In this volume the author develops and applies methods for proving,
from large cardinals, the determinacy of definable games of countable
length on natural numbers. The determinacy is ultimately derived
from iteration strategies, connecting games on natural numbers with
the specific iteration games that come up in the study of large cardinals.
The games considered in this text range in strength, from games of
fixed countable length, through games where the length is clocked
by natural numbers, to games in which a run is complete when its
length is uncountable in an inner model (or a pointclass) relative to
the run. More can be done using the methods developed here,
reaching determinacy for games of length $\omega_1$.
The book is largely self-contained. Only graduate level knowledge
of modern techniques in large cardinals and basic forcing is assumed.
Several exercises allow the reader to build on the results in the text,
for example connecting them with universally Baire and
homogeneously Suslin sets. Overall it is intended that the book
should be accessible both to specialists and to advanced graduate
students in set theory