Cycling of simple rules in the spatial model

McKelvey [4] proved that for strong simple preference aggregation rules app lied to multidimensional sets of alternatives, the typical situation is tha t either the core is nonempty or the top-cycle set includes all available a lternatives. But the requirement that the rule be strong excludes, inter al ia, all supermajority rules. In this note, we show that McKelvey's theorem further implies that the typical situation for any simple rule is that eith er the core is nonempty or the weak top-cycle set (equivalently, the core o f the transitive closure of the rule) includes all available alternatives. Moreover, it is often the case that both of these statements obtain.