Hierarchies of Beliefs and Measurable Uniformizations
Stuart Zoble
TL;DR
The paper investigates the Fundamental Theorem of Epistemic Game Theory under determinacy axioms, extending the equivalence between rationalizability and rationality with common belief in rationality to broader payoff classes and strategy spaces. It leverages interactive epistemics, universal type spaces, and measurable uniformization (MUP/MEA) to connect determinacy with real Gale-Stewart games and uniformization arguments. Under AD$_ ext{R}$ and its projective/determinacy variants, RAT$(E)=RCBR$(E) for analytic and projective payoffs, with extensions to Baire class one payoffs and Polish spaces; under Continuum Hypothesis, independence results yield a counterexample to the equivalence. The work highlights a deep interplay between descriptive set theory and epistemic game theory, identifying the limits and optimality of measurability assumptions via Solovay-model uniformization and large-cardinal considerations.
Abstract
We extend the Fundamental Theorem of Epistemic Game Theory to games with Baire class one payoffs and locally compact Polish strategy spaces, and under Projective Determinacy, to games with analytically measurable payoffs and arbitrary Polish strategy spaces. We show that in full generality, the statement that rationalizable strategies are consistent with rationality and common belief in rationality follows from the Axiom of Real Determinacy, has a characterization in terms of real Gale-Stewart games, fails under the Continuum Hypothesis, and in the framework of interactive epistemics is equivalent to the Measurable Uniformization Principle from the Solovay model.