Box A or Box B?Date: 11/21/2000 at 21:42:27 From: Joe Subject: Logic Riddle Hello, and thank you for your help. This riddle has me completely stumped. I would choose to take the contents of both boxes, but I'm not sure if this is the correct way to go, and how to create a valid argument for this. Below is the exact riddle I have been given. A psychology professor at ASU has a reputation for being brilliant as well as possessed of an enormous fortune she has dedicated to her research. One day you get a request to report to her office at a certain hour. On a table are two boxes. One of them, labeled A, is transparent; in it you can see an enormous pile of $100 bills. The other, labeled B, is opaque. She tells you that there is $10,000 in transparent Box A and that in Box B there is either $1,000,000 or nothing. She tells you that she is going to give you a choice between taking just what is in Box B or taking what is in both boxes. Then she tells you that this is part of an experiment. During registration at the beginning of the semester, you walked under a peculiar device that reminded you of the machines at airports that are used to prevent hijacking. You didn't think much about it at the time. But she now informs you that this machine was something she designed and that it recorded an instant profile of your basic personality and character traits. On the basis of your profile, she made a prediction about what choice you would make, and she decided what to put in Box B on the basis of this prediction. If she predicted you would take both, she put nothing in Box B. If she predicted you would take only Box B, she put $1,000,000 in it. At this point, you ask her how accurate her predictions have been. She says that 5000 students have been given the choice, and she has only been wrong once. In all the other cases, students who chose both boxes got only $10,000, whereas those who chose only box B got $1,000,000. Then she tells you to choose. What do you do? You are to decide what you should do in the imagined circumstances and to construct a valid argument justifying your choice in two pages or less. Date: 11/27/2000 at 11:23:18 From: Doctor Shawn Subject: Re: Logic Riddle Joe, What a fun question! I'm almost 100% sure that this one doesn't have an absolute right answer, but I'll tell you what I'd do in this situation. Let's assume that the professor is telling the truth, that she's been very accurate in the past, and that she has some kind of mind-pattern on file for you. (If she's lying, that adds another element to the puzzle.) In this puzzle, you are given a gamble. You can either keep the ten grand for sure, or risk it for the million dollars. Your first inclination might be to play it safe and choose the two boxes. But I think that the professor would have recognized you as a play-it-safe type and box B won't have a bonus million for you. Or you could be a risk-taker and go for the gamble, which she'd also spot and reward with a million (you hope). This seems to me to actually be a problem of decision theory, which deals with puzzles like this. Psychology as well as mathematics shows up in this subject - just like the professor from ASU. There are several different strategies that people can take, and it depends on your own personal preferences. The two key strategies for this problem: MAXIMAX -- MAXImizing MAXimum payoffs. You are a gambler. You always choose the option that gives you the greatest potential payoff if things go well, even if you might lose everything. You choose this option even if you have a very, very small chance of winning. If I asked a maximaxer to choose between $995 and a lottery that offered a 10% chance of $1000 and a 90% chance of 0, they'd take the lottery. MINIMAX -- MINImizing MAXimum losses. You play it safer. Given a choice, you will always pick the option that leaves you the greatest payoff if things go bad. For example, if I asked a minimaxer to choose between $5 and a lottery that offered a 90% chance of $1000 and a 10% chance of nothing, they would take the $5. These are obviously stupid strategies to play ALWAYS, but they are your choices here. Do you take the sure thing, or do you gamble at your chance for a cool million? I would choose box B, but that is because I am a gambler - I enjoy games of chance, and I would figure that the good doctor's brain machine would have known this about me, and that my reward would be a million. However, if you can't stomach the thought of risking $10,000 on a stupid bet, then both boxes is the way to go. Sorry this is such a long reply for such an inconclusive answer, but I figured that I should explain the basics of decision theory for my conclusion to make any sense, to wit: Go with your gut feeling. If you're the kind of person who wouldn't take stupid bets, then don't. If you are, do. There isn't a CORRECT answer, but there is an answer for YOU. (Different for everyone, and equally valid; people just have different utilities for different things, and look at these problems with different perspectives.) I hope that helps. If you have any other questions, feel free to ask Dr. Math. - Doctor Shawn, The Math Forum http://mathforum.org/dr.math/ Date: 11/27/2000 at 11:25:02 From: Doctor Ian Subject: Re: Logic Riddle Hi Joe, Regardless of what happened at registration, box B _now_ contains either $1M or $0, and box A contains $10K. So here is your payoff matrix: Box B contains $1M $0 You take +------------------------- B only | $1M $0 | A and B | $1M+$10K $10K That is, regardless of what's in box B, you're better off taking both boxes. There is no scenario in which you're not worse off if you take only the second box. The rest of the problem description is noise. In fact, it sounds like an experiment designed to test superstition. Anyone who chooses to take only box B obviously believes that the choice he makes _now_ will have an effect on what happened _before_ he entered the room. In other words, if you choose to take only box B, you believe that causality travels backwards in time. Something else to consider is that if a student had actually walked out of this experiment with either $10K or $1M, everyone on campus would have heard about it already. But you haven't heard about it. So no one has walked away with any money. What does _that_ tell you? I hope this helps. Write back if you'd like to talk about this some more, or if you have any other questions. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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