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?**