Axiom of Determinacy - a Journey in Game Theory

Specialeforsvar ved Theis Kehlet Nielsen

Titel: Axiom of Determinacy - A Journey in Game Theory

  

Abstract: In this thesis we study the axiom of determinacy. We show that the axiom implies that every subset of a perfect Polish space has regularity properties, namely the property of Baire, the perfect set property and Lebesgue measurability. By doing this, we demonstrate that axiom of determinacy is in some sense more convenient than axiom of choice. Using axiom of choice we construct a Bernstein set, showing that axiom of determinacy and axiom of choice are incompatible. In the recent years, efforts has been made to show that axiom of determinacy implies Blackwell determinacy. Where axiom of determinacy is a statement about perfect information games, Blackwell of determinacy is a statement about imperfect information games. We follow Donald A. Martins article from 1998 showing that axiom of determinacy implies Blackwell determinacy. Martin also make the conjecture that Blackwell determinacy implies axiom of determinacy but the truth of this theorem is still open. We then follow Vervoort’s proof, that Blackwell determinacy implies that every subset of a perfect Polish space has Lebesgue measurability, thus giving a new proof that axiom of determinacy implies Lebesgue measurability. Whether Blackwell determinacy implies the property of Baire or the perfect set property is still to this day unknown.

  

Vejleder:   Asger D. Törnquist
Censor:     David Kyed, SDU