Is it consistent that for all ordinals $α$ and $λ$ and infinite regular cardinals $κ$, games on $V_λ$ with game length $κα$ and $\mathrm{OD}(\mathrm{On}^κ)$ payoff that depends only on the set of all sets played (and not by whom and in what order) are determined?

If not, can we get indeterminacy for such games<br/>
\- of length $ω$ with OD payoff ?<br/>
\- on ordinals with OD payoff ?<br/>
\- of length $ω_1$ on ordinals with $\mathrm{OD}(\mathrm{On}^{ω_1})$ payoff (assuming CH if needed) ?

<i>Notes:</i>
- As is standard, the games are two player perfect information games in which the players take turns (and exactly one player wins at the end).  In the question, every $f∈V_λ^{κα}$ is a valid run of the game, with arbitrary $\mathrm{OD}(\mathrm{On}^κ)$ payoff that depends only on $\mathrm{ran}(f)$.
- Proving consistency of the determinacy is likely beyond current techniques (and for the top question, I suspect the consistency strength is at least that of a proper class of supercompacts), but given the extremely broad class of games, there might be simple examples of undetermined games.
- Given a stationary co-stationary $A⊂ω_1$, the following game is undetermined:  game of length $ω$ on countable ordinals with player I winning iff $\sup(\text{moves})∈A$.
- I require $κ$ to be regular because for singular strong limit $κ$ of uncountable cofinality, a well-ordering of $P(κ)$ is definable from a subset of $κ$, giving a 'trivial' indeterminacy.
- A natural extension is to allow the payoff to depend on the sets of moves before $κ$ cardinal number of checkpoints of cofinality $≥κ$.

<i>Consequences of the determinacy</i>

Determinacy of the games implies very strong symmetry principles.  For the games of countable (limit) length and payoff $A$,<br/>
\- player I wins iff all sufficiently closed countable $M⊂V_λ$ are in $A$ (equivalently, $A$ includes a club subset of $P_{ω_1}(V_λ)$)<br/>
\- player II wins iff none of such $M$ are in $A$.

Thus, the determinacy implies that all sufficiently closed countable $M⊂V_λ$ satisfy the same $\text{Theory}(V_λ,∈,M)$ (which depends on $λ$).

For uncountable sizes, we have similar correspondences, but with subtleties involving cofinality.  For $|M|=ω_1$, there are exactly four distinct structure types (without the extension), corresponding to game lengths $ω_1$, $ω_1+ω$, $ω_1ω$, and $ω_1ω+ω$, and one can analogously classify $M$ of higher cardinality.

Given determinacy of the games in the question on $V_{λ+1}$ with length $ω$ and OD payoff, we get determinacy of the games on $V_λ$ with length $ω_1ω$ and OD payoff (take the union of all sets of size $≤ω_1$ that were played), and a weakening for the determinacy for length $ω_1$ (combine played strategies for length $ω_1$ that depend only on move timings up to multiples of $ω$, and see who wins), and analogously with other ordinals.

Using the determinacy for games of length $κ$, we get a $κ^+$-complete ordinal definable normal fine $\mathrm{OD}(\mathrm{On}^κ)$-ultrafilter on $P_{κ^+}(V_λ)$.  Thus, $κ^+$ has properties resembling supercompactness.  The resemblance with supercompactness also motivated me in a previous question ([Independence through forcing vs generic collapses](https://mathoverflow.net/questions/384685/independence-through-forcing-vs-generic-collapses)).