at 100%.

Suppose 10% of the people choose AB. Then for reaching 90% accuracy, all the Being has to do is predict A all the time. Again, not a very convincing predictive ability.

Now let’s continue for various levels of percentages of real choices. There is (almost) always a percentage that the Being can choose, without any prescience, to achieve a 90% fit. Noting X the percentage of players choosing AB and Y the percentage of times the Being empties A, independently, then all X,Y satisfying XY+(1-X)(1-Y)=90% are possible.
The info 90% does not give info on any correlation between the Being and you. It may be zero, it may be small, it may be large. There is no basis for selecting.

Conclusion: 90% of accurate technical predictions is NOT 90% of accurate true anticipation of players’ decisions. The number 90% gives you no information at all about the Being’s ability. In your words, it should give you no particular ‘’respect’’ for it.
Even if you know X (say, from statistical observations) and compute Y -i.e. making the assumption of independence- the optimal strategy would NOT be to randomize at level X, NOR would it be, in most cases, choosing deterministically one or the other option.

And finally, considering you have a unique opportunity to play the game, your result does not affect the 90% if they are based on a sufficiently large sample (and even more so if that information came from another source). One more reason to do just what you like.

Quote: kubikulann

Let’s study a few scenarios.

The way I interpreted the 90% is the NB (Newcomb Being) didn't track past performance or human behavior. He simply picked A if the player picked A 90% of the time and picked AB if the player picked AB 90% of the time. Why muddy the waters with a NB with an adaptive strategy?

So, much like I would respect somebody how could handicap against the spread with a 90% success, so do I respect the NB.

Of course I know you understand that the crux of my argument is the Bayesian paradox: the « 90% » info is useless if you don’t know the prior probability of behaviour.
And the second line argument: the probability knowledge is to be accounted differently according to whether you look at the situation from outside [for example, you are betting on another player’s result] or you are part of the experiment (self-reference).


.note 1.
The initial wording did not imply your interpretation. It sounds like it said: « the NB has a record of 90% correct prediction. » Which is not the same as « 90% of the time player chooses 1, the NB predicted 1 AND 90% of the time player chooses 2, the NB predicted 2. »
Nor is it the same as « 90% of the time NB predicts x, the player chooses x. »
But whatever the meaning, that info is insufficient if you don’t have the priors.


.note 2.
Finally, I’m not sure « respect » is the adequate concept here. I respect my father, that does not mean I believe he is infallible. In my view, the concept is rather, « Do you trust the info you were given about its aptitude ».
And I do not, because I feel I showed part of it is impossible (the Future Prediction part) and part of it is tautological, hence useless (the 90% part).

Just seen this thread for the 1st time.

There is no reason to assume that people that have had this choice in the past believed that the NB has a 90% correct predictive capability.

Most people would have realized that there has been money already placed in the two boxes. The obvious choice for most people in the past is to take both boxes, because you will gain more money.

The NB has understood that, and has probably always "predicted" that people will take both boxes, and placed cash accordingly -hence a roughly 90% predictive capability.

I think the NB predicts that you also will take two boxes. Act accordingly.

Quote: gordonm888

I think the NB predicts that you also will take two boxes. Act accordingly.

Exactly!

An interesting aside is the question that helped solve the Bernoulli paradox.

Where does the money come from?
Why should the Boeing want to offer me (or all those previous players) free cash?
Who are all those previous players alluded to? How many?
Or, if not statistical and there are no precedents, where does trust in the Being’s ability come from? (What Wizard calls ‘‘respect’’)

And that is also part of my solving of the paradox in my paper (which was about economics): thought experiments are very fine, but when they lead to paradox, the only sound response is to say: « That doesn’t exist in the real world. »

I still don't get what is so hard about taking on faith that the NB will get any prediction right 90% of the time. This includes predictions of picking A, AB, and what the weather will be like tomorrow.

Here is an example that may help. The NB has ten identical looking crystal balls. Nine are always right and one is always wrong. He picks one at random with every prediction.

The example does not solve the question. The impossibility of the NB’s existence just transfers to the impossibility of the crystal balls being always right (and always wrong, by the same token).

What confirms my doubt about any good rational (epistemological) reason to accept the NB aptitude (or the 90% figure), is your use of the word ‘’faith’’. To me, faith is precisely non rational, non evidence based, non logic based.

You can’t establish serious results on the argument of faith, can you?