Game characterizations of function classes and Weihrauch degrees

Games are an important tool in mathematics and logic, providing a clear and intuitive understanding of the notions they define or characterize. In particular, since the seminal work of Wadge in the 1970s, game characterizations of classes of functions in Baire space have been a rich area of research, having had significant and far-reaching development by van Wesep, Andretta, Duparc, Motto Ros, and Semmes, among others. In this thesis we study the connections between these games and the notion of Weihrauch reducibility, introduced in the context of computable analysis to express a particular type of continuous reducibility of functions. Especially through the work of Brattka and his collaborators, Gherardi, de Brecht and Pauly, among others, several functions — commonly referred to as choice principles — have been isolated that capture the complexity of classes of functions with respect to Weihrauch reducibility. In particular, we use the games for the corresponding classes to provide new proofs of the Weihrauch-completeness of countable choice for the Baire class 1 functions and of discrete choice for the functions preserving ∆2 under preimages, and to introduce new Weihrauch-complete choice principles for the class of functions preserving ∆3 under preimages and for a particular class Λ2,3 characterized by a games of Semmes. In the process, we recast some of these games in a different style, which also allows for a uniform intuitive view of the way each game presented is related to the class of functions it characterizes.