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Topic: Deus Ex Machina (Read 2650 times) |
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BMAD
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Deus Ex Machina
« on: Jun 4th, 2014, 4:58am » |
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Five years ago, your rich uncle secretly commissioned the construction of a supercomputer! The purpose of this binary behemoth (the "Very Good Predictor") was to predict, as accurately as possible, every action you would take over the course of these five years. It has managed to do so with 100% accuracy for its entire runtime.
Now, your uncle has approached you with this information, and an offer to participate in a game involving the supercomputer's final prediction.
He has placed two envelopes in a room: Envelope A and Envelope B.
Envelope B contains exactly 1,000 dollars. Envelope A either contains 10,000, 10, or 0 dollars. If the supercomputer correctly predicted your selection of Envelope A alone, it will contain $0. If the supercomputer correctly predicted your selection of both A and B then Envelope A contains only $10. Otherwise the supercomputer predicted that you would not choose A and told your uncle that he could safely put $10,000 in the envelope (which he would oblige).
Now since this is essentially a contest between you and the supercomputer, if the supercomputer correctly predicts your choice, the amount selected is the amount you owe your uncle, otherwise you keep the amount in the envelopes (both). The values in both envelopes is revealed once a selection is made.
What should you do?
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Grimbal
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Re: Deus Ex Machina
« Reply #1 on: Jun 4th, 2014, 6:56am » |
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What are my choices?
Select one, both or none of the envelopes?
Is the computer's prediction exclusively about which envelope(s) I will select?
If I toss a coin (or something similar using quantum measurement), can the computer predict it?
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| « Last Edit: Jun 4th, 2014, 7:01am by Grimbal » |
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rmsgrey
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Re: Deus Ex Machina
« Reply #2 on: Jun 4th, 2014, 7:53am » |
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So, if you pick Envelope A, either you owe your uncle $1000 or your uncle owes you $1010 or $11,000
Pick both and you get $1000 or $11000 or lose $1010
Pick just B and you get $1000 or $1010 or lose $11000
If you believe that the amounts in the envelopes are determined by your choice, you should pick envelope A and minimise your losses. Actually, you should choose not to play, and save yourself $1000.
If you believe the outcome is chosen uniformly at random, then you should choose envelope A and maximise your expected gain.
It gets interesting when you assume that the machine predicts your choice of weighting for a random decision, but cannot predict what the outcome of that random event will be.
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BMAD
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Re: Deus Ex Machina
« Reply #3 on: Jun 4th, 2014, 7:54am » |
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on Jun 4th, 2014, 6:56am, Grimbal wrote:What are my choices?
Select one, both or none of the envelopes?
Yes
Is the computer's prediction exclusively about which envelope(s) I will select?
Yes based on its 100% accuracy in your past decisions
If I toss a coin (or something similar using quantum measurement), can the computer predict
The computer can predict all your actions.
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towr
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Re: Deus Ex Machina
« Reply #4 on: Jun 4th, 2014, 1:06pm » |
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on Jun 4th, 2014, 7:54am, BMAD wrote:| The computer can predict all your actions. |
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Yeah, but not their outcome. It may predict I'll flip a quantum-mechanical coin, but it can't predict if it comes up cats or tails or a superposition.
Maybe I'll just ask my uncle to tell me what it predicted and then do something else. Though I suppose I should then also ask what it predicted I'd do if I knew what it predicted I'd do if i didn't know. (Repeat ad infinitum)
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| « Last Edit: Jun 4th, 2014, 1:06pm by towr » |
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BMAD
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Re: Deus Ex Machina
« Reply #6 on: Jun 4th, 2014, 2:10pm » |
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on Jun 4th, 2014, 1:06pm, towr wrote:
Yeah, but not their outcome. It may predict I'll flip a quantum-mechanical coin, but it can't predict if it comes up cats or tails or a superposition. |
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I agree fully.
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Maybe I'll just ask my uncle to tell me what it predicted and then do something else. Though I suppose I should then also ask what it predicted I'd do if I knew what it predicted I'd do if i didn't know. (Repeat ad infinitum) |
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I don't think he would tell you.
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BMAD
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Re: Deus Ex Machina
« Reply #7 on: Jun 4th, 2014, 2:12pm » |
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on Jun 4th, 2014, 1:29pm, dudiobugtron wrote:
Wow. I am smarter than I thought! I honestly did not know of this question here.
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| « Last Edit: Jun 4th, 2014, 2:12pm by BMAD » |
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dudiobugtron
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Re: Deus Ex Machina
« Reply #8 on: Jun 4th, 2014, 5:14pm » |
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on Jun 4th, 2014, 1:06pm, towr wrote:
Yeah, but not their outcome. It may predict I'll flip a quantum-mechanical coin, but it can't predict if it comes up cats or tails or a superposition. |
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While the computer can't directly predict their outcome, it can predict your actions that followed from that outcome. So I don t see how you could use it to your advantage in this case.
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towr
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Re: Deus Ex Machina
« Reply #9 on: Jun 4th, 2014, 11:11pm » |
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on Jun 4th, 2014, 5:14pm, dudiobugtron wrote:| While the computer can't directly predict their outcome, it can predict your actions that followed from that outcome. So I don't see how you could use it to your advantage in this case. |
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Well it would be a bit of a paradox if it can't predict the outcome but can predict the action dependent on the outcome (in a way other than probabilistically)
Suppose I flip a coin, and if it's head I pick A, and if it's tails I pick B. It will be able to predict "if H then A" and "if T then B", but it won't be able to specifically predict A (or B), because it can't predict that H (or T) will happen. So there may be some advantage to be had there if you can beat the odds. Since there are three choices, and the machine can only specifically predict one, the advantage is yours.
As long as physics isn't broken, there should always be such an escape route into randomness. (Though you may need props, because the human mind isn't random.)
If physics is broken, and we're dealing with a perfect oracle, then you're screwed.
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dudiobugtron
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Re: Deus Ex Machina
« Reply #10 on: Jun 5th, 2014, 12:13am » |
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on Jun 4th, 2014, 11:11pm, towr wrote:
Well it would be a bit of a paradox if it can't predict the outcome but can predict the action dependent on the outcome (in a way other than probabilistically)
Suppose I flip a coin, and if it's head I pick A, and if it's tails I pick B. It will be able to predict "if H then A" and "if T then B", but it won't be able to specifically predict A (or B), because it can't predict that H (or T) will happen. So there may be some advantage to be had there if you can beat the odds. Since there are three choices, and the machine can only specifically predict one, the advantage is yours.
As long as physics isn't broken, there should always be such an escape route into randomness. (Though you may need props, because the human mind isn't random.)
If physics is broken, and we're dealing with a perfect oracle, then you're screwed.
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Here's how it looks from the computer's point of view, as it relates to it's prediction:
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I predict you will say: "I'll choose A if I flip heads, and B if I flip tails"
I predict you will then flip the coin.
I predict you then look at the the coin to see the result of the flip.
I predict you will then choose envelope B.
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The computer doesn't need to know what result the coin got to predict you will choose B. It can't even work it out for sure; just because you say you will choose A if you get heads, doesn't mean you actually have to once you've flipped it.
Unless the rules of the game specified that you must choose a result that corresponds exactly to your flip in a pre-determined way, then there's no recourse to absolute randomness.
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0.999...
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Re: Deus Ex Machina
« Reply #11 on: Jun 5th, 2014, 4:43am » |
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Supposing the supercomputer can predict the precise way you will toss the coin and the precise way you will catch the coin, it has the means to predict the result of your coin toss. Perhaps we can communicate with a friend who performs the actual random event to avoid this problem.
I keep on going back and forth between following dudiobugtron's reasoning and not. However, I am not willing to take one's word for it that there are no stimuli which cause the same mental response in you every time.
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| « Last Edit: Jun 5th, 2014, 4:47am by 0.999... » |
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towr
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Re: Deus Ex Machina
« Reply #12 on: Jun 5th, 2014, 9:03am » |
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on Jun 5th, 2014, 12:13am, dudiobugtron wrote:| just because you say you will choose A if you get heads, doesn't mean you actually have to once you've flipped it. |
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I know I don't have to, but i know i will if that's what I say I do. So the computer would be wrong, and I win.
If my action is dependent on the result of the coin, as I posited, then I have a chance to win.
Considering the case where my action isn't dependent on the result of the coin doesn't affect that conclusion.
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| « Last Edit: Jun 5th, 2014, 9:04am by towr » |
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dudiobugtron
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Re: Deus Ex Machina
« Reply #13 on: Jun 5th, 2014, 12:58pm » |
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@0.999...: Even if the computer can infallibly predict your exact movements, it can't infallibly predict exactly the movement of the coin in the air. If it could, as towr said, physics would be broken.
on Jun 5th, 2014, 9:03am, towr wrote:
I know I don't have to, but i know i will if that's what I say I do. So the computer would be wrong, and I win.
If my action is dependent on the result of the coin, as I posited, then I have a chance to win.
Considering the case where my action isn't dependent on the result of the coin doesn't affect that conclusion. |
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Like 0.999... is saying, we aren't supercomputers with heretofore infallible prediction algorithms. You can't know for sure how you will act.
Some scenarios which might affect your actions. (Note that by 'you', I mean the protagonist of the puzzle):
Maybe you will misread the coin, or mis-remember the match-up between coin-sides and envelopes. Maybe the coin will land in some state (as you suggested earlier) which is neither heads nor tails. Maybe it wall fall out of your reach, and you will be unable to read its result. Maybe you will start to second-guess the result, and flip again. Maybe the coin will land up heads for A, but it will land exactly in the centre of envelope B, and you'll take that as a sign. Maybe it will land underneath envelope B, and so you'll have to pick up B to see the result. Maybe it will land in some potentially unambiguous way that you hadn't considered beforehand; eg if it hit the ceiling and become stuck with heads touching the ceiling, and with tails being the side facing you. etc etc...
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towr
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Re: Deus Ex Machina
« Reply #14 on: Jun 5th, 2014, 1:21pm » |
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Once you say "tails=A, heads=B, anything else is A+B and that's final"*) then you've already made the choice; you may not know what you've chosen exactly, but the choice has been made and it will lead to a clear unambiguous result once the coin is tossed. You don't have to choose (as a separate action) the envelope(s) after the coin toss based on the result of the coin-toss (or what you think the result is). And things like misreading the coin or whatever do not change what you've really chosen.
If you're allowed to change your mind whenever you want after already having made a choice, then why not just open the envelopes and choose one then?
Probably a good idea anyway, since the puzzle doesn't forbid it.
*) Or set up a single-photon double-slit experiment where you divide the screen into three equally likely**) sections to represent A, B and A+B; such that you have a true quantum-mechanically random result.
**) Or whatever probability distribution would give the best payout.
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| « Last Edit: Jun 5th, 2014, 1:28pm by towr » |
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wakiza33
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Re: Deus Ex Machina
« Reply #15 on: Sep 16th, 2014, 8:40am » |
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I think you guys are right on the money with the randomized approach. While the computer may predict your action--choosing a random experiment--it will not be able to predict the outcome effectively.
The very first answer, too, quite impressive.
Since Radiactive decay is considered random, with the associated formula not equal to any predictable probability, I would use that formula to assign a judgment.
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| « Last Edit: Sep 16th, 2014, 8:41am by wakiza33 » |
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