My understanding is Nash equilibrium exists in any game with finite states, regardless of perfect or imperfect information.
The bit about announcing your strategy holds by definition - the equilibrium is defined such that no player can improve, thus it doesn't matter if the other player's strategy is known. Also, I don't think there's a requirement that Nash strategies be reachable by iteration.
What I wanted to say is that methods for finding it via iteration and self-play exist and are well defined but only for perfect information games.
The research itself is here: https://ai.facebook.com/blog/rebel-a-general-game-playing-ai...
I am way out of it though. I suspect anything more than 2 players is unsolvable when you factor in the stochastic nature of the drunk guy at the table with more than 2 players. Then based on his play he subtracts or adds EV to us randomly.
I won't even let myself think past this though with poker because IMO it is a waste of time.