### Newcomb’s Problem (Math Circle Teens 4)

NOVEMBER 26, 2013

You have a choice of boxes.  Box A is transparent and contains \$1,000.  Box B is opaque, and contains either \$0 or \$1,000,000.  You may take just the opaque box (Box B), thereby “one-boxing,” or you may take both boxes, thereby “two-boxing.”  A prediction has already been made about whether you will one-box or two-box.  If that prediction was that you would two-box, then Box B is empty.  It that prediction was that you would one-box, then box B contains the million dollars.1

I actually presented a slight variation of this problem written by lukeprog on the lesswrong wiki.2  In this version, an alien robot named Omega presents the box choice.  Omega has always correctly predicted which box people will choose.  You’ve seen it a thousand times.

“I would two-box,” said two of the students pretty quickly.  Things got sticky when I asked for reasons.

The group stumbled through a mathematical approach that was totally consistent with the result obtained when applying causal decision theory.3

At this point, everyone was satisfied that the solution was clear.  Then I threw a monkey wrench into things:  since you’ve seen Omega play this game a thousand times and Omega always predicted correctly, you realize that everyone who two-boxed received \$1,000 while everyone who one-boxed got \$1,000,000.  Doesn’t the evidence suggest that you should one-box?5

The group concurred.  Everyone wanted to one-box. I talked briefly about how we got different decisions using different algorithms (causal decision making vs. evidential decision making), but time was up so got nowhere with this discussion.  To be continued…

Rodi

1 This is the classic Newcomb’s Problem in decision theory.  The problem is explained well on one of my favorite websites, Stanford’s Encyclopedia of Philosophy.

2 The Less Wrong wiki is fascinating.  Don’t look at it unless you have a lot of time.  But if you do have the time, or know something about it or its co-founder Eliezer Yudkowsky, let me know.  My curiosity has been aroused.

3The Stanford site explains this approach clearly and concisely.

4Salman Rushdie, The Ground Beneath Her Feet, p213.  Henry Holt and Company, 1999

5lukeprog’s Less Wrong post explains this well.

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