Extremely Counterfactual Mugging or: the gist of Transparent Newcomb

Bongo

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:

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:

I prefer this interpretation: P(a=X) means how sure the agent is it will X. If it flips a coin do decide whether X or Y, P(a=X)=P(a=Y)~=0.5. If it's chosen to "just X", P(a=X) ~= 1. Omega for his part knows the agent's surety and uses a randomizing device to match his actions with it.

ETA: if interpreted naively, this leads to Omega rewarding agents with deluded beliefs about what they're going to do. Maybe Omega shouldn't look at the agent's surety but the surety of "a perfectly rational agent" in the same situation. I don't have a real solution to this right now.