Taking the view that infinite plays are draws, we study Conway
non-terminating games and non-losing strategies. These admit a
sharp coalgebraic presentation, where non-terminating games are seen as a
final coalgebra and game contructors, such as disjunctive
sum, as final morphisms. We have shown, in a previous paper,
that Conway’s theory of terminating games can be rephrased naturally in terms of game
(pre)congruences. Namely, various conceptually independent notions of
equivalence can be defined and shown to coincide on Conway’s
terminating games. These are the equivalence induced by the ordering on surreal
numbers, the contextual equivalence determined by observing
what player has a winning strategy, Joyal’s categorical
equivalence, and, for impartial games, the denotational
equivalence induced by Grundy semantics. In this paper, we
discuss generalizations of such equivalences to non-terminating games and
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 fair strategies and a
category of fair pairs of strategies, both generalizing Joyal’s category
of Conway games and winning strategies. Interestingly, the category of fair pairs captures
the equivalence defined by Berlekamp, Conway, Guy on loopy games.