We study repeated normal form games where the number of players is large and suggest that it is useful to consider a neighbourhood structure on the players. The structure is given by a graph G whose nodes are players and edges denote visibility. The neighbourhoods are maximal cliques in G. The game proceeds in rounds where in each round the players of every clique X of G play a strategic form game among each other. A player at a node v strategises based on what she can observe, i.e., the strategies and the outcomes in the previous round of the players at vertices adjacent to v. Based on this, the player may switch strategies in the same neighbourhood, or migrate to another neighbourhood. Player types, giving the rationale for such switching, are specified in a simple modal logic.We show that given the initial neighbourhood graph and the types of the players in the logic, we can effectively decide if the game eventually stabilises. We then prove a characterisation result for these games for arbitrary types using potentials. We then offer some applications to the special case of weighted co-ordination games where we can compute bounds on how long it takes to stabilise.