Taking the view that infinite plays are \emph{draws}, we study \emph{Conway infinite games} and \emph{non-losing strategies}. 
These admit a sharp \emph{coalgebraic} presentation, where infinite games are seen as a \emph{final coalgebra} and game 
contructors, such as \emph{disjunctive sum}, as \emph{final morphisms}. We have shown, in a previous paper, that Conway`s theory of terminating games can be rephrased naturally in terms of game \emph{congruences}. Namely, various conceptually independent notions of \emph{equivalence} can be defined and shown to coincide on Conway's terminating games. These are
the equivalence induced by the ordering on \emph{surreal numbers}, the \emph{contextual equivalence} determined by  observing what player has a  \emph{winning strategy}, Joyal`s 
\emph{categorical} equivalence, and, for impartial games, the \emph{denotational} and \emph{operational} equivalence induced by \emph{Grundy semantics}. 
In this paper we discuss generalizations of such equivalences to \emph{infinite games} and \emph{non-losing strategies}. The scenario is even more rich and intriguing in this case. In particular, 
we investigate efficient characterizations of the contextual equivalence, and we introduce a category of \emph{balanced strategies}, generalizing
Joyal's category of Conway's games and winning strategies. Interestingly, the category of balanced strategies captures the equivalence defined by Berlekamp, Conway, Guy on \emph{loopy games}.