Omega will either award you $1000 or ask you to pay him $100. He will award you $1000 if he predicts you would pay him if he asked. He will ask you to pay him $100 if he predicts you wouldn't pay him if he asked.

Omega asks you to pay him $100. Do you pay?

This problem is roughly isomorphic to the branch of Transparent Newcomb (version 1, version 2) where box B is empty, but it's simpler.

Here's a diagram:

This is also isomorphic to the absent-minded driver problem with different utilities (and mixed strategies*), it seems. Specifically, if you consider the abstract idealized decision theory you implement to be "you", you make the same decision in two places, once in omega's brain while he predicts you and again if he asks you to pay up. Therefore the graph can be transformed from this

into this

which looks awfully like the absent minded driver. Interesting.

Additionally, modifying the utilities involved ($1000 -> death; swap -$100 and $0) gives Pa... (read more)

I contend it's also isomorphic to the very real-world problems of hazing, abuse cycles, and akrasia.

The common dynamic across all these problems is that "You could have been in a winning or losing branch, but you've learned that you're in a losing branch, and your decision to scrape out a little more utility within that branch takes away more utility from (symmetric) versions of yourself in (potentially) winning branches."

I have sympathy for the commenters who agreed to pay outright (Nesov and ata), but viewed purely logically, this problem is underdetermined, kinda like Transparent Newcomb's (thx Manfred). This is a subtle point, bear with me.

Let's assume you precommit to

not pay if asked. Now take an Omega that strictly follows the rules of the problem, but also has one additional axiom:I will award the player $1000 no matter what. This Omega can easily prove that the world in which it asks you to pay is logically inconsistent, and then it concludes that in that world y... (read more)By paying, you reduce probability of the low-utility situation you're experiencing, and correspondingly increase the probability of the counterfactual with Omega Award, thus increasing overall expected utility. Reality is so much worse than its alternatives that you're willing to pay to make it less real.

Of course I'd pay.

'a' should use a randomizing device so that he pays 51% of the time and refuses 49% of the time. Omega, aware of this strategy, but presumably unable to hack the randomizing device, achieves the best score by predicting 'pay' 100% of the time.

I am making an assumption here about Omega's cost function - i.e. that Type 1 and Type 2 errors are equally undesirable. So, I agree with cousin_it that the problem is underspecified.

The constraint

P(o=AWARD) = P(a=PAY)that appears in the diagram does not seem to match the problem statement. It is also ambiguous. ... (read more)Nice diagram. By the way, the assertion "Omega asks you to pay him $100" doesn't make sense unless your decision is required to be a mixed strategy. I.e., P(a = PAY) < 1. In fact, P(a = PAY) must be significantly less than the strength of your beliefs about Omega.

Of course I don't pay. Omega has predicted that I won't pay if he asked, and Omega's predictions are by definition correct. I don't see how this is a decision problem at all.

Would you agree that, given that Omega asks you, you are guaranteed by the rules of the problem to not pay him?

If you are inclined to take the (I would say) useless way out and claim it could be a simulation, consider the case where Omega makes sure the Omega in its simulation is

alsoalways right - creating an infinite tower of recursion such that the density of Omega being wrong in all simulations is 0.