The Newcomb Problem
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The being, whom in deference to tradition we will call the Newcomb Being, will allow you to either take both boxes, or just Box A.
Twenty-four hours ago, the Newcomb being made a prediction about what you would choose. At this point, it placed the money in the boxes. If the Newcomb Being predicted that you would take both boxes, it left nothing for you in Box A and the $1000 in Box B. If the Newcomb Being predicted that you would only take Box A, it would leave the million dollars in it, as well as the thousand dollars in Box B. If the being predicts you will base your choice on a random event (like flipping a coin), it will leave Box A empty.
Now you have grounds to believe that in such matters, the Newcomb Being has a success rate on predictions of about 90%.
Do you choose box A only, or both?
Wording above taken from greylabyrinth.com.
Quote: Wizard
Do you choose box A only, or both?
.
Oviously you take both so you have a guarantee of getting at least a grand. Always go for the sure thing, gambling is for idiots.
By taking both, I feel that I will either get $1,000 or $1,001,000 - by taking just A then I stand to get $0 or $1,000,000.
Of course, a lot may depend on how the predictive powers were utilised - for example, stepping into the future and returning back to present would give 100% correct prediction and would alter my decision.
Also, the question states 'That you believe ...' - quite a few people think that John Edward can 'connect' with 'the other side' so I suppose it would also depend on the correlation between your belief and the probability of it being accurate.
Anyway, I still pick both boxes for now.
(Any chance you can 'switch' if you don't like it :-) ).
Quote: SwitchOf course, a lot may depend on how the predictive powers were utilised - for example, stepping into the future and returning back to present would give 100% correct prediction and would alter my decision.
We should not assume time travel, for purposes of the problem. The question doesn't state how you know the Newcomb Being is correct 90% of the time. For the sake of argument, Let's assume that it has done this thousands of times before, and the well-documented results show a predictive success rate of 90%.
Jerry, I agree with your first sentence. I may agree with the second sentence, but stating it that way is not my style.Quote: JerryLoganI take both. To do anything else means you are dumber than a rock.
Quote: MathExtremistMy initial reaction is to take just box A,
There is probably some convoluted reason to take A, but people who pass up a sure thing for a possible pig in a poke usually get what they deserve. Lets see what Mister 190 IQ says, this would be an easy problem if there was a Will Hunting on here. Oh wait, we have the next best thing.
Quote: MathExtremistMy initial reaction is to take just box A, because the Newcomb being will be 90% right in predicting that I'd try to maximize my EV, which means I'm going for the 1M rather than being greedy, trying for 1,001,000 and getting only 1000. I'm sure there's more to it, but this is a 30-second impression.
How is the problem a random variable? It states the boxes have already been filled. Could this not be reworded that you get box A no matter what, the choice is whether or not you refuse box B?
Quote: WizardHow is the problem a random variable? It states the boxes have already been filled. Could this not be reworded that you get box A no matter what, the choice is whether or not you refuse box B?
Sure, but the rewording doesn't change the nature of the prediction 24 hours before, or what the NB did based on that. If he thought I'd take box B, box A is empty. Same situation.
I'm ruling out deception or other fraud - I just buy into the premise that NB knows what I'm thinking 90% of the time. I know what I'm thinking all the time in situations like this - maximize my EV. So he knows that 90% of the time too. That means if I'm greedy, 90% of the time I get $1000, while if I'm sensible, 90% of the time I get $1000000. I know that, and so does he - and he based his decision 24 hours ago based on that, which means there's $1000000 in box A because that's the only box I'm going to pick.
I think the problem still seems to stem from the predictive qualities, despite being 90%. For example, if you stated that there were 9 red balls and 1 white ball in a bag and the predictive success depended on picking out a red ball, then you would have a mathematical reason to pick just Box A with a 90% probability of getting $1,000,000 and just a 10% chance of getting $1,001,000 (if you were to pick both).
However, without going backwards and forwards into the future, it almost seems like that as the 'predictor' is possibly affecting your style of decision making. Another example, if the Newcomb also stated that you would be wearing black shoes when you make your choice you can affect that outcome by wearing brown moccasins (or pink stilhetto's depending which box you ticked in another thread :-) ).
So, what seems obvious (increased probability by picking Box A only) is not clear as you feel that you can affect the prediction by being unpredictable :-) Sort of a double-bluff.
I think it boils down to 2 groups ??? (I reserve the right to be wrong :- ) :-
Group A (picking both) who's motto is that as the prediction has already been made then they can make a decision which is not affected by the prediction and which shows the better ev.
Group B (picking Box A) who feels that their future thoughts of what they would do have all been catered for and the end result is 90% taking into account double-bluffs, changes of mind etc etc (bit like picking a red ball out) so that this choice gives the better overall ev.
Most paradoxes seem to end like this with no definitive answer. I'm less sure than I was when I first looked at the question :-)
Quote: SwitchI think it boils down to 2 groups ??? (I reserve the right to be wrong :- ) :-
Group A (picking both) who's motto is that as the prediction has already been made then they can make a decision which is not affected by the prediction and which shows the better ev.
Group B (picking Box A) who feels that their future thoughts of what they would do have all been catered for and the end result is 90% taking into account double-bluffs, changes of mind etc etc (bit like picking a red ball out) so that this choice gives the better overall ev.
By my reading, the problem statement clearly indicates Group B, and as a result picking just box A is the correct choice. I basically read it as "whatever you're thinking about, 24 hours ago NB probably knew it and acted accordingly". So the question boils down to whether, 24 hours ago, you wanted him to be acting accordingly because you were going to win $1000 or win $1000000. If you go for the whole pot, you'll only get it 10% of the time.
Quote: MathExtremistIf you go for the whole pot, you'll only get it 10% of the time.
But you only get to do it once, go for the sure thing. Don't even bother opening A, its empty.
Quote: MathExtremistBy my reading, the problem statement clearly indicates Group B, and as a result picking just box A is the correct choice. I basically read it as "whatever you're thinking about, 24 hours ago NB probably knew it and acted accordingly". So the question boils down to whether, 24 hours ago, you wanted him to be acting accordingly because you were going to win $1000 or win $1000000. If you go for the whole pot, you'll only get it 10% of the time.
However, if you're the type to box A, then you have possessed such tells 24 hours ago, and the Newcomb Being would have picked up on that. Couldn't it be said that there is no harm in changing your thinking now, and take the extra $1000 in box B. If that isn't convincing, again, the money has been sitting there for 24 hours already. How can what you are thinking now change what is in the boxes?
If the Newcomb Being has a success rate on predictions of 100%. Do you choose box A only, or both?
Then apply it to the success rate on predictions of about 90%.
Quote: WizardHowever, if you're the type to box A, then you have possessed such tells 24 hours ago, and the Newcomb Being would have picked up on that. Couldn't it be said that there is no harm in changing your thinking now, and take the extra $1000 in box B. If that isn't convincing, again, the money has been sitting there for 24 hours already. How can what you are thinking now change what is in the boxes?
But he knew I'd do that with 90% probability, no? In other words, if that was going to be my thinking at time t+24, then 9 out of 10 times at time t=0 NB knows that and puts nothing in the box. I think the crux of the issue is that the NB is 90% right about my thinking even before I have the thoughts. Or 90% omniscient, or something.
I think the point is that, if you buy into the premise that NB can do what he's purported to be able to do, then you have to buy into the premise that if you even consider changing your mind, NB knew you would consider changing your mind, etc...
Counterfactual: Take the 90% away - make it 100%, and make NB = an all-knowing supreme omniscience with time-spanning knowledge. Or just call him God. If God is running the game, do you pick box A or both boxes?
Quote: WizardNow you have grounds to believe that in such matters, the Newcomb Being has a success rate on predictions of about 90%.
I vote to take only Box A. If NB is 90% correct in his predictions, then by my thinking that means that there is a 90% chance he correctly predicted my choice of only Box A, thus a 90% chance I get $1,001,000 out of the deal.
Quote: WizardCan I have an explanation from somebody who would choose both boxes?
Its not like the Monty Hall problem, when switching doors would always be right. You have to take both.
1) It's Monday. You know that on Wednesday you'll meet the NB and he'll give you the choice. That means on Tuesday he'll put the money in the boxes.
2) You call a friend over and tell him "be at the park at 12:00 noon on Wednesday, and at 12:05 scream out 'PICK BOTH BOXES!'
3) Then, you write out a Post-It note that says "Only Box A" and stick it to your mirror.
4) Then you have your friend club you over the back of the head and put you in bed.
5) You wake up Tuesday with a splitting headache and no memory of the day before, nor do you remember that you'll meet the NB the next day. You see the Post-It note and have no idea what it means, but you make a note of it.
6) Wednesday rolls around. You're at the park at noon playing Frisbee with friends when the NB approaches you with the boxes and gives you the choice. You remember the Post-It note telling you to pick only box A, and since that agrees with your normal sense of the puzzle (to maximize your EV), you start to announce your decision, when
7) Your friend screams out "PICK BOTH BOXES", which you do, and you find both have money within.
Of course, this only works if you know in advance you'll meet the NB, and then again only if he doesn't have the ability to see into your past. Also, it might not be worth taking a whack on the head for only an extra $1000.
Quote: EvenBobIts not like the Monty Hall problem, when switching doors would always be right. You have to take both.
How would you counter MathExtremist's argument?
Quote: WizardHow would you counter MathExtremist's argument?
I can't counter it. My philosophy is never gamble when you don't have to, always take the sure thing. Never play unless you think you've got the best of it.
Quote: WizardA being, who you believe to have superior predictive powers, makes you an offer. Before you are two boxes, identical in appearance. Within one box, let us call it 'Box A', there is either one million dollars or nothing. Within the other box, 'Box B', there is definitely one thousand dollars.
The being, whom in deference to tradition we will call the Newcomb Being, will allow you to either take both boxes, or just Box A.
Twenty-four hours ago, the Newcomb being made a prediction about what you would choose. At this point, it placed the money in the boxes. If the Newcomb Being predicted that you would take both boxes, it left nothing for you in Box A and the $1000 in Box B. If the Newcomb Being predicted that you would only take Box A, it would leave the million dollars in it, as well as the thousand dollars in Box B. If the being predicts you will base your choice on a random event (like flipping a coin), it will leave Box A empty.
There are four possibilities:
I choose box A only, and the Newcomb predicted that correctly.
I choose box A only, and the Newcomb predicted that incorrectly.
I choose both boxes, and the Newcomb predicted that correctly.
I choose both boxes, and the Newcomb predicted that incorrectly.
If I choose Box A, 90% of the time, the Newcomb predicted correctly, and I get $1,000,000. 10% of the time, the Newcomb predicted incorrectly, and I get nothing.
If I choose both boxes, 90% of the time, the Newcomb predicted correctly, and I get $1,000. 10% of the time, the Newcomb predicted incorrectly, and I get $1,001,000.
The EV of choosing only Box A is therefore +$900,000, and the EV of choosing both boxes is just a tad over $100,000. Therefore, choosing Box A is by far the superior option.
Quote: mkl654321There are four possibilities:
I choose box A only, and the Newcomb predicted that correctly.
I choose box A only, and the Newcomb predicted that incorrectly.
I choose both boxes, and the Newcomb predicted that correctly.
I choose both boxes, and the Newcomb predicted that incorrectly.
If I choose Box A, 90% of the time, the Newcomb predicted correctly, and I get $1,000,000. 10% of the time, the Newcomb predicted incorrectly, and I get nothing.
If I choose both boxes, 90% of the time, the Newcomb predicted correctly, and I get $1,000. 10% of the time, the Newcomb predicted incorrectly, and I get $1,001,000.
The EV of choosing only Box A is therefore +$900,000, and the EV of choosing both boxes is just a tad over $100,000. Therefore, choosing Box A is by far the superior option.
Wow, thats almost word for word what the solution says! You must be a Newcomb..
If the Newcomb Being has a success rate on predictions of 100%.
Do you choose box A only, or both?
then apply it to the success rate on predictions of about 90%.
To pick A only sounds like an endorsement of time travel. That a decision you make now will affect your actions 24 hours ago. You can't help what Newcomb observed 24 hours ago. His decision has been sealed. The question is would you prefer A or A+$1000?
This answer (and maybe I'm just playing the devil's advocate) is the same even with god/100% Newcomb.
I yam what I yam. -- Popeye
Quote: WizardThe expected value argument is an answer, but I wouldn't call it the solution. Recall also how the expected value argument leads us to a paradox with the Two Envelope problem.
To pick A only sounds like an endorsement of time travel. That a decision you make now will affect your actions 24 hours ago. You can't help what Newcomb observed 24 hours ago. His decision has been sealed. The question is would you prefer A or A+$1000?
This answer (and maybe I'm just playing the devil's advocate) is the same even with god/100% Newcomb.
I yam what I yam. -- Popeye
Not of time travel, of prescience -- but that's the foundation of the question. The decision you make now won't affect *your* actions 24 hours ago, they'll affect the NB's actions 24 hours ago. Or they did - because he already knew what you'd pick. Bumping up the rate to 100% makes it even more obvious because it utterly eliminates two of the four possibilities (ref: many-worlds hypothesis). Either NB foretold that you'll pick both and you get $1000, or NB foretold that you'll pick A only and you get $1000000. There is no other option, no other possibility. It effectively turns the question into "If you pick both, you get $1000; if you pick just A, you get $1000000. What's your pick?" Phrased like that, it's an easy question. It also fits in with our standard, time-forward sense of causality. But if this NB has predictive skills superior to us puny humans, then he knows what you're going to pick before you consider it, because he knows *how* you're going to consider it -- including any changing-of-mind that you might do.
It may sound like a denial of free will, but it's not -- it's your free will to make the choice. It's just the result of the choice that's been predetermined. But if you're dealing with a "being with superior predictive powers" (or God), then there's no paradox in suggesting that the result of that choice has to be constant across universes. To use your phrasing, the question is whether you would prefer A (where A = 1000000) or A+1000 (where A = 0). The chances of A being anything other than those values are only 10% in the original statement, and 0 in the alternative.
On the other hand, if someone is having a hard time accepting the prescience of the NB, even at 90%, then I can understand the discomfort in getting to the conclusion above. To that, I can only recommend reading more sci-fi. It's good for stretching one's ability to suspend disbelief. :)
For example, I think there is a difference if I state to a friend "I will predict what color shoes The Wizard will wear tomorrow" (without telling The Wizard), compared with telling The Wizard that I can predict what color shoes he will wear tomorrow. The latter allows The Wizard to be more obscure with his choice of shoes if he wishes to try and make himself more unpredictable.
An interesting twist on this would be if you were on that game show (called 'Golden Balls in the UK, not sure in US), where the last 2 contestants had to decide what to do with the prize money. Both contestants had to decide whether to 'Steal' or 'Share' the prize money, let's say it's $200,000. If you both 'steal' then nobody gets any money, if one 'steals' and the other 'shares' then the 'stealer' gets all the money and if you both 'share' then you both get 1/2 the money each.
So, if you had Newcomb as your fellow contestant and was told that he would choose what he predicted that you would choose (say 90% success rate again) then would you 'Steal' or 'Share' ?
Generally, the 'correct' way to play the game is to always 'steal' but now it would appear that mathematically you should 'share' if you was facing the Newcomb ... BUT ... the Newcomb would know this (or would he ?) so, by 'stealing' you get all the money so that's the right choice ...BUT ... the Newcomb would also predict this too (maybe?) :-) :-) ...etc etc ... STEAL or SHARE ? :-)
Quote: Switch
An interesting twist on this would be if you were on that game show (called 'Golden Balls in the UK, not sure in US),
If you appear on that show, do you only win if you have golden balls?
Quote: SwitchAn interesting twist on this would be if you were on that game show (called 'Golden Balls in the UK, not sure in US), where the last 2 contestants had to decide what to do with the prize money. Both contestants had to decide whether to 'Steal' or 'Share' the prize money, let's say it's $200,000. If you both 'steal' then nobody gets any money, if one 'steals' and the other 'shares' then the 'stealer' gets all the money and if you both 'share' then you both get 1/2 the money each.
Good point. That show is known as Friend or Foe in the U.S., and has been discussed here before. I agree, that if you could, the perfect choice (capturing both altruism and revenge) on that show would be "I pick what you pick," which, of course, is not an option. Said answer was given on The Apprentice once, but in a different kind of situation.
Quote: WizardThe expected value argument is an answer, but I wouldn't call it the solution. Recall also how the expected value argument leads us to a paradox with the Two Envelope problem.
To pick A only sounds like an endorsement of time travel. That a decision you make now will affect your actions 24 hours ago. You can't help what Newcomb observed 24 hours ago. His decision has been sealed. The question is would you prefer A or A+$1000?
This answer (and maybe I'm just playing the devil's advocate) is the same even with god/100% Newcomb.
I yam what I yam. -- Popeye
With no explanation for the Newcomb's 90% prediction accuracy rate, there IS no solution, only an answer.
Quote: mkl654321If you appear on that show, do you only win if you have golden balls?
I think that some of the contestants had them removed by the audience after they stole from their poor and honest female opponents :-)
I don't think my answer depends on whether the sage's predictive ability is 90%, 10%, 0.05%, or 99.99% correct. What is important to me is that he/she/it acted on the prediction two days ago. Now it's time for me to actually make my decision and act on it myself. If -- and it may be a very big if - if I have free will in my choice, then my decision will determine whether the prediction actually was correct or not, but it cannot change what action was taken with the money and boxes two days ago. I cannot do worse now by taking both boxes than by taking just one, because my actual decision does not retroactively change a prediction and action of two days ago, only the correctness of the prediction.
On the other hand, if I do not have free will, if my decisions, choices, and actions are predestined by fate, then what the heck -- I don't need to analyze it, I just watch myself and see what I do. (By the way, for those Presbyterians who hold predestination as a tenet of their faith and might initially be offended by that comment, remember that I am using the term here as the equivalent of fatalism, a belief that each and every individual action is set in advance, which is not a part of that faith.)
Quote: DocI see that most of my thinking has already been reported, endorsed, and discredited all at the same time. Being late to the post, I'll offer my say anyway.
I don't think my answer depends on whether the sage's predictive ability is 90%, 10%, 0.05%, or 99.99% correct.
Don't you think it matters whether the sage is at least 50% correct or not?
No, not at the time I make my choice. Whatever the prediction was, the money in the boxes is already set, and my decision can no longer change that. I might as well take all I can get, whether it is a whole bunch or just a piddlin. It certainly does matter what the sage thought my choice would be, but no matter how good or bad the predictive ability may be, my actual choice now cannot change a two-day-old prediction/action. Note: I am speaking in terms of reality, not in the SciFi terms of time travel or non-sequential time experiences and such.Quote: mkl654321Don't you think it matters whether the sage is at least 50% correct or not?
Quote: mkl654321Don't you think it matters whether the sage is at least 50% correct or not?
Yes, the % does matters.
Think about it this way:
If the Newcomb Being has a success rate on predictions of 100%. Do you choose box A only, or both?
In this case the Newcomb knows 100% what you are going to choose.
As long as you choose box A you will get the million dollars (it don't matter why you choose box A).
And if you choose box A & B (it don't matter why you choose box A & B) you will get $1000.
Then use the formula for success rate on predictions of 90%.
Now, If the Newcomb Being has a success rate on predictions of 0%.
Do you choose box A only, or both?
why and at what % will you get the million dollars + $1000?
If the being has a 100% success rate at prediction and made the prediction of my actions two days in advance, then I do not actually have any freedom of choice in the matter. The prediction itself is already set, and I must choose what the being predicted. (My earlier comment was that I should just watch and see what I do.) What would happen if I were somehow to do the opposite? It just can't be, under this scenario.Quote: MrCasinoGamesYes, the % does matters.
Think about it this way:
If the Newcomb Being has a success rate on predictions of 100%. ...
This is, of course, the absurdity that arises from an assumption of 100% accurate predictions of future actions by a person. Such an assumption makes the entire problem absurd. Unless, of course, you assume that the being actually controls the future by making predictions that must come true, but I have already stated that I am speaking of the world of reality instead of SciFi. In reality, the person choosing the box or boxes is in control at that moment -- their actions can make the prediction correct or wrong (and affect the being's percentage), but they cannot influence how much money was placed where two days earlier.
Quote: DocIf the being has a 100% success rate at prediction and made the prediction of my actions two days in advance, then I do not actually have any freedom of choice in the matter. The prediction itself is already set, and I must choose what the being predicted. (My earlier comment was that I should just watch and see what I do.) What would happen if I were somehow to do the opposite? It just can't be, under this scenario.
This is, of course, the absurdity that arises from an assumption of 100% accurate predictions of future actions by a person. Such an assumption makes the entire problem absurd. Unless, of course, you assume that the being actually controls the future by making predictions that must come true, but I have already stated that I am speaking of the world of reality instead of SciFi. In reality, the person choosing the box or boxes is in control at that moment -- their actions can make the prediction correct or wrong (and affect the being's percentage), but they cannot influence how much money was placed where two days earlier.
The being wouldn't be a predictor if his predictions have a success rate of 100%. He would be a seer. He sees what you are going to do before you do it, he does not influence you in any way what so ever. For you not to have a choice in the matter, he would have to be a higher being. You choose yourself. If you choose the box or boxes he predicted, he wouldn't have predicted it, he would have SEEN the future correctly because his prediction are 100%, instead of influence your choice. And 100% accurate predictions are not an assumption, they are fact. SO, how can such a FACT make the entire problem absurd. He see's things that will come true, not because he decides them to. Also, his actions cant make the predictions wrong because a 100% success rate on predictions would not be a prediction at all, but future fact.
Quote: blackjackgoldenQuote: DocIf the being has a 100% success rate at prediction and made the prediction of my actions two days in advance, then I do not actually have any freedom of choice in the matter. The prediction itself is already set, and I must choose what the being predicted. (My earlier comment was that I should just watch and see what I do.) What would happen if I were somehow to do the opposite? It just can't be, under this scenario.
This is, of course, the absurdity that arises from an assumption of 100% accurate predictions of future actions by a person. Such an assumption makes the entire problem absurd. Unless, of course, you assume that the being actually controls the future by making predictions that must come true, but I have already stated that I am speaking of the world of reality instead of SciFi. In reality, the person choosing the box or boxes is in control at that moment -- their actions can make the prediction correct or wrong (and affect the being's percentage), but they cannot influence how much money was placed where two days earlier.
The being wouldn't be a predictor if his predictions have a success rate of 100%. He would be a seer. He sees what you are going to do before you do, it he does not influence you in any way what so ever. For you not to have a choice in the matter he would have to be a higher being. You choose yourself. If you choose the box or boxes he predicted he wouldnt have predicted it he would have SEEN the future correctly because his prediction are 100% instead of influence your choice. And 100% accurate predictions are not an assumption, they are fact. SO how can such a FACT make the entire problem absurd. He see's things that will come true not because he decides them to. Also his actions cant make the predictions wrong because a 100% succees rate on predictions would not be a prediction at all but future fact.
There for 90% is still a FACT.
and the % does matters.
If, during the two days in between, the being were to tell me what he saw, could I not do anything different? If I could change the future (as seen by him), how would I go about doing something different, and what would be the repercussions? (Of course, this would imply that he is not 100% accurate.)
If it is indeed not possible for me to act differently from what he "saw", then I guess I will still just watch and see for myself in real time just what I do, since I no longer have any free will. No need for me to do any analysis of what I "should" do.
But I don't think this fits with reality, and I contend that the 100% accurate prediction or seeing of the future renders the entire analysis absurd and meaningless. In that sense, the percentage actually is important: 100% (or 0%) results in a meaningless problem, and anything in between gives a situation where my actions affect the being's accuracy rate, and one in which it is better for me to choose both boxes.
Quote: DocIf he "sees" something (with 100% clarity and accuracy) two days in advance, does that mean my final action is definitely determined at that time? I no longer have any possibility of doing the opposite?
If, during the two days in between, the being were to tell me what he saw, could I not do anything different? If I could change the future (as seen by him), how would I go about doing something different, and what would be the repercussions? (Of course, this would imply that he is not 100% accurate.)
If it is indeed not possible for me to act differently from what he "saw", then I guess I will still just watch and see for myself in real time just what I do, since I no longer have any free will. No need for me to do any analysis of what I "should" do.
But I don't think this fits with reality, and I contend that the 100% accurate prediction or seeing of the future renders the entire analysis absurd and meaningless. In that sense, the percentage actually is important: 100% (or 0%) results in a meaningless problem, and anything in between gives a situation where my actions affect the being's accuracy rate, and one in which it is better for me to choose both boxes.
I think what renders this whole question somewhat moot is that trying to say that a successful prediction rate of 100% essentially FORCES the player's hand constructs a tautology. If he predicted what I would choose, and he's deadly accurate, then my choice doesn't matter, therefore I have no choice, therefore he didn't really predict my CHOICE, since I never had any choice in the first place...and round and round.
Quote: mkl654321I think what renders this whole question somewhat moot is that trying to say that a successful prediction rate of 100% essentially FORCES the player's hand constructs a tautology. If he predicted what I would choose, and he's deadly accurate, then my choice doesn't matter, therefore I have no choice, therefore he didn't really predict my CHOICE, since I never had any choice in the first place...and round and round.
What comes to my mind is a post about a month ago that asked (and I'm paraphrasing), "There is an urn with some combination of black and white marbles. You draw 100, with replacement, and they are all black. What is the probability the urn is all black?"
I was very unsatisfied with the answer given. My point being that just because Newcomb was 100% in the past, doesn't mean he will get YOU right. This still seems to me like a choice between $A and $(A+1000).
mkl, I suspect that you and I can agree that a guaranteed prediction/seeing accuracy of either 100% or 0% makes the problem analysis impossible or absurd because it predetermines exactly what action will be taken and eliminates the “choice”. The important issue is when the probability of an accurate prediction is 0<P<1. In that case, the probability cannot be a fixed number if there are additional events: a basketball player who has hit 80 of 100 free thows is considered an 80% shooter, but what happens on his very next free throw? His percentage must go either above or below 80%. The same is true of the being in this problem. When the chooser makes a final selection, then that selection establishes the prediction as correct or incorrect, and the percentage adjusts at least slightly. It is the probability that changes based on the final choice of box(es), not the action of where the money was placed two days earlier.
OK, so this next paragraph is off-topic, but it relates to the assumption that the being’s prediction rate could be fixed in its accuracy. I am reminded of a case study we covered in a graduate management course I took long ago. The company had determined that indirect costs were 35% of direct manufacturing costs, and they were allocated to each product at that rate to obtain a fully-allocated cost of manufacturing each product. That is common in manufacturing industries. In the case study, a cost reduction team had determined a way to reduce the direct manufacturing cost of one of the products by X dollars per unit. They felt that implementation of their plan should be credited with saving 1.35X dollars per unit, to represent savings of fully-allocated costs. Something close to the 1.35X savings was necessary in order to justify their changes in the process. One of the student groups fell for that notion, which I had actually seen people try to do in industry. I pointed out that the extra 0.35X savings could only be claimed if they also implemented some actual improvement in their indirect costs. Otherwise, saving on direct costs alone just increases the indirect cost percentage.
The way I described that experiment from the one time I tried to teach statistics was that the samples were not all black or all white, but were indeed one color or the other. After having the students take numerous samples, I asked the probability that the next marble drawn would be black, white, or red. I admit I don't remember exactly what equations we used, but I remember that they showed a 0% probability of the marble being red and a 0% uncertainty in that estimate. Then I revealed the lone red marble in the bag, and we were left with a bit of a paradox.Quote: WizardWhat comes to my mind is a post about a month ago that asked (and I'm paraphrasing), "There is an urn with some combination of black and white marbles. You draw 100, with replacement, and they are all black. What is the probability the urn is all black?" ...
