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I'm interested to know results on repeated games indexing each stage game by $\mathbb Z$ which contrast with those indexed by $\mathbb Z_+$.
It seems to me this could be quite different from repeated games with a starting stage, because there's no need to concern ourselves about whether or not a node could be obtained. Is it the case that we only need to consider stationary equilibrium?
What you're asking seems to me just a matter of interpretation. Note that $\mathbb Z$ and $\mathbb Z_+$ have the same cardinality. So it makes no substantive difference which index set you use.
Moreover, in a continuation game with finite history, any strategy can be interpreted as a (possibly non-stationary) Markov strategy in a continuation game with infinitely long history.
If a history of infinite length is what you really care about, then may be looking into the literature of continuous-time repeated game will provide some insight.
I do not know of any theory of repeated games without a beginning. However, some formalization may help regarding your question about stationary equilibria. I will answer your question with respect to the simple Nash equilibrium, but my remarks should also apply to more refined equilibrium concepts. Note first that we do not need to change our definition of a Nash equilibrium, as it is abstractly defined for any strategy space.
A Nash equilibrium in a repeated game with a starting point does not need to be stationary. For example, an infinitely repeated prisoners dilemma can have any sequence of "cooperate" and "defect" for both players given players do not discount future payoffs too much. This is the same for a game without a starting point. Suppose that the players have access to a binary randomization device to coordinate their strategies. Suppose they both play "cooperate" if the randomization device is 1 and "defect" otherwise. Any player deviating from this, will be punished with the other player playing "defect" for the remainder of the game. Now set the randomization device to probability $\frac{t^2}{t^2-1}$ of being 1 in period t. Clearly, the equilibrium strategies are non-stationary. However, players have no incentive to deviate as long as their time discounting is not too high.
