Games in Set Theory and Logic Daisuke Ikegami Abstract: In this dissertation, we discuss several types of infinite games and related topics in set theory and mathematical logic. Chapter 1 is devoted to the general introduction and preliminaries. The rest is organized as follows: Chapter 2: It is known that the Baire property is one of the nice properties for sets of reals called regularity properties and that it can be characterized by Banach-Mazur games. We characterize almost all the known regularity properties for sets of reals via the Baire property for some topological spaces and use Banach-Mazur games to prove the general equivalence theorems between regularity properties, forcing absoluteness, and the transcendence properties over some canonical inner models. With the help of these equivalence results, we answer some open questions from set theory of the reals. Chapter 3: We discuss the connection between Gale-Stewart games and Blackwell games where the former are infinite games with perfect information coming from set theory and the latter are infinite games with imperfect information coming from game theory. The determinacy of Gale-Stewart games has been one of the main topics in set theory and one could also consider the determinacy of Blackwell games. We compare the Axiom of Real Determinacy (AD_R) and the Axiom of Real Blackwell Determinacy (Bl-AD_R). We show that the consistency strength of Bl-AD_R is strictly greater than that of the Axiom of Determinacy (AD) and that Bl-AD_R implies almost all the known regularity properties for every set of reals. We discuss the possibility of the equivalence between AD_R and Bl-AD_R under the Zermelo-Fraenkel set theory with the Axiom of Dependent Choice (ZF+DC) and the possibility of the equiconsistency between AD_R and Bl-AD_R. Chapter 4: We work on the connection between the determinacy of Gale-Stewart games and large cardinals. Iteration trees are important objects to prove the determinacy of Gale-Stewart games from large cardinals and alternating chains with length \omega are the most fundamental iteration trees connected to the determinacy of Gale-Stewart games. We investigate the the upper bound of the consistency strength of the existence of alternating chains with length \omega. Chapter 5: Wadge reducibility measures the complexity of subsets of topological spaces via the continuous reduction of a subset of a topological space to another one in descriptive set theory corresponding to many-one reducibility in recursion theory. With the help of the characterization of the Wadge reducibility for the Baire space in terms of Wadge games, one can develop the beautiful theory of the Wadge reducibility for the Baire space (e.g., almost linearity, wellfoundedness) assuming the Axiom of Determinacy (AD). We study the Wadge reducibility for the real line which cannot be characterized by infinite games in a similar way. We show that the Wadge Lemma for the real line fails and that the Wadge order for the real line is illfounded and investigate more properties of the Wadge order for the real line. Chapter 6: Modal fixed point logics are modal logics with fixed point operators and they enjoy several nice properties as first-order logic has. We define a product construction of an event model and a Kripke model and discuss the product closure of modal fixed point logics. We show that PDL, the modal \mu-calculus, and a fragment of the modal \mu-calculus are product closed. Keywords: