  We show the equivalence between the existence of winning strategies for
$G_{\delta \sigma}$ (also called $\Sigma^{0}_{3}$) games in Cantor or Baire
space, and the existence of functions generalized-recursive in a higher type-2
functional. (Such recursions are associated with certain transfinite
computational models.) We show, inter alia, that the set of indices of
convergent recursions in this sense is a complete $\Game \Sigma_{3}^{0}$ set:
as paraphrase, the listing of those games at this level that are won by player
I, essentially has the same information as the `halting problem' for this
notion of recursion. Moreover the strategies for the first player in such games
are recursive in this sense. We thereby establish the ordinal length of
monotone $\Game \Sigma^{0}_{3}$-inductive operators, and characterise the first
ordinal where such strategies are to be found in the constructible hierarchy.
  In summary:
  Theorem (a) The following sets are recursively isomorphic.
  (i) The complete ittm-semi-recursive-in-${eJ}$ set, $H^{eJ}$;
  (ii) the $\Sigma_{1}$-theory of $( L_{\eta_{0}} , \in ) $, where $\eta_{0}$
is the closure ordinal of $\Game \Sigma_{3}^{0}$-monotone induction;
  (iii) the complete $\Game \Sigma_{3}^{0}$ set of integers.
  (b) The ittm-recursive-in-${eJ}$ sets of integers are precisely those of
$L_{\eta_{0}}$.
