TITLE: Games that never begin
QUESTION [18 upvotes]: 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?
Edits:  By the cold light of morning, I see that I abused the words "mutatis mutandis."  A la Kechris, Alice's strategy beats Bob's if the the unique run of the game falls in the payoff set.  What I had in mind here was that Alice's strategy beats Bob's
if the run set (consistent with both strategies) is a subset of the payoff set.  Joel David Hamkins' clever answer remains trenchant, only now with the import that Alice always wins by playing a strategy with empty run sets regardless of Bob's strategy.
Joel's Alice needs an infinite memory, but if her strategy at each move consists of increasing her previous move by 1, that also necessarily produces an empty run set regardless of Bob's choice.
Possible fix 1  Alice must play an engaged strategy, a strategy that produces a nonempty run set against at least one strategy of Bob's.
Possible fix 2  The residual game at any turn only has a finite future, so one player has a winning strategy from that point on.  Call a strategy rational if it requires the player to play a winning move in the residual game whenever one exists.  Call a strategy strongly rational if it requires the player to play the least possible winning move whenever one exists.  To avoid easy disengaged strategies that don't even reference the payoff set, insist on rational, or even strongly rational strategies.
Possible fix 3  Combine the previous fixes.

REPLY [4 votes]: A slightly more tangential answer, but one which I hope is still useful: there is a well-known connection between infinite games and infintary logic. In the usual context of games with no ending, determinacy principles can be viewed as versions of De Morgan's Law for certain infinitary sentences: for $\Gamma$ a pointclass, $\Gamma$-Det is the statement that each of the disjunctions $$ \forall x_0\exists x_1\forall x_2\exists x_3 . . . . ((x_0, x_1, x_2, x_3, . . . )\not\in X) \vee \exists x_0\forall x_1\exists x_2\forall x_3 . . . ((x_0, x_1, x_2, x_3, . . . ) \in X)$$ for $X\in\Gamma$ is true. Similarly, games with no beginnings should be connected to the semantics of infinitary sentences with ill-founded strings of quantifiers. In the paper "On languages with non-homogeneous strings of quantifiers" (https://doi.org/10.1007/BF02771553), Saharon Shelah did some work on the behavior of such sentences (his semantics for these sentences is in terms of Skolem functions; it appears to avoid Joel's observation by requiring that a strategy for one player look only at moves made by the other player, but I'm not certain of this - please correct me if I'm wrong!). The main result is that "every linear string of quantifiers can be replaced by a well-ordered sequence of quantifiers," which goes some way towards reducing the study of beginningless games to the study of endless games.
However, it should be noted that non-linear "strings" (posets?) of quantifiers have also been studied (cf. "Dependence Logic"), and I have no idea what happens if we look at branching, ill-founded collections of quantifiers, or if this has been looked at in the past (although I vaguely recall a paper by either Hintikka or Vaananen on the subject, but I can't find it, so maybe it doesn't exist). I also don't know a good game-theoretic interpretation of such collections of quantifiers, but I imagine one would not be too hard to come by.