As outlined in the last paragraph of the post. I want to convince people that TDT-like decision theories won’t give a “neat” game theory, by giving an example where they’re even less neat than classical game theory.
Hum, then I’m not sure I understand in what way classical game theory is neater here?
I think you’re thinking about a realistic case (same algorithm, similar environment) rather than the perfect symmetry used in the argument. A communication channel is of no use there because you could just ask yourself what you would send, if you had one, and then you know you would have just gotten that message from the copy as well.
As long as the probabilistic coin flips are independent on both sides (you also mention the case where they’re symmetric, but let’s put that aside for the example), then you can apply the basic probabilistic algorithm for leader election: both copies flip a coin n times to get a n-bit number, which they exchange. If the numbers are different, then the copy with the smallest one says 0 and the other says 1; otherwise they flip a coin and return the answer. With this algorithm, you have probability ≥1−12n of deciding different values, and so you can get as close as you want to 1 (by paying the price in more random bits).
I’d be interested. I think even just more solved examples of the reasoning we want are useful currently.
Do you have examples of problems with copies that I could look at and that you think would be useful to study?
Hum, then I’m not sure I understand in what way classical game theory is neater here?
As long as the probabilistic coin flips are independent on both sides (you also mention the case where they’re symmetric, but let’s put that aside for the example), then you can apply the basic probabilistic algorithm for leader election: both copies flip a coin n times to get a n-bit number, which they exchange. If the numbers are different, then the copy with the smallest one says 0 and the other says 1; otherwise they flip a coin and return the answer. With this algorithm, you have probability ≥1−12n of deciding different values, and so you can get as close as you want to 1 (by paying the price in more random bits).
Do you have examples of problems with copies that I could look at and that you think would be useful to study?
Changing the labels doesn’t make a difference classically.
Yes.
No, I think you should take the problems of distributed computing, and translate them into decision problems, that you then have a solution to.