We provide new evidence about US monetary policy using a model that: (i) estimates time-varying monetary policy weights without relying on stylized theoretical assumptions; (ii) allows for endogenous breakdowns in the relationship between interest rates, inflation, and output; and (iii) generates a unique measure of monetary policy activism that accounts for economic instability. The joint incorporation of endogenous time-varying uncertainty about the monetary policy parameters and the stability of the relationship between interest rates, inflation, and output materially reduces the probability of determinate monetary policy. The average probability of determinacy over the period post-1982 to 1997 is below 60% (hence well below seminal estimates of determinacy probabilities that are close to unity). Post-1990, the average probability of determinacy is 75%, falling to approximately 60% when we allow for typical levels of trend inflation.

A short core model induction proof of $\mathsf {AD}^{L(\mathbb {R})}$ from $\mathsf {TD} + \mathsf {DC}_{\mathbb {R}}$.

Work by Chomsky et al. (2019) and Epstein et al. (2018) develops a third-factor principle of computational efficiency called “Determinacy”, which rules out “ambiguous” syntactic rule-applications by requiring one-to-one correspondences between the input or output of a rule and a single term in the domain of that rule. This article first adopts the concept of “Input Determinacy” articulated by Goto and Ishii (2019, 2020), who apply Determinacy specifically to the input of operations like Merge, and then proposes to extend Determinacy to the labeling procedure developed by Chomsky (2013, 2015). In particular, Input Determinacy can explain restrictions on labeling in contexts where multiple potential labels are available (labeling ambiguity), and it can also provide an explanation for Chomsky's (2013, 2015) proposal that syntactic movement of an item (“Internal Merge”) renders that item invisible to the labeling procedure.

Assume $\mathsf {ZF} + \mathsf {AD}$ and all sets of reals are Suslin. Let $\Gamma $ be a pointclass closed under $\wedge $, $\vee $, $\forall ^{\mathbb {R}}$, continuous substitution, and has the scale property. Let $\kappa = \delta (\Gamma )$ be the supremum of the length of prewellorderings on $\mathbb {R}$ which belong to $\Delta = \Gamma \cap \check \Gamma $. Let $\mathsf {club}$ denote the collection of club subsets of $\kappa $. Then the countable length everywhere club uniformization holds for $\kappa $: For every relation $R \subseteq {}^{<{\omega _1}}\kappa \times \mathsf {club}$ with the property that for all $\ell \in {}^{<{\omega _1}}\kappa $ and clubs $C \subseteq D \subseteq \kappa $, $R(\ell ,D)$ implies $R(\ell ,C)$, there is a uniformization function $\Lambda : \mathrm {dom}(R) \rightarrow \mathsf {club}$ with the property that for all $\ell \in \mathrm {dom}(R)$, $R(\ell ,\Lambda (\ell ))$. In particular, under these assumptions, for all $n \in \omega $, $\boldsymbol {\delta }^1_{2n + 1}$ satisfies the countable length everywhere club uniformization.

Schmidt’s game and other similar intersection games have played an important role in recent years in applications to number theory, dynamics, and Diophantine approximation theory. These games are real games, that is, games in which the players make moves from a complete separable metric space. The determinacy of these games trivially follows from the axiom of determinacy for real games, $\mathsf {AD}_{\mathbb R}$ , which is a much stronger axiom than that asserting all integer games are determined, $\mathsf {AD}$ . One of our main results is a general theorem which under the hypothesis $\mathsf {AD}$ implies the determinacy of intersection games which have a property allowing strategies to be simplified. In particular, we show that Schmidt’s $(\alpha ,\beta ,\rho )$ game on $\mathbb R$ is determined from $\mathsf {AD}$ alone, but on $\mathbb R^n$ for $n \geq 3$ we show that $\mathsf {AD}$ does not imply the determinacy of this game. We then give an application of simple strategies and prove that the winning player in Schmidt’s $(\alpha , \beta , \rho )$ game on $\mathbb {R}$ has a winning positional strategy, without appealing to the axiom of choice. We also prove several other results specifically related to the determinacy of Schmidt’s game. These results highlight the obstacles in obtaining the determinacy of Schmidt’s game from $\mathsf {AD}$ .

We consider a real-valued function f defined on the set of infinite branches X of a countably branching pruned tree T. The function f is said to be a limsup function if there is a function $u \colon T \to \mathbb {R}$ such that $f(x) = \limsup _{t \to \infty } u(x_{0},\dots ,x_{t})$ for each $x \in X$ . We study a game characterization of limsup functions, as well as a novel game characterization of functions of Baire class 1.

We show that, assuming the Axiom of Determinacy, every non-selfdual Wadge class can be constructed by starting with those of level $\omega _1$ (that is, the ones that are closed under Borel preimages) and iteratively applying the operations of expansion and separated differences. The proof is essentially due to Louveau, and it yields at the same time a new proof of a theorem of Van Wesep (namely, that every non-selfdual Wadge class can be expressed as the result of a Hausdorff operation applied to the open sets). The exposition is self-contained, except for facts from classical descriptive set theory.

It is shown that the determinacy of $G_{\delta \sigma }$ games of length $\omega ^2$ is equivalent to the existence of a transitive model of ${\mathsf {KP}} + {\mathsf {AD}} + \Pi _1\textrm {-MI}_{\mathbb {R}}$ containing $\mathbb {R}$ . Here, $\Pi _1\textrm {-MI}_{\mathbb {R}}$ is the axiom asserting that every monotone $\Pi _1$ operator on the real numbers has an inductive fixpoint.

We analyze the hereditarily ordinal definable sets $\operatorname {HOD} $ in $M_n(x)[g]$ for a Turing cone of reals x, where $M_n(x)$ is the canonical inner model with n Woodin cardinals build over x and g is generic over $M_n(x)$ for the Lévy collapse up to its bottom inaccessible cardinal. We prove that assuming $\boldsymbol \Pi ^1_{n+2}$ -determinacy, for a Turing cone of reals x, $\operatorname {HOD} ^{M_n(x)[g]} = M_n(\mathcal {M}_{\infty } | \kappa _{\infty }, \Lambda ),$ where $\mathcal {M}_{\infty }$ is a direct limit of iterates of $M_{n+1}$ , $\delta _{\infty }$ is the least Woodin cardinal in $\mathcal {M}_{\infty }$ , $\kappa _{\infty }$ is the least inaccessible cardinal in $\mathcal {M}_{\infty }$ above $\delta _{\infty }$ , and $\Lambda $ is a partial iteration strategy for $\mathcal {M}_{\infty }$ . It will also be shown that under the same hypothesis $\operatorname {HOD}^{M_n(x)[g]} $ satisfies $\operatorname {GCH} $ .