Something about your proposed decision problem seems cheaty in a way that the standard Newcomb problem doesn’t. I’m not sure exactly what it is, but I will try to articulate it, and maybe you can help me figure it out.
It reminds me of two different decision problems. Actually, the first one isn’t really a decision problem.
Omega has decided to give all those who two box on the standard Newcomb problem 1,000,000 usd, and all those who do not 1,000 usd.
Now that’s not really a decision problem, but that’s not the issue with using it to decide between decision theories. I’m not sure exactly what the issue is but it seems like it is not the decisions of the agent that make the world go one way or the other. Omega could also go around rewarding all CDT agents and punishing all FDT agents, but that wouldn’t be a good reason to prefer CDT. It seems like in your problem it is not the decision of the agent that determines what their payout is, whereas in the standard newcomb problem it is. Your problem seems more like a scenario where omega goes around punishing agents with a particular decision theory than one where an agent’s decisions determine their payout.
Now there’s another decision problem this reminds me of.
Omega flips a coin and tell you “I flipped a coin, and I would have paid you 1,000,000 usd if it came up heads only if I predicted that you would have paid me 1,000 usd if it came up tails after having this explained to you. The coin did in fact come up tails. Will you pay me?”
In this decision problem your payout also depends on what you would have done in a different hypothetical scenario, but it does not seem cheaty to me in the same way your proposed decision problem does. Maybe that is because it depends on what you would have done in this same problem had a different part of it gone differently.
I’m honestly not sure what I am tracking when I judge whether a decision problem is cheaty or not (where cheaty just means “should be used to decide between decision theories”) but I am sure that your problem seems cheaty to me right now. Do you have any similar intuitions or hunches about what I am tracking?
I had already proved it for two values of H before I contracted Sellke. How easily does this proof generalize to multiple values of H?
I see. I think you could also use PPI to prove Good’s theorem though. Presumably the reason it pays to get new evidence is that you should expect to assign more probability to the truth after observing new evidence?
I honestly could not think of a better way to write it. I had the same problem when my friend first showed me this notation. I thought about using "E[P(H=htrue)]" but that seemed more confusing and less standard? I believe this is how they write things in information theory, but those equations usually have logs in them.
I didn’t take the time to check whether it did or didn’t. If you would walk me through how it does, I would appreciate it.
Luckily, I don’t know much about genetics. I totally forgot that, I’ll edit the question to reflect it.
To be sure though, did what I mean about the different kinds of cognition come across? I do not actually plan on teaching any genetics.