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
As students were thinking through the problem aloud, I was recording their thoughts on the board. Without realizing it, however, I had put their thoughts down into a visual organizer that they had never seen before: the probability tree. Fortunately one student asked what that was (“It reminds me of something we did in school”), opening up the discussion to the freedom to use any visual organizer – such as the decision matrix, which we’ve been using for weeks – or even none. Unfortunately, I was not doing a good job of becoming invisible, an important role of the math circle leader. Becoming invisible has been on my mind lately, ever since I read Rushdie’s description of a photographer becoming invisible. “I tried to find narratives, mysteries, in the come and go at the doors of great hotels. After a time I no longer knew why I was doing these things, and it was at this point, I believe, that the pictures started to improve, because they were no longer about myself. I had learned the secret of becoming invisible, of disappearing into the work.”4 (I think it would help my math circle to get back to topics where I don’t know what I’m talking about.)
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…
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.