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/