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Some Cute Philosophical Puzzles About Rational Choice
MrMister
Jesus dying on the cross in pain? Morally better than us. One has to go "all in".Registered User regular
Jesus dying on the cross in pain? Morally better than us. One has to go "all in".Registered User regular
This thread is as it's title suggests. Rather than give a treatise about some obscure and fairly technical topic, as I did last time with Frege-Geach, I figure I'd canvas some fun philosophical puzzles. Each is, in some loose sense, related to rational choice; and although they are all somewhat fanciful, the consequences of the answers one gives will have implications for what we take to be more normal instances of rational reasoning. They are also all scenarios that embody questions. What should someone do under these odd circumstances? So it will be interesting to see what people have to say.
Without further ado:
The Ever-Better Bottle of Wine:
Newcombe's Box:
Kavka's Toxin:
And, finally:
Sleeping Beauty:
In case you're curious, my answers:
I can get into it a little more over the answers, although I don't have particularly extensive arguments--these are not my area of expertise. But first I'm curious what people think. Feel free to discuss with abandon!
Edit: per (what seems like good) advice, I'm splitting the scenarios and explanations.
Without further ado:
The Ever-Better Bottle of Wine:
Suppose that you're immortal, and, furthermore, that time is infinite. Now also suppose that you have an ever-better bottle of wine, which is a bottle of wine such that each day you let it age it gets twice as good as it was the day before. When should you drink it?
The puzzling thing about this scenario is that it seems that you always have a reason to prolong your gratification: after all, tomorrow it will be twice as good! Yet if you always so choose, then it follows that you will never drink the wine at all, and thus literally get the minimum possible enjoyment. So that seems bad. Then when should you drink it?
This puzzle is fanciful in of that all three conditions are obviously false in the actual world: no one is immortal, time is not (as far as I understand) infinite, nor do any bottles of wine get ever-better. But the puzzle is not actually that far from certain choices we might have to make with regards to distributing goods across future generations. The more we save and scrimp, the better their (and their children's) lives will be in perpetuity; but if at each stage they save and scrimp too, then... well, I imagine you see the parallel.
This puzzle is fanciful in of that all three conditions are obviously false in the actual world: no one is immortal, time is not (as far as I understand) infinite, nor do any bottles of wine get ever-better. But the puzzle is not actually that far from certain choices we might have to make with regards to distributing goods across future generations. The more we save and scrimp, the better their (and their children's) lives will be in perpetuity; but if at each stage they save and scrimp too, then... well, I imagine you see the parallel.
Newcombe's Box:
Suppose there is a game you can play that goes as follows: a machine puts out two boxes, each containing money. You can then choose to take either box A, or both box A and box B.
Before you play, the machine decides what money to put in both boxes. If it predicts that you are only going to take box A, then it puts a million dollars in box A and two million dollars in box B. On the other hand, if it predicts that you are going to take both boxes, then it puts a hundred dollars in box A and two hundred dollars in box B. Suppose furthermore that the machine, as a matter of historical fact, is extremely good at predicting what people are going to do. It gets it right in the vast majority of cases.
So now you're standing in front of the machine. Do you choose to take just box A, or both boxes?
Before you play, the machine decides what money to put in both boxes. If it predicts that you are only going to take box A, then it puts a million dollars in box A and two million dollars in box B. On the other hand, if it predicts that you are going to take both boxes, then it puts a hundred dollars in box A and two hundred dollars in box B. Suppose furthermore that the machine, as a matter of historical fact, is extremely good at predicting what people are going to do. It gets it right in the vast majority of cases.
So now you're standing in front of the machine. Do you choose to take just box A, or both boxes?
This is puzzling because no matter what, there's more money in both boxes than there is in just box A. And yet, it seems like the people who one-box are going to walk away with more money, and, hence, that it's rational to one-box.
Kavka's Toxin:
"An eccentric billionaire places before you a vial of toxin that, if you drink it, will make you painfully ill for a day, but will not threaten your life or have any lasting effects. The billionaire will pay you one million dollars tomorrow morning if, at midnight tonight, you intend to drink the toxin tomorrow afternoon. He emphasizes that you need not drink the toxin to receive the money; in fact, the money will already be in your bank account hours before the time for drinking it arrives, if you succeed. All you have to do is. . . intend at midnight tonight to drink the stuff tomorrow afternoon. You are perfectly free to change your mind after receiving the money and not drink the toxin" (from the original paper, via wikipedia)
What do you do? What can you do?
What do you do? What can you do?
Can you sincerely intend to drink the toxin, knowing full well that when the appointed hour arrives you'll have no need to? Can you sincerely intend to drink it even knowing that, as a factual matter, you simply won't? If you can sincerely intend to drink it, can you stop intending just as soon as you get the money? None of these questions have an obvious answer.
And, finally:
Sleeping Beauty:
This one has gotten a lot of attention recently.
Suppose that experimenters tell a woman--call her Sleeping Beauty--the following: we are going to put you to sleep. Then we are going to flip a fair coin. If the coin comes up heads, then we'll wake you up once and the experiment will be over. If the coin comes up tails, however, we'll wake you up once, then put you back to sleep again, then wake you up a second time before ending the experiment. When you are awoken the second time, you will have no memory whatsoever of your first waking; this will be a side effect of the drugs we administer to put you back to sleep.
Knowing full well this experimental set-up, Sleeping Beauty gets put under. Later on, she has the experience of being awakened, somewhat groggily, by the scientists. When she has her wits about her, they ask her the following question:
"What are the chances that the fair coin we said we'd flip landed heads?" What should Sleeping Beauty say?
Suppose that experimenters tell a woman--call her Sleeping Beauty--the following: we are going to put you to sleep. Then we are going to flip a fair coin. If the coin comes up heads, then we'll wake you up once and the experiment will be over. If the coin comes up tails, however, we'll wake you up once, then put you back to sleep again, then wake you up a second time before ending the experiment. When you are awoken the second time, you will have no memory whatsoever of your first waking; this will be a side effect of the drugs we administer to put you back to sleep.
Knowing full well this experimental set-up, Sleeping Beauty gets put under. Later on, she has the experience of being awakened, somewhat groggily, by the scientists. When she has her wits about her, they ask her the following question:
"What are the chances that the fair coin we said we'd flip landed heads?" What should Sleeping Beauty say?
Some people say one half. It's a fair coin, and the chance that a fair coin lands heads is one-half. Certainly, the chances she should have assigned to the coin landing heads before starting the experiment should have been one half. But it's not like she learned anything new about it or what it did by going to sleep and waking up. So she should keep that initial judgment. Others say one third. When she wakes up, she could be in any one of three indiscernible situations: she could be in the only waking on the heads branch, or in the first waking on the tails branch, or in the second waking on the tales branch. Since these situations are indiscernible for her, she should assign each of them an equal probability. But if that's so, then there's only a one-out-of-three chance she's on the heads branch: so, one-third.
In case you're curious, my answers:
Ever-Better Wine: it is in principle rationally permissible to drink it at any arbitrary time t, although for pragmatic reasons one might as well drink it straightaway
Newcombe: I'm a one-boxer
Kavka: Undecided
Sleeping Beauty: I'm a thirder
Newcombe: I'm a one-boxer
Kavka: Undecided
Sleeping Beauty: I'm a thirder
I can get into it a little more over the answers, although I don't have particularly extensive arguments--these are not my area of expertise. But first I'm curious what people think. Feel free to discuss with abandon!
Edit: per (what seems like good) advice, I'm splitting the scenarios and explanations.
MrMister on
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Box) If I know the rules, Box A every time.
Toxin) Drink it. Otherwise, you're not truely sincere.
Sleeping Beauty) 50%. You have two outcomes, wake once to heads, or wake twice to tails. The fact that you have two wakings for tails makes no difference to the initial condition. Both of the wakings are covered by one flip of tails.
1. You never drink the wine. You start enjoying another alcohol and convince yourself that wine doesn't exist.
2. You do whatever you want because "extremely" good doesn't accurately provide information.
3. Just take the thing. The condition is irrelevant upon decision to simply take it.
4. 18.3% or less!
I am a two boxer. By the time I am opening boxes, the computer has already decided how much to give me, so I might as well open both. If in one particular instance I would not choose to open both, that doesn't really change the decision of the computer or whatever it is basing it's prediction on.
I suppose I could convince myself that there is some sort of moral need to do what I say absent obvious external motivation, and I don't know... like when I make pledges to make donations I do actually have the intent to give, even though once I am dealing with a bit of paper some organization sent me, rather than a person, that doesn't always occur.
I'm pretty firmly in the 1/2 camp. "Since these situations are indiscernible for her, she should assign each of them an equal probability. But if that's so, then there's only a one-out-of-three chance she's on the heads branch: so, one-third." doesn't seem to actually follow. Ignorance or knowledge of math or results doesn't really change the probability, so long at the choice is only being made one time(compared to the Monty Hall problem).
Boxes: If I knew all of what you posted, then I would take only Box A, of course. If I didn't then I would likely take both. The knowledge you provided is a very important element of that hypothetical, isn't it? Because then the two choices are not between definitely 1 million dollars or possibly 3 million dollars, it's only between definitely 1 million dollars or definitely 300 dollars.
Toxin: Nope. I couldn't sincerely intend to do something harmful to myself if I know I will be rewarded equally by not harming myself.
Sleeping Beauty: I don't see how Sleeping Beauty could say anything but "one third" in the hypothetical as presented. "When you are awoken the second time, you will have no memory whatsoever of your first waking; this will be a side effect of the drugs we administer to put you back to sleep." From this, I infer that being put back after being awoken will erase the memory of being put to sleep and awoken for the second time after flipping tails. There is nothing to suggest that she would be incapable of communicating after being awoken the first time after flipping tails. So how could it be "one-half"?
Wine
Box
Toxin
Sleeping Beauty
edit: Yeah, I'm changing my answer to 50/50. If there's one flip, then the chance of the coin flip itself is equal. What that coin flip means to Sleeping Beauty is a different matter.
Not sure if this is intentional or not, but the problem as presented only indicates one coin was flipped, after the first time she was put to sleep. I'd say that's a pretty clear 50%.
EDIT: Beaten! Curse my tardiness.
2. One box - you're essentially trusting the machine to accurately read the future.
3. Possibly, to the extent that I can convince myself that I will drink it even if I don't have to.
4. I'm in the 50/50 camp. You know no more on waking than you knew before the coin was flipped. You describe it as three branches, but there are only two - the two 'tails' situations are just different time points on the same branch. That you wouldn't know which time point you were at in that branch shouldn't change your opinion about that branch's probability.
No! Plenty of things are indiscernible for us, but we should not assign them equal probability. A simple course in probability should answer this question.
the answer: The answer is 1/2. The answer is 1/2 in a whichever philosophical framework of statistics you want to work under. For a Bayesian they believe there is a 1/2 chance and waking up provides no new information so the answer is still 1/2. For a frequentist the answer is one half because if you repeat the experiment no matter how many times you do it half the time it will be heads and half the time it will be tails. For a "x" where x is the framework that assumes that there is an absolute real probability but i forgot what those people call themselves then the answer is 1/2 because the answer is 1/2.
The proper way to ask the question is this:
The reason this works is that the population of answers will be correct 2/3rds of time if you say heads because you're double sampling the heads probability
I too am undecided on the toxin, though it strikes me as an almost identical puzzle to Newcombe's Demon.
Sleeping Beauty I need to consider, because I am tired and it is maths.
Regarding the wine, I say at any power of 10 days, because binary, otherwise the same as MrMr.
I may have had as near an approximation of this experience as it is possible to have in the real world, at the Retzlaff vineyard in Livermore, CA. During a tour, they drew off a tiny plastic cup of some white wine that was still fermenting. It was... magically crisp, cold, just lightly alcoholic, a little sweet, a little fruity... purely amazing. A unique taste in every sense of the word... that flavor changes within days, and can only come from those specific grapes on those specific days. They will never exist again! I remember it wistfully and with a little regret, because it is the now impossible standard by which all other liquids are judged, and to which none can measure up. Literally, it is the best thing I have ever drank, and will likely remain so forever.
So, having done this once, I'd wait for a memorable occasion and drink the wine! Sure, you'll wonder forever what it might've been like if you waited another day, but that's the nature of life in general. It's better to craft a satisfying memory than to always be waiting for some mythical perfect time. The memory and the enjoyment are, for me, more than enough to counterbalance the wondering about what might have been.
The Box Problem:
It seems very difficult to beat the machine, so Just going for box A is the rational choice. The solution appears to rely on how much you know about the machine's track record, and well the machine knows you... and therefore you need to know yourself. If you're the sort of person who nearly always settles on the safe bet, especially when you've passed up many chances, two boxes seems best! Absent any knowledge of the machine, I'd say that by the time you're choosing, the machine has already filled the boxes. The choice is between $300 and $3m if you don't know that it's possible to try and "beat" the machine's perception of you. So both boxes regardless. If you do know that it's possible to throw off the machine, look back with an honest eye to the sorts of choices you make, and do the opposite.
The Poison Bottle:
Foreknowledge that you will be rewarded regardless breaks the ability to honestly intend to do anything, or at least it seems to. This one, of all the problems, seems to have the most frequent analogs in real life! Paying bills, doing unsolicited favors or buying gifts for a spouse or a relative, even getting your oil changed. Maybe the hardest question of the four, IMHO.
Sleeping Beauty:
This seems to be a variation of the Monty Hall problem. Just as in Monty Hall, you have two sets: the set a, of being awakened once, and the set b, of being awakened twice. Whether you are a member of set a or b is determined by a single coin flip. Everything after the flip is meaningless, just as when Monty Hall's lovely assistant opens a goat door for you. The flip is the point of determination, so the chance is 1/2.
Boxes: Box A. I'm not a gambler, which you don't even need to be a godlike box-picking-predictor to know.
Vial: I'd drink it at the appointed time, even knowing I didn't have to; that's a good payoff for one day of feeling like shit and I'd prefer to be honest. I feel like shit for more than 24 hours every year in cold and flu season for free.
Sleeping Beauty: As others have said, the question is not "What are the odds that this is the final time we are waking you?" or similar, they're "What are the odds that the coin flip was tails?" which is always 50/50.
Drinking the wine therefore condemns you to regret.
Box: My plan would be to determine which option I choose with a coin toss.
It's always more reasonable to choose both boxes because they always contain more money than Box A. Knowing that, all I can do is increase the odds that the machine will incorrectly guess my decision by removing my rational conclusion about the two boxes from the decision making process.
Since the machine's odds of predicting human behavior are extremely high, that means transferring the decision to a harder to predict variable like the outcome of a coin toss.
This should give me a 50/50 shot on getting a million dollars or more, which is much higher than what I'd get by picking both boxes in order to get all of the money available on the table.
Toxin: I'd write a $100,000 check as payment to someone who will force me to consume the poison, but only if that person is willing to be tried for poisoning me afterwards. The person will of course not poison me until after he has received the money, and he can only receive the money if the millionaire has fulfilled his end of the bargain.
As a result, I've tied taking the poison to receiving the money rather than my rational faculties, which raises my odds of taking the poison to 100% if the millionaire is fair.
Sleeping Beauty: 1/3
https://twitter.com/Hooraydiation
More pedantically, assuming that the bottle has a volume of wine of 1, and an initial quality of 1, and that the quantity increases the enjoyment in a proportional relationship, you could just drink one tenth of the amount currently in the bottle at a certain interval and gain an ever-increasing amount of enjoyment at each interval. When to start and the interval are both still arbitrary, plus at some point you'll be drinking a single particle that doesn't count as wine anymore, I guess...
Well, what if you only drink half the wine at each sitting? Then half of that, then half of that...
Infinite wine to go with your immortality!
One box, definitely. If the machine is that good at predicting what people will do, then what's to make me think that I could 'beat' it?
The toxin: I don't think that I could actually convince myself of this, knowing that I'd just toss the poison. It's a million dollars; just drink it!
As for sleeping beauty, I don't believe her current state has anything to do with the original coin flip. If the question is "what are the chances that the first coin landed heads?" then the answer is "50%". If the question is "did the coin land heads", then your answer can be predicated on the three indiscernible states.
Even if you change it to just being "a whole more delicious" each day, I'm still not sure what that would mean. Mostly because I don't think there's any such thing as "infinite deliciousness". At some point, the wine will reach a point where it just can't get any better, or if it did you wouldn't even notice it.
So I think there's two valid answers. 1) Drink it whenever, it doesn't matter, because at some point it'll stop getting noticeably better. Or 2) Never drink it, because the intellectual satisfaction that you get from having this magical bottle of wine will be so much more than the brief satisfaction of actually drinking it.
that was what i said, only a tenth instead of half because i am not a lush :P
really i'd probably just drink it after a year, because if it's 2^365 times better than it was to begin with, I think it's way past the point at which quality actually makes a difference for any normal person's palate, and you'll hit severely diminishing returns.
edit: also i agree with Goumindong re: how the question is phrased and the answers for those different phrasings for Sleeping Beauty.
Based on the assumption that the computer is guessing whether you'll one box or two, if it decides you'll two box and you pick one, you're out 1/3 or 2/3 of whatever is on the table. Why would you ever one box, knowing that?
Mmmmm....toasty.
This reminds me of the teacher who promises a pop-quiz will occur on one day in the semester, and that it will be a surprise when it happens. She can't do it on the last day of the semester, because everyone would know it had to be that day and then it wouldn't be a surprise. Since everyone knows she can't do it on the last day, then she can't do it on the 2nd to last day either, because again, it wouldn't be a surprise. And so on, meaning there is no day she can give the quiz that will be a surprise. Not the same quandary, of course, just reminiscent.
As for the wine, your opening assumptions already tell us that a "best wine" doesn't exist; much like a "largest integer" doesn't exist. So asking us to solve for the best wine is a contradiction. Drink it whenever you feel you've had a particularly bad day and need some cheer.
But yeah, it does make a good point about spending your whole life planning or preparing or saving, etc.
You say that it seems that no matter what, there is more in A+B than in just A. But that's not accurate. There are really four boxes, A1 ($1MM), A2 ($100), B1 ($2MM) and B2 ($200). And your only options are either A1 ($1MM) or A2 + B2 ($300). If this were a gambling problem, and I could play over and over, then there is a formula here that depends on the cost of a wager and exactly how "good" this machine is at predictions, such that it might be worth it to risk going for both boxes. But if we're talking about a single-shot at it, then yeah, the obvious choice is just box A.
But the simple meta-answer is that the use of a "future-predicting" machine, or particularly a "choice-predicting" machine, is a paradox along the lines of dividing by zero. It can be used in any number of hypotheticals to futz with the application of basic logic to the real world.
Forget boxes of money, whatever the choices are, what if I just asked the machine "which choice did you predict I'd make?" and then I commit to making the opposite choice of whatever the machine answered? The choice-prediction thing is its own conundrum.
Oh yeah, case-in-point. Anything involving predicting choices will open up a hypothetical paradox. In this case, the act of depositing money simply because of an honest intent to drink is, effectively, a choice-predicting calculation. The calculation to determine whether or not someone honestly intends to drink must take into consideration itself (that is, the calculation must take into consideration the results of calculation). This is because the result of this calculation will itself be a factor in the ultimate decision and thus must be a factor in determining an honest intent to drink. This is an infinte recursion with no base. It's like trying to put an equation into a cell in a spreadsheet which uses the value of the cell itself as a variable in the equation. Can't work.
I see both sides, and it's interesting. Here's where I think some of you are missing the trick: When she is awakened, there is (arguably) a probability as to whether or not this is the first or second time she has been awakened. She doesn't know if it's first or second, but there is a probability that it is the second time. If it is the second time, then that means that the coin was tails.
Knowledge affects probability evaulation. In this case, she has knowledge of being awakened, though not how many times.
In case no one else has said it yet, let me say where I think MrMr is going with Sleeping Beauty. Imagine that this experiment was repeated over and over, and each time someone was awakened, they were asked to state whether they thought the coin landed heads or tails. If someone says "heads", how likely are they to be correct? How about someone who says, "tails"? It would seem that if you are awakened, and asked to guess, you better guess tails. Odds are 2:1 that it was tails. I think there are some similar probability problems like this one.
In other words, there is a chance that it is the first awakening, at which point there is a 50/50 chance heads v. tails, but there is also a chance that it is the 2nd awakening, at which point there is a 100% probability of tails. Her awakening and being questioned is an event that could happen in one of three possible scenarios, 2 of which are tails and one of which is heads. When awakened, she knows that this could be the second time she's been awakened, which means tails could be a certainty. At no point could heads be a certainty. But, as others pointed out, there is only a 50% chance of the first scenario and a 50% chance of the latter two scenarios together... the odds of the coin flip are what they are. As with all probability problems, it depends on how you group the events... but this hypothetical makes it challenging.
In this case I think we have another interesting method of generating paradoxes - delving into the gap between conscious experience and abstract knowledge (i.e., math). Probability is a tricky sort of math because it leans heavily on a very philosophical issue - "knowledge." All math relies on philosophically tricky assumptions, but since questions of probability often deal with a subjective point-of-view, like what a subject knows and when he knows it, they can really bring that math-philosophy interaction into the forefront. By screwing around with consciousness and knowledge here (sleep, memory, etc.), we've screwed up our ability to apply questions of probability in a recognizably consistent manner.
The chances are always 50/50 for fair coins.
However, like the Monte Carlo problem, the intended result is demonstrated most easily by asking the correct question and extrapolating.
Scenario:
We will put you to sleep. We will flip a fair coin. If it comes down heads we will wake you up, ask you a question about the coin toss and the experiment is over. If it comes down tails, we will wake you up, ask you a question about the coin toss, and then put you back to sleep; then we will do this fifty more times before the experiment ends.
The question is "Did the coin land heads up?"
There is only one coin flip, but there are fifty-two awakenings, for fifty-one of which the correct answer is "No."
For the machine problem, you can screw with the system by introducing a coin flip; however good the machine is at predicting human nature (even if it predicts you will decide to use random chance to make the decision for you), it cannot predict random results. So the machine cannot predict your results since it knows you will be using random chance, and cannot decide what to put into the boxes. Buggery.
Aside from that, compare the choices against the other choice for the answer:
Pick A, it was A. You're down 2 million.
Pick A, it was A+B: you're down 200.
Pick A+B, it was A. You're up 2 million.
Pick A+B, it was A+B. You're up 200.
Choose both. The contents of the boxes are fixed, you're comparing your choice against your other possible choice.
The machine never comes into it unless we're given a statistic for its accuracy.
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In the sleeping beauty problem, assuming you are asking which coin it was, there's no compelling reason to answer heads at all. If you take the halfer position, heads and tails have an even chance, wheras if you take the thirder position, you're better off choosing tails. There's no rational reason to choose heads.
Lets put it another way:
Sleeping beauty will be put to sleep. After she is put to sleep, a D6 will be rolled. On a roll of 2-6, she will be woken up. However, on a roll of 1, she will be put back to sleep, and woken up again each day for 10 days. On each of these days she will have no memory as to any previous days, and will be asked whether the roll of the die was a 1.
Should sleeping beauty answer upon awakening that the roll was a 1 or not a 1?
That's not exactly true. If I flip a coin and you see that it lands tails, and then I ask you, "what are the odds that it landed tails?" the answer is 100%. The coin has already been flipped. Known vs. unknown. When Sleeping Beauty is awakened, she knows that the coin has already been flipped, and based on her knowledge that she has been awakened and is being asked to guess, she therefore knows that there is a 2/3 chance it was tails.
So if the question is, "did the coin land heads?" then the answer is "no" with a 2/3 likelihood of being correct. But if the question is, "what is the likelihood that the coin was heads?" then the answer is 50/50? That does not seem to be a consistent stance.
So you're giving yourself a 50% chance at $3MM vs. $300, whereas I'm taking a guaranteed $1MM. Technically the gambler math is on your side with that equation, since your average expected winnings is over $1.5MM, but I can't see many rational people taking that chance. The effect of winning $1MM is so much greater than $300 that the math is irrelevant. I'll take the $1MM.
Interesting question, but still, not a 1 gives her the best chance of being right. Sure, 1/6 times she'd be wrong 10 times in a row, but unless there's some punishment for being wrong, the best bet's playing the field.
If, however, there was a reason for her to care about being wrong 10 times in a row, then that would change things. What if the question was the same as yours, but with the added punishment of: Each time you are wrong, we'll shoot a puppy.
Then the risk / reward is different. She is more likely to be right if she plays the field, but if she is wrong and it's 1, they'll shoot 10 puppies.
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One's a question of a coin's potential, the other's a question of the entire problem.
Monte Carlo's the same; given two boxes and having chosen one, what's the chance that the prize is in the other? 50%. Then if you add the rest of the Monte Carlo framework on, the other box is favoured.
The random chance thing is just a demonstration of how you can foil any philosophical problem regarding determinism, rather than a serious point. ^^
Apart from her experience, the answer is either 100% or 0, not 50/50. The outcome has already been decided, after all.
She's ignorant of the outcome, though, so she can only give her answer as a reflection of her unique, limited understanding rather than as a prediction.
Of course, if she understands that heads is either 100 or 0, the answer that she'd give is 50/50. That's only if she understands the problem as a coin toss separate from it's context, though, and assigning any chance except for 100 and 0 relies on the context of her experience.
Considering that, I was encouraged to incorporate her experience of the probleminto the response. She perceives three separate possibilities and only one entailing Heads, so she says 1/3.
So basically it comes down to whether the problem calls for you to give the most accurate guess at the coin toss' outcome or, in my opinion, and answer that captures the Woman's experience within her response to the question of odds.
Basically, the problem is about determining what's more important, the woman or the coin.
https://twitter.com/Hooraydiation
I thinIt's not about whether it's better to pick one box, but rather how people convince themselves that one box is better because their incorrect conclusion yields a better outcome.
https://twitter.com/Hooraydiation
Though it would get big enough to be an inconvenience, in which case you drink it when it starts cluttering up the universe.
https://twitter.com/Hooraydiation
It's true, that variation would require infinite space, unless you're willing to imagine a bag-of-holding sort of situation.
Beat'd! Also yeah, on this volume-based variation it has to stop improving as soon as you open it (although, I suppose, on the original formulation it doesn't have to, so long as wine is discrete and hence the infinite division trick doesn't work).
If your conclusion is that one box will give you the better outcome and the prediction machine is sufficiently accurate then the conclusion isn't incorrect. Picking both boxes is only the better approach if you assume that the machine has a degree of inaccuracy, which we are not told that it does.
Paradox and so forth don't come into it. There was a thread last week or so about tests indicating that scientists could tell when someone had made a decision with regard to pressing a button up to seven seconds before they pressed it. That's plenty of lead time for a machine to swap the contents of some boxes. The moment that you decide to pick one box, it loads box A with $1mil. The moment you decide to pick two boxes, it loads both with $100 and $200. Taking the machine's capabilities as a given, without supposing some degree of failure, there is no reason to believe that you can benefit by taking both boxes.
As to the box/toxin problems, which are basically identical, I would go to bed with the intention of one-boxing/drinking toxin, and then I would wake up and do it. Whether or not your actions can alter reality at that point, I don't think it's possible to genuinely intend to do something with the idea that later you will genuinely not do it after all. In both cases, the system seems based on what you decide ahead of time. And if your intention is to cheat, the magical system we've invented should, reasonably, pick up on that. So just take one box and drink the damned poison.
As to the SB problem, I'd go with 1/2 for the reasons Goum suggested.
The problem with a bag of holding is that when you exceed the bag's maximum volume it will rupture and all of your wine will be lost in astral plane.
/geek