Tweeter, Claus Metzner (@cmetzner) alerted me to this cool area of study with this paper.
Suppose you meet a Wise being (W) who tells you it has put $1,000 in box A, and either $1 million or nothing in box B. This being tells you to either take the contents of box B only, or to take the contents of both A and B. Suppose further that the being had put the $1 million in box B only if a prediction algorithm designed by the being had said that you would take only B. If the algorithm had predicted you would take both boxes, then the being put nothing in box B. Presume that due to determinism, there exists a perfectly accurate prediction algorithm. Assuming W uses that algorithm, what choice should you make?
Ultimately one is lead to understand that the paradox is a manifestation of different interpretations of the problem definition (aren’t all paradoxes though?) If you interpret the setup one way, then you should choose just B and you will net $1M. If another way, then you should choose both and net either $1000 or $1,001,000 depending on W’s unknowable prediction. As the authors conclude:
Newcomb’s paradox takes two incompatible interpretations of a question, with two different answers, and makes it seem as though they are the same interpretation. The lesson of Newcomb’s paradox is just the ancient verity that one must carefully deﬁne all one’s terms.
The authors suggest combining Bayesian nets with game theory is what yields this resolution. And at first I thought they missed the obvious further conclusion from Bayes, which is that you should clearly choose just B. Here was my reasoning. The key clue is in this piece of information: “people seem to divide almost evenly on the problem”. I.e. your Bayesian priors should now be set to 50% on either interpretation. Now, we know that the expected value (EV) of the “just B” scenario is $1M, but we don’t really know what the EV is for the “both boxes” scenario in which “your choice occurs after W has already made its prediction”:
…if W predicted you would take A along with B, then taking both gives you $1,000 rather than nothing. If instead W predicted you would take only B, then taking both boxes yields $1,001,000….
Since in this scenario you are choosing after W’s prediction, is there any way you can “predict” what W’s choice might be? No, of course not, it’s a variant of the Liar’s Paradox where if you predict one thing, the answer is the other. Thus, if we are using a probabilistic approach (as the authors have laid out for us), we must conclude there is no information to be gleaned on W’s prediction and we are forced to assign 50% likelihood of either choice. Hence, the EV of the “both boxes” interpretation is $501,000.
Putting both meta-Bayesian analyses together, we can conclude that since the “just B” interpretation yields $1M and the “both boxes” interpretation yield’s an EV of a little over half that, it’s a no-brainer to choose just B. Which means your EV is exactly $500,000. But wait! We just concluded that the EV for “both boxes” is $501,000, which is clearly better!!!
Newcomb’s paradox will probably crack my list of Top 10 Paradoxes of All-Time (unless I figure out how to solve it after it does).