Hierarchies of Beliefs and Measurable Uniformizations

TL;DR

This paper extends the Fundamental Theorem of Epistemic Game Theory to broader classes of payoffs and strategy spaces, linking rationalizability to determinacy axioms and measurable uniformization principles.

Contribution

It generalizes the theorem to Baire class one and analytic payoffs, and connects epistemic concepts with determinacy axioms and set-theoretic principles.

Findings

Rationalizable strategies align with rationality and common belief under the Axiom of Real Determinacy.

The statement fails under the Continuum Hypothesis.

In the framework of interactive epistemics, it is equivalent to the Measurable Uniformization Principle.

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.