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Thus, the determinacy implies that all sufficiently closed countable $M⊂V_λ$ satisfy the same $\text{Theory}(V_λ,∈,M)$ (which depends on $λ$). Many things are definable from such $M$, including for every ordinal $δ<λ$ of uncountable cofinality, a cofinal sequence (which can depend on $M$) of length $\mathrm{cf}(δ)$.

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 $ω_1, \, ω_1\!+\!ω, \, ω_1ω, \, ω_1ω\!+\!ω$, and one can analogously classify $M$ of higher cardinality.

Thus, the determinacy implies that all sufficiently closed countable $M⊂V_λ$ satisfy the same $\text{Theory}(V_λ,∈,M)$ (which depends on $λ$). Many things are definable from such $M$, including for every well-ordering $δ∈∪M$ with $\mathrm{cf}(δ)>ω$, a cofinal sequence (which can depend on $M$) of length $\mathrm{cf}(δ)$, and hence for $λ>ω+2$, $ω_2$ disjoint stationary subsets of $ω_2$ of uncountable cofinality, and similarly for larger ordinals.

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 $ω_1, \, ω_1\!+\!ω, \, ω_1ω, \, ω_1ω\!+\!ω$, and one can analogously classify $M$ of higher cardinality.

The determinacy likely has key implications for inner model theory. Even with moves restricted to ordinals, length $ω$, and OD payoff, it would refute Woodin's HOD Conjecture (assuming, per the conjecture's assumptions, an extendible cardinal).

• 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 countably many checkpoints at ordinals of cofinality $κ$ or limits of such ordinals.

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

• 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 restriction is $α=1$.
• A natural extension is to allow countably many checkpoints $κβ_γ$, and given a run $f$, allow payoff to depend on $\mathrm{ran}(f|κβ_γ)$ for each $γ$, with the intervals between successive checkpoints (counting game start, end, and limits of checkpoints as checkpoints) additively indecomposable and with non-increasing cardinality.

Thus, the determinacy implies that all sufficiently closed countable $M⊂V_λ$ satisfy the same $\text{Theory}(V_λ,∈,M)$ (which depends on $λ$). Many things are definable from such $M$, including for every ordinal $δ<λ$ of uncountable cofinality, a cofinal sequence (which can depend on $M$) of length $\mathrm{cf}(δ)$.

• 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 $≥κ$.

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).

• 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 countably many checkpoints at ordinals of cofinality $κ$ or limits of such ordinals.

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 $ω_1, \, ω_1\!+\!ω, \, ω_1ω, \, ω_1ω\!+\!ω$, and one can analogously classify $M$ of higher cardinality.

Extending lengths: If games in the question on $V_{λ+2}$ (in most cases, $V_{λ+1}$ suffices) with length $κ$ and $\text{OD}(X)$ payoff are determined, then for such games on $V_λ$ with $\text{OD}(X)$ payoff:
- for arbitrary length $κα$, we get a weakening of determinacy: combine played strategies that at each stage $κβ$, give the next $κ$ moves as a function of $\mathrm{ran}(f|κβ)$ where $f$ is the play of the game, and see who wins.
- for game length $α$ with $\mathrm{cf}(α)=κ$ and $∀β<α \; β+|α|κ≤α$, we get full determinacy (take the union of all sets of size $≤|α|$ that were played (in the game of length $κ$) to see who wins).

Supercompactness: Using the determinacy for games of length $κ$, we get an ordinal definable $κ^+$-complete 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).