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
- of length $ω$ with OD payoff ?
- on ordinals with OD payoff ?
- of length $ω_1$ on ordinals with $\mathrm{OD}(\mathrm{On}^{ω_1})$ payoff (assuming CH if needed) ?
Notes:
- 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.
Consequences of the determinacy
Determinacy of the games implies very strong symmetry principles. For the games of countable (limit) length and payoff $A$,
- player I wins iff all sufficiently closed countable $M⊂V_λ$ are in $A$ (equivalently, $A$ includes a club subset of $P_{ω_1}(V_λ)$)
- 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 $λ$). 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).
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).
