uy and R are independently chosen from well-defined distributions. Regardless of sequence, neither knows the other and CANNOT be chosen based on the other. I’ll see if I can find time tonight to figure out whether I’m saying your claim 1 is wrong (it dropped epsilon too soon from the floor value, but I’m not sure if it’s more fundamentally problematic than that) or that your claim 2 is misleading.
My current expectation is that I’ll find that your claim 2 results are available in situation 1, by using your given function with a pre-agreed value rather than a random one.
True, they will fail to cooperate for some R, but the values of such R have a low probability. (But yeah, it’s also required that uy and R are chosen independently—otherwise an adversary could just choose either so that it results in the players choosing different actions.)
The smoothness comes in from marginalising a random R. The coordination comes from making R and ε common knowledge, so they cooperate using the correlation in their observations—an interesting phenomenon.
uy and R are independently chosen from well-defined distributions. Regardless of sequence, neither knows the other and CANNOT be chosen based on the other. I’ll see if I can find time tonight to figure out whether I’m saying your claim 1 is wrong (it dropped epsilon too soon from the floor value, but I’m not sure if it’s more fundamentally problematic than that) or that your claim 2 is misleading.
My current expectation is that I’ll find that your claim 2 results are available in situation 1, by using your given function with a pre-agreed value rather than a random one.
The theorems are of the form “For all uy, you get good outcomes” or “There exists a uy that causes bad outcomes”.
When you want to prove statements of this form, uy is chosen adversarially, so it matters whether it is chosen before or after R.
What distribution is uy chosen from? That’s not specified anywhere in the post.
True, they will fail to cooperate for some R, but the values of such R have a low probability. (But yeah, it’s also required that uy and R are chosen independently—otherwise an adversary could just choose either so that it results in the players choosing different actions.)
The smoothness comes in from marginalising a random R. The coordination comes from making R and ε common knowledge, so they cooperate using the correlation in their observations—an interesting phenomenon.
(How can I write LaTeX in the comments?)
ctrl-4