  The object of this study are countably infinite games with perfect
information that allow players to choose among arbitrarily many moves in a
turn; in particular, we focus on the generalisations of the finite board games
of Hex and Draughts.
  In chapter 1 we develop the theory of transfinite ordinal game values for
open infinite games following Evans and Hamkins (arXiv:1302.4377), and we focus
on the properties of the omega one, that is the supremum of the possible game
values, of classes of open games; we moreover design the class of
climbing-through-T games as a tool to study the omega one of given game
classes.
  The original contributions of this research are presented in the following
two chapters.
  In chapter 2 we prove classical results about finite Hex and present Infinite
Hex, a well-defined infinite generalisation of Hex. We then introduce the class
of stone-placing games, which captures the key features of Infinite Hex and
further generalises the class of positional games already studied in the
literature within the finite setting of Combinatorial Game Theory. The main
result of this research is the characterization of open stone-placing games in
terms of the property of essential locality, which leads to the conclusion that
the omega one of any class of open stone-placing games is at most $\omega$. In
particular, we obtain that the class of open games of Infinite Hex has the
smallest infinite omega one, that is $\omega_1^{\rm Hex}=\omega$.
  In chapter 3 we show a dual result; we define the class of games of Infinite
Draughts and explicitly construct open games of arbitrarily high game value
with the tools of chapter 1, concluding that the omega one of the class of open
games of Infinite Draughts is as high as possible, that is $\omega_1^{\rm
Draughts}=\omega_1$.
