Your formulation doesn't work well if games are allowed to be infinite, because standard propositional logic doesn't handle formulae of infinite length.
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.
Neither propositional logic nor first-order logic (FOL) allow formulae of infinite length.
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).
You can convert any terminating formula (including recursive) to a finite logic representation in second order logic with exponential number of terms. This by use of Peano numbering.
Proving whether the given (inductive or recursive) formula is actually terminating is the halting problem though.
Linear logic is closely related to martingales too.
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.