“…Speci…cally, by linking response functions across partitions, compatibility ensures functional dependence between …ne and coarse partition e¤ects, generalizing the e¤ect linkages studied in the classical identi…cation problem for linear structural systems (e.g., Fisher, 1966 Suppose that action spaces are compact convex locally convex semilattices or compact locally complete metric semilattices, that type spaces are partially ordered probability spaces, that payo¤s are continuous in actions for each type vector, and that the joint distribution over types induces atomless marginals for each player assigning positive probability only to sets that can be order-separated by a …xed countable set of his types. 13 If, whenever the others employ monotone pure strategies, each player's set of monotone pure-strategy best replies is non-empty and join-closed, 14 then a monotone pure-strategy equilibrium exists.…”