  Spoiler-Duplicator games are used in finite model theory to examine the
expressive power of logics. Their strategies have recently been reformulated as
coKleisli maps of game comonads over relational structures, providing new
results in finite model theory via categorical techniques. We present a novel
framework for studying Spoiler-Duplicator games by viewing them as event
structures. We introduce a first systematic method for constructing comonads
for all one-sided Spoiler-Duplicator games: game comonads are now realised by
adjunctions to a category of games, generically constructed from a comonad in a
bicategory of game schema (called signature games). Maps of the constructed
categories of games are strategies and generalise coKleisli maps of game
comonads; in the case of one-sided games they are shown to coincide with
suitably generalised homomorphisms. Finally, we provide characterisations of
strategies on two-sided Spoiler-Duplicator games; in a common special case they
coincide with spans of event structures.
