When considering (set-theoretic) games, we have three parameters we can adjust:

- Definability of the payoff set
- The set of legal moves
- The length of the game

When working in $\textsf{ZFC}$, what are our limitations on the above three parameters? When do we reach inconsistencies? Some more specific subquestions could be:

- What is the least $\alpha$ such that open (or analytic) determinacy on $\omega$ of length $\alpha$ is inconsistent, if such an $\alpha$ exists?
- Same question as above but instead consider analytic determinacy on $\alpha$ of length $\omega$.

I have found some partial results:

- If we have no definability requirement then we can't even have determinacy of length $\omega$ games on $\omega$ (as this is just $\textsf{AD}$).
- If we consider $\textsf{OD}$ games then (according to wiki) it's consistent relative to the sharp of a Woodin limit of Woodins that games on $\omega$ of length $\omega_1$ are determined, but it's inconsistent to consider length $\omega_1+\omega$ games. Is this relative to $\textsf{ZFC}$? (EDIT: Yes it is)
- Caicedo mentions here that it's consistent that all $\textsf{OD}$ games on ordinals of length $\omega$ are determined - but again, is this relative to $\textsf{ZFC}$? I suppose not.

Also, here Noah Schweber asks the same question, but where we fix the length to be $\omega$, the legal moves to be $\mathbb R$ and then ask about the definability.

**EDIT**: I've written up an overview based on Juan's answer below and by taking a closer look at the complexity of the payoff sets in the non-determinacy proofs - this can be found here. Here are a few diagrams to illustrate.