Limitations of ZFC determinacy

I was recently playing a (set-theoretic) game and the question of whether it was determined slowly emerged. As I was working in a ZFC context, most of the determinacy results were of no use to me, so I tried to investigate how much we really know about ZFC determinacy. Of course we can’t have full determinacy (AD), but how about definable variants, where we alter both the objects played and the length of the game?

We can consider games of type (X,\alpha,\Gamma), where X is the set of objects played, \alpha is the length of the game and \Gamma\subseteq \mathcal P(X^\omega) is a pointclass in which our payoff sets lie. My question was then

Question. What are the boundaries to ZFC determinacy? For what game types (X,\alpha,\Gamma) do we reach an inconsistency? And also the dual question, what game types (X,\alpha,\Gamma) are consistently determined relative to large cardinals?

To enforce some kind of ordering on the pointclasses, I’ll work with the commonly used boldface variants in determinacy contexts:

{\bf\Delta}^1_1, {\bf\Sigma}^1_1, {\bf\Sigma}^1_2, {\bf\Sigma}^1_3,\dots,{\bf\Sigma}^2_1,{\bf\Sigma}^2_2,\dots,\textsf{OD}(\mathbb R).

Just to be clear, I’m only considering these classes with respect to V_\omega, which means that {\bf\Delta}^1_1 will for instance mean a single existential quantifier over the reals followed by an arithmetical formula, with real parameters. When we’re playing games on \omega this is the same thing as the descriptive set theoretic notions of taking projections and complements of closed sets, but as soon as we move away from playing integers, these pointclasses are a lot smaller than their descriptive set theoretic analogues. Also, in this post I’ll consider the following variants of objects played:

\omega, \mathbb R=\mathcal P(\omega_0), \mathcal P(\omega_1), \mathcal P(\omega_2),\hdots

As we got three dimensions I’ll be working in two-dimensional cross-sections to make things a bit simpler. Let’s start with length \omega games. To start things off we have the incredible Borel determinacy result by Martin, giving us determinacy of all Borel games of length \omega, no matter what objects are played. As I mentioned above the Borel sets are really a much larger pointclass than the {\bf\Delta}^1_1 class we’re considering, making it a bit of an overkill, but it works.

When we move to playing games on \mathcal P(\omega_1) we reach an inconsistency. Indeed, if we let player I play an \omega_1-sequence X of reals as his first move (which is possible as such a sequence can be encoded as an \omega_1-sequence of integers). If X has a perfect subset then it encodes a well-order of the reals and the proof that AD+AC is inconsistent gives us our non-determined game. If X does not have a perfect subset then the perfect set game on X is non-determined, as X then doesn’t have the perfect set property.  This is a (lightface) \Delta^2_2 definition, giving us a lower complexity bound for inconsistency when playing games of length \omega.

When it comes to longer length games it’s quite analogous, where determinacy of \Sigma^2_2 games of length \omega_1+\omega on \omega is inconsistent, where the first \omega_1 many moves are simply used to reconstruct the above sequence X. On the other hand, Woodin has shown that it’s consistent (relative to large cardinals) that we have determinacy of \textsf{OD}(\mathbb R) games of length \omega_1 on the reals (this can be found in Neeman’s book on long games, exercise 7F.15).

When we get to play subsets of the reals, we can simply play an undetermined set A\subseteq 2^\omega of reals, whereafter they play either \emptyset or \mathbb R, encoding an element of A. Here the payoff set is (at least) \Pi^1_2, so we get a lower inconsistency bound in this case. The analogous case is for length \mathfrak{c}+\omega games on \omega. In total, we end up with the following diagrams:

Here the red colour symbolises an inconsistency and the blue colour that the determinacy is consistent modulo large cardinals. I’m not sure about what happens in the white area. Again, the inconsistency lower bounds are quite naive – they might be a lot lower.