Games that never end play a major role in descriptive set theory. See for example Kechris' GTM.

Question: Does there exist a literature concerning games that never begin?

I have in mind two players, Alice and Bob, making alternate selections from ${\Bbb N}$, their moves indexed by increasing non-positive integers, the game terminating when Bob plays his move 0.

As for payoff sets and strategies, define these as for games that never end, mutatis mutandis.

One major difference: a pair of strategies, one for Alice, one for Bob no longer determines a unique run of the game, but rather now a set of runs, possibly empty. Even so one may still say that Alice's strategy beats Bob's if every compatible run of one strategy against the other belongs to the payoff set.

Another major difference involves the set-theoretic size of strategies. Now Alice and Bob play every move in the light of infinite history. So size considerations mean that certain familiar arguments, for example non-determined games from the axiom of choice, don't work in any obvious way?

Question: What payoff sets give determined games that never begin?