This is an exploration of a way of looking at Newcomb’s Problem that helped me understand it. I hope somebody else finds it useful. I may add discussions of other game theory problems in this format if anybody wants them.

Consider Newcomb’s Problem:: Omega offers you two boxes, one transparent and containing $1000, the other opaque and containing either $1 million or nothing. Your options are to take both boxes, or only take the second one; but Omega has put money in the second box only if it has predicted that you will only take 1 box. A person in favor of one-boxing says, “I’d rather have a million than a thousand.” A two-boxer says, “Whether or not box B contains money, I’ll get $1000 more if I take box A as well. It’s either $1001000 vs. $1000000, or $1000 vs. nothing.” To get to these different decisions, the agents are working from two different ways of visualising the payoff matrix. The two-boxer sees four possible outcomes and the one-boxer sees two, the other two having very low probability.

The two-boxer’s payoff matrix looks like this:

Box B

|Money | No money|

Decision 1-box| $1mil | $0 |

2-box | $1001000| $1000 |

The outcomes $0 and $1001000 both require Omega making a wrong prediction. But as the problem is formulated, Omega is superintelligent and has been right 100 out of 100 times so far. So the one-boxer, taking this into account, describes the payoff matrix like this:

Box B

|Money | No money|

Decision 1-box| $1mil | not possible|

2-box | not possible| $1000 |

If Omega is really a perfect (nearly perfect) predictor, the only possible (not hugely unlikely) outcomes are $1000 for two-boxing and $1 million for one-boxing, and considering the other outcomes is an epistemic failure.