Using Kelley's intersection number (and a variant of it) we define two classes of simple games, the regular and the strongly regular games. We show that the strongly regular games are those in which the set of winning coalitions and the set of losing coalitions can be strictly separated by a finitely additive probability measure. This, in particular, provides a combinatorial characterization for the class of finite weighted majority games within the finite simple games. We also prove that regular games have some nice properties and show that the finite regular games are exactly those simple games which are uniquely determined by their counting vector. This, in particular, generalizes a result of Chow and Lapidot.