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Some Cute Philosophical Puzzles About Rational Choice
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Money boxes: Flip a coin. The odds then become heavily bent in your favor and you're able to avoid the endless loop of prediction. Alternative answer is to always choose both boxes because the only way to actually pick the million dollar choice is to not understand the implications of the question in the first place and then actually misunderstand your misunderstanding so your pick is counter-intuitive to your internal reasoning.
(edit: didn't remember the question properly. corrections are below.)
Toxin: Just mentally consider it to be the true cost of your financial gain. You endure one day of suffering for a lifetime of freedom. Alternative just develop a sense of honor for that day along with moral obigation and the problem solves itself.
The whole point is that your choice changed the amounts before the fact on account of the computer being so good at predicting.
Yeah. But now both boxes in front of me contain cash and I will take both. Me knowing how the system works before I make my choice is responsible for me probably not becoming a millionaire.
But the computer is NOT GUESSING. That's not the same thing.
This computer has shown historically with extreme accuracy that it can predict what people will take. The computer VERY LIKELY can easily predict when people will go in thinking they'll take one and decide to take two. If it didnt, it wouldnt be the machine described.
Think of it this way, my roommate works in a lab where something being off by .0001% in his data is going to fuck retardedly with it's results. So he must be extremely accurate.
Lets say extreme accuracy in this case is 99.999%, that's fair from the scientific perspective of an extremely accurate machine.
Whether or not the machine makes the choice before you, whether or not you think "Oh, two boxes are better.". This machine, by definition has given 99.999% of people who took one box a million dollars, and 99.999% of people who took 2 boxes 300 dollars.
Therefor taking one box gives you more money than in 99.999% percent of cases of taking two boxes.
You have to tell me how you can prove that you would walk in there, take two boxes, and have the computer not have determined you would do that. There is nothing in the problem to suggest it would not.
knows that you know that it knows what you would pick normally and will pick the opposite of your natural inclination
or
if it knows that you know that it knows what you would pick normally and will pick the opposite of the opposite of your natural inclination to beat the machine.
ad infintium.
By your logic the machine should know that you'd select one box. If it is able to accurately predict what you will select and that is your first argument you should be walking out of there with $100.
You get more money if your choice is a single box than if your choice is two boxes.
It accurately predicts I will take out one, I take out one, I have a million dollars versus three hundred.
I had to do a double take in case I was reading it backwards too ^^;
I'm just chillin with the box question because sleeping beauty is a whole bag of worms that I don't even want to claim a right answer for.
The only way to really lose is if the machine predicts you will take two boxes and you actually do take two boxes. Which is a gamble that would earn you 33% more if you win or cause you to lose 99.97% of your profit.
It doesn't seem logical.
Please elaborate on this "meaningful analysis" bit. I don't see how it follows that because she was asked the question she must assume that she is asked the same question on each waking.
What is the winning strategy for sleeping beauty's guess?
and
What is the probability the coin was tails?
Become obviously different questions.
Note that the winning strategy also depends on the probability of the researcher's asking her the question as a function of the coin landing heads or tails.
It's how probability works. I've explained it a hojillion times now but ok, why not again: Say you're going to flip a coin. The odds are 50% heads, right? Well, 50% means 50/100. 50 heads out of 100 times. If you want, you can say that this "assumes" I'm going to flip a coin 100 times, and it "assumes" that every time I flip it, I'll write down whether or not it was heads and add that to the total. Because if I don't do that, then I won't get 50/100 heads and therefore I can't say it's 50%. If I flip it 100 times, but I only write down every second result, then I'll end up with 25 heads out of 100 flips, which is 25%, right?
Except that's hogwash. Even if you just flip it once, the odds are 50%. I am not required to say, "the odds are 50%, but only assuming that I flip it 100 times and record the result each time and add it to the total." That "assumption" is inherent within the very concept of probability. It isn't really a required assumption for any particular coin-flipping scenario in order to say 50%. Barring any additional information otherwise, it is naturally assumed that I am not talking about some strange way of doing probability in which I pretend that I only sometimes write down the result, such that the odds of a coin flip isn't 50% heads.
Similarly, when SB wakes up and is asked the question, the odds are 1/3 heads. I am not required to base this on some "assumption" that she will be asked every time she wakes. That assumption is inherent within the very fact that I'm trying to answer a probability question. The odds are 1/3 regardless of how many times she's been woken up or how often she's asked the question. The "1/3" means that 1 out of every 3 potential times she might wake up and be asked the question, it will be heads. Just as with the coin toss, I am not required to base this on some assumption of how many times she'll actually wake and be asked, it's just the answer based on the analysis of the current situation extrapolated over many identical trials.
How about this: I've got a sum of money to award. I'm going to flip a coin, and it's heads, I'll give one of you all the money, and if it's tails, I'll split it between both of you. Congrats, here's $40. What are the odds it was heads?
Except in probability problems those are in fact the same question.
Again, I believe this is not the right way to approach the question. Just because there are 3 possible waking states does not mean that they are automatically equally likely. You are approaching the problem as though there is one initial branch (H/T), and then T branches once more (T1/T2), but in fact there is only one branch (H/T1) and one of the branches has an additional step (T2), which I will hastily illustrate below.
Flip H T T1 T2as opposed to
Flip H T1 T2A node partway down the branch cannot be treated as though it were identical to a node at the end of a branch.
Describe the information that you seem to think Sleeping Beauty has which leads her to believe anything about which of three possible waking states she is in.
By virtue of the fact that she just woke up and the study isn't over, she's either H1, T1, or T2, but not H2. At that point, there's a 2/3 likelihood of tails. Once the study is over and she's home, she adds in two more possible Hs to get back to a 50/50 chance. In fact, if SB answers "1/3" and then the scientist says, "ok, we're all done, you can leave," SB should then say, "wait, I want to change my answer to 1/2."
Every event in the universe is a node partway down a branch of some previous event. When we do probability, we distill and isolate reasonable knowns and unknowns. If the theoretical branch your node is on isn't known, then the probability associated with being on that branch is about as meaningful to the problem as the probability that life evolved on Earth instead of Mars. When SB awakens, she doesn't know anything that could help her know which branch she's on, so the nature of that branch doesn't factor in. What she knows is that this might not be the first she's been awakened, so there's a good chance the flip was tails.
She's in a room with 3 doors, all leading to the same other room. If it's heads she goes through door 1, if it's tails, she goes through either door 2 or 3 (chosen by some other random process). Once she's in the other room, she won't remember which door she came through. She's then asked what's the chance she came through door 1. Same answer or different answer?
What are the odds the coin was tails?
I believe this is how most of you are seeing the Sleeping Beauty problem. And yes, the odds here are 50/50. It doesn't matter that Bob gets $2 for tails. In either case he will have a $ to give you. Just as you are thinking that since SB gets waked up either way, then it doesn't matter and the odds are 50/50.
HOWEVER
Now what are the odds?
See, the thing that changed is that now you know that there's a chance someone else already came in and got a dollar from Bob. The fact that you were invited in the room at all means it was likely tails.
The fact that Sleeping Beauty forgets waking up the first time effectively means we could view her as two people. The SB who is woken up the first time might as well be a different person from the one who gets woken up the second time, since no consciousness whatsoever passes from one to the next. The very act of getting woken up is something that happens a finite number of times and yet twice as often on tails, so getting woken up at all is odds on tails. SB knows she might have been woken up once before, and the fact that she's sitting here having just woken up, instead of at home and the study over, by itself means the result was likely tails.
Good one. I think it's 50/50 here. Mainly because in your scenario, the game is over. The system is whole. There isn't a fourth scenario that can be ruled out. For SB, the game is still on, the 50/50 system isn't closed yet, and her position in the game is information that points to one coin result over the other.
I think the better analogy would be that if it's tails, she'll walk through door 2, then back out, then forget, then through door 3. At some point she finds herself in the second room and is asked the odds of the flip.
Take the common error people make with dice. If you roll two dice, the chance you will roll two sixes is 1 in 36. After you roll one die and get a six, what's the chance of rolling another six? 1 in 6, because you have new information about your location in the tree.
The issue with this puzzle is that there is only one branch, and no information can do any pruning. But it makes you think it does because of the double counted results.
What is essential is her amnesia - any given waking state must be treated as though it is the only one that has occurred. It is unmoored, in her experience, from previous events, and can thus only refer back to the original flip for statistical information. Nothing she knows can give her information about the coin flip itself, and thus none of her information can be relevant to the decision for that one, independent waking moment.
In each waking state, there has still only been one coin flip, and there are two possible answers, and they are equally likely to have happened. This cannot be altered.
Since a "waking state" is something that is not equally likely to happen based on the flip, you cannot say that on each waking state, the two sides of the coin are equally likely to have happened.
It's like my dice-rolling example before. Heads, I roll a d12, tails, a d6. There are only 12 possible outcomes, right? 1 - 12? And no matter what, the coin toss is still 50/50 heads/tails?
But what if I roll a 10? You still going to answer 50/50?
What if I roll a 4?
Sleeping Beauty rolled a 4. That's more likely to happen after a tails than a heads. The coin was thus more likely tails.
In order to frame the entire system in the manner you are attempting, you must be able to 1) describe the resulting states such that they are divided into n equally likely states that all add up to 100%, and 2) be able to apply the given knowledge in order to prune that set to a fraction of the 100%
In other words, you aren't using enough characteristics to describe your states in order to answer the question. Simple math fails you here... how do you get three discrete waking states from a 50/50 coin flip? If you drew a grid with 4-squares, heads and tails along one axis, and three waking states in the squares, what is the other axis? What's in the fourth square?
If your analysis stops at "she's in one of three waking states, one with a 50% probability and two with a 25% probability each, but she has no idea which one" then you haven't framed the probability completely yet. You're fixating on the coin toss and not using enough variables to actually make this an answerable problem. You haven't given Sleeping Beauty a set of equally likely results that she can apply knowledge to in order to prune.
That's why an abstract time variable is suggested as the other factor. Your grid is tails and heads across one axis, T1 and T2 across the other. Both axes are "known unknowns" i.e., they are things SB knows are possibilities. The flip could have been heads or tails, and she could have been woken up once before or not. The fourth square involves the study being over and SB didn't just wake up and isn't being asked about the flip. All four are equally likely, two of them heads and two of them tails, and all add up to 100%. But if she just woke up and is being asked about the flip, she's in square 1, 2, or 3, but not 4.
If you listened to me, you'd have $3 million, and if I listened to you, I would have $100.
There's a strong correlation between picking one box and there being $1 million being in the box, but it isn't causation. There's a confounding factor of if you are the type to choose one box, it will likely put $1 million in the box, and if you are the type to choose one box, you will likely choose one box. The computer has a strong historical record. That's correlation. Can you think of a single time in your life where you were able to change the past?
Choice prediction is pretty much time travel. I mean, you're here on the forums, arguing for picking both boxes. The machine probably won't need to research very long before it knows you're a two-boxer. You get $300 and I get $1MM.
Drink it the first day.
Box:
Take only box A
Toxin:
Drink the toxin, get $1,000,000 and get to call in sick for work? It's like eating bad Chinese take out, but instead of explosive diarrhea, you get a million bucks. Worth it, considering I've eaten bad Chinese take out enough, with out getting $1,000,000 in exchange for being sick. Also, that which doesn't kill you, only makes you stronger.
This is only equivalent to the SB problem if there is no chance that you won't be invited into the room with Bob. In this case there's no argument: you got invited in, so probably Bob had more dollars. Your odds are N:M with N the number of Bob's dollars and M the number of possible dollar-receivers, so the overall odds of the coin flip being tails, given that you received a dollar, are (N+1):M.
But in the SB problem there is no pool of possible sleepers. If there were a line outside Bob's door to get dollars, instead of a random pool, SB would be the first one in line: she gets a dollar whether Bob has one to give out or a million.
If you were to say: Then the best strategy is to guess tails, obviously, for any N.
If you replace *1,2 with: Then there is no strategy better than a coin toss of your own.
SB doesn't know where she is in the line. She knows that it's more likely that she is further down the line than the head position, but she doesn't actually know that she is. So while her best strategy to be correct is to say that the coin landed on tails, she doesn't have enough information to actually say that she knows the odds to be different from 50/50.
The box would predict that one of you would listen to the other, and put money in accordingly. What makes you think you can fool the box?
The box knows all.
For instance, what if we perfected a form of suspended animation / cryogenics as well as a viable long term spaceship that could colonize a distant world. If you did this, you would be asleep for decades, you would leave behind your life and family and likely never see them again...would you do it?
Same circumstances, but your choice is to not go off to a distant world...just to freeze yourself for a hundred years to see how the world changes in that time. The same cost...your life and family would be completely gone but you have a new start. would you choose to do it?
What if we slightly alter the question and say that the machine is never wrong?
This is not true. SB could be at home and the study already over, rather than having just been woken up and asked about odds. Just as you could have not been invited into Alice and Bob's room at all.
SB is not in the immediate category, she is in the line category. Upon waking, she has no idea if she's been woken once before or not, just as you have no idea if you're the first one in the room or not. You seem to be implying that SB knows whether or not this is the first time she's woken up, which is clearly altering the scenario. You do realize that the Alice and Bob answer is still to guess tails, even if you're at the front of the line, so long as you don't know that you're at the front of line, right?
And, again, saying her best strategy is to say tails is the same thing as saying odds. It is an unacceptable contradiction to say that she knows that tails is twice as likely to be the result, and yet she also knows that the odds of the result are 50/50 heads/tails. You need to resolve this somehow.
Heads: you wake up. Tails: you wake up.
Correct. You should not look, and pick the first box. The option to look I would imagine just make the machine a lot more likely to label people as two-boxers.
Tails: you wake up again. Two points for tails, one for heads.
"We are going to run this experiment 1,000 times, and every time we wake you up we will ask you, 'What was the result of the coin flip?' For every correct answer, we will give you $1."
SB should clearly answer "tails" every single time. This would result in her making, on average, $667.
Does this mean that the probability of a "tails" result on the coin flip is 2/3? I'm not sure that it does, because that's fundamentally a semantic argument - what does it mean when we describe the probability of an event?
Where I lean is that the probability of a tails result on the coin flip is different than the probability of tails being a correct answer to the question, because the behavior of the questioner changes based on the coin flip result.
An analogous situation might be if we changed the question in the Monty Hall problem. If the question asked in the Monty Hall problem is not "should you switch doors?" but rather, "what was the probability, before any doors were revealed, that the car was behind door #1?" I see these as two very different questions.
But I'm not totally certain of my position here.
the "no true scotch man" fallacy.
Uh Feral, I think your math is off there. running the experiment 1,000 times means you get asked 1,500 times what the result was, and earn $1,000 if you answer tails each time.
That's the crux I think. You get woken up more often than a coin is flipped, so naturally the odds of being in the path where you get woken up more often is likelier. It has nothing to do wither the fact that being woken up and not sitting at home means it's likely you got tails. There is just the fact that if you get tails you are woken up more times.
The odds of you being in that path are only more likely if you are in fact still in the study and just woken up, rather than at home after the study ends. When you're at home after the study is over, the odds are 50/50.
I have no idea what you're saying here. If I wasn't invited into the room at all then I don't know anything about the outcome. The Alice and Bob scenario only imparts information to me if I enter the room, because entering the room means that odds are better that Bob had more money to give away, allowing me to get picked. If I don't get picked then I have no idea whether he picked many people or one person and no way of judging.
Saying that the odds in the coin flip aren't 50/50 if I never enter the room at all is equivalent to saying that if I flip a coin and ask you the odds of tails while, silently, telling myself that I will give away $1000 to 1000 individuals on tails or $1 to one person on heads, without telling you about it, the answer is no longer 50/50.
Please read the bolded part of the quote above again.
There is no answer if you're at the front of the line. You can guess tails, but it has no greater odds of being correct than guessing heads if you're at the front of the line.
She doesn't know that tails is twice as likely to be the result. She knows that there are twice as many situations in which tails is the correct answer. The result is equally likely to be heads or tails, but, based on her situation, there are multiple, differently fit, strategies for guessing correctly.
If the goal of the exercise is for SB to guess correctly, then she should always answer tails. If the goal of the exercise is to accurately answer the question of "what is the probability that a fair coin landed on tails?" she doesn't have enough information to differentiate between a situation where she knows the probability to differ from 50/50 and one where she does not.
She doesn't know she isn't at the head of the line, so she can, at best, say that the probability of tails is at least 50%.
It's not a question of "try to guess the right answer with the best frequency of success", it's a question of "what is the accurate answer, based on available information that you know to be true". Those are different questions. It's the difference between "what is probably true?" and "what do you know to be true?"
Modifying the problem to arbitrarily many wakings on the tails-branch lets her have arbitrarily high confidence that she is in one of the later, tails-branch, waking cycles, but she does not actually know that to be true and so cannot answer the latter of those two questions any way except "at least 50% probability of tails". The former, which allows guessing based on confidence, is a probability dependent on the number of wakings.
The situation is hypothetical. The bottle becomes increasingly better and there is no upper-limit to the betterness. Moreover, the person drinking it will experience the betterness of the bottle in its entirety, even if the geometric progression described in the puzzle has the bottle reach mind-boggling greatness after only a few months.
I hate it when people try to beat logic puzzles into submission through sheer pedantry. If you're going to side-step the essential logic of the puzzle, why even bother with it at all. Yes, there are real-world limitations to a person's sense of taste. If that particular point bothers you that much, just pretend that the hypothetical person in the puzzle is a space god with an infinite capacity to enjoy the unspecified betterness imbued in the bottle of wine.
Though, I must admit, I don't have a satisfying answer to that puzzle.
Drink it the day I realize I'm bored with being alive and no longer want to live, then blow my brains out.
Yes, you're right about the math. :P
the "no true scotch man" fallacy.