I'm writing up some math results developed on LW as a paper with the tentative title "Self-referential decision algorithms". Something interesting came up while I was cleaning up the Loebian cooperation result. Namely, how do we say precisely that Loebian cooperation is stable under minor syntactic changes? After all, if we define a "minor change" to program A as a change that preserves A's behavior against any program B, then quining cooperation is just as stable under such "minor changes" by definition. Digging down this rabbit hole, I seem to have found a nice new reformulation of the whole thing.
I will post some sections of my current draft in the comments to this post. Eventually this material is meant to become an academic paper (hopefully), so any comments on math mistakes, notation or tone would be much appreciated! And yeah, I have no clue about academic writing, so you're welcome to tell me that too.
Your linked result seems to talk about average utilities in the long run, which corresponds to a discount factor of 1. In the general case it seems to me that discount factors can change the outcome. For example, if the benefit of unilaterally defecting instead of cooperating on the first move outweighs the entire future revenue stream, then cooperating on the first move cannot be part of any Nash equilibrium. I found some results saying indefinite cooperation is sustainable if the discount factor is below a certain threshold.
That sounds reasonable. If v is the expected discounted utility at minmax, w the expected discounted utility according to the cooperative strategy, then whenever the gain to defection is less than w-v, we're fine.