In this paper we point out that standard
-like logics with intersection are useful for reasoning about game structures. In particular, they can express coalitional ability operators known from coalition logic and
. An advantage of standard, normal, modal logics is a well understood theoretical foundation and the availability of tools for automated verification and reasoning. We study a minimal variant, multi-modal
with intersection of modalities, interpreted over models corresponding to game structures. There is a restriction: we consider only game structures that are injective. We give a complete axiomatisation of the corresponding models, as well as a characterisation of key complexity problems. We also prove a representation theorem identifying the effectivity functions corresponding to injective games.