The Generic Existence of a Core for q-Rules

A q-rule is where, for n-voters, a winning coalition consists of q or more voters. An important question is to determine when, generically, core points exist; that is, when the core exists in other than highly contrived settings. As known, the answer depends upon the dimension of issue space. McKelvey and Schofield found bounds on these dimensions, but Banks found a subtle, but critical error in their proofs. The sharp dimensional values along with results about the structure of the core are derived.