  We revisit the crucial issue of natural game equivalences, and semantics of
game logics based on these. We present reasons for investigating finer concepts
of game equivalence than equality of standard powers, though staying short of
modal bisimulation. Concretely, we propose a more finegrained notion of
equality of "basic powers" which record what players can force plus what they
leave to others to do, a crucial feature of interaction. This notion is closer
to game-theoretic strategic form, as we explain in detail, while remaining
amenable to logical analysis. We determine the properties of basic powers via a
new representation theorem, find a matching "instantial neighborhood game
logic", and show how our analysis can be extended to a new game algebra and
dynamic game logic.
