Not sure what you mean here. The axiom of choice just says that the Cartesian product of a set of nonempty sets is nonempty. Can you explain what you mean by "best possible option" and "worst possible options"?
This is a silly example but compare playing chess single player vs multiplayer. In the single player chess, you still have two sides, you are still trying to make your side to win, however you decide all the moves (even for the other party). IRL, a lot of choices are outside of your control. So in some sense, you need to account for your opponent's best moves (which are generally the worst moves for you).
Two player games to me are all about quantifiers. If the opponent plays, it means we have to prove a “for all”. (This includes all worst case scenarios obviously.) And own plays are “exist” (just the best option at that time). No axiom of choice involved, just FO logic.
There is something called the Axiom of Determinacy which states that all two-player games of perfect information have a winning strategy for one player. This axiom applies to games of any length, including all types of uncountable infinities. The Axioms of Choice and Determinancy are incompatible; if one of them is true, the other can be proven false.
I don't know if this is related to what GP was talking about, but it reminded me.
While there are logics with formulae of infinite length (see [1, 2] for an overview), I would not consider them logics on par with propositional or FOL. Why? Because as a finite human you cannot actually write down infinite formulae! Instead, you use some finite abbreviation mechanism (typically some variant of FOL, extended with set theoretic axioms like ZFC) so you can denote (in a finite way) those infinite formulae. I'd suggest to see infinite formulae as a semantics gadget that is sometimes useful in model-theoretic investigations.
It's not surprising that we encounter questions independent from ZFC (or whatever your preferred foundation of mathematics) when we look at infinite games. It's but an instance of our inability to define infinite sets in an impredicative way (i.e. the usual inductive definition of the natural numbers as the least set closed under successor).
Proving whether the given (inductive or recursive) formula is actually terminating is the halting problem though.
Linear logic is closely related to martingales too.
What's the connection with martingales?
For all is a clutch. There exists is much more manageable. However you need both the worst and the best “there exists”.
https://www.semanticscholar.org/paper/Chu-spaces-as-a-semant...
Fundamentally, think of the minimax algorithm. You have two sides each optimizing for victory.