Probably going to reply to the rest later (and midco can as well, of course), but regarding:
Coming back after reading more, do you use σ−i to mean “the strategy profile for every process except i”? That would make more sense of the formulas (since you fix ai, there’s no reason to have a σi) but if it’s the case, then this notation is horrible (no offense).
By the way, indexing the other strategies by −i instead of, let’s say j or k is quite unconventional and confusing.
Using “σ−i” to mean “the strategy profile of everyone but player i” is common notation; I remember it being used in 2-3 game theory textbooks I read, and you can see its prominence by consulting the Wikipedia page for Nash equilibrium.
Do I agree this is horrible notation? Meh. I don’t know. But it’s not a convention we pioneered in this work.
Ok, that’s fair. It’s hard to know which notation is common knowledge, but I think that adding a sentence explaining this one will help readers who haven’t studied game theory formally.
Maybe making all vector profiles bold (like for the action profile) would help to see at a glance the type of the parameter. If I had seen it was a strategy profile, I would have inferred immediately what it meant.
Probably going to reply to the rest later (and midco can as well, of course), but regarding:
Using “σ−i” to mean “the strategy profile of everyone but player i” is common notation; I remember it being used in 2-3 game theory textbooks I read, and you can see its prominence by consulting the Wikipedia page for Nash equilibrium.
Do I agree this is horrible notation? Meh. I don’t know. But it’s not a convention we pioneered in this work.
Ok, that’s fair. It’s hard to know which notation is common knowledge, but I think that adding a sentence explaining this one will help readers who haven’t studied game theory formally.
Maybe making all vector profiles bold (like for the action profile) would help to see at a glance the type of the parameter. If I had seen it was a strategy profile, I would have inferred immediately what it meant.