Triple-consistent social choice and the majority rule

We define generalized (preference) domains as subsets of the hypercube {-1,1} (D) , where each of the D coordinates relates to a yes-no issue. Given a finite set of n individuals, a profile assigns each individual to an element of . We prove that, for any domain , the outcome of issue-wise majority voting phi (m) belongs to at any profile where phi (m) is well-defined if and only if this is true when phi (m) is applied to any profile involving only 3 elements of . We call this property triple-consistency. We characterize the class of anonymous issue-wise voting rules that are triple-consistent, and give several interpretations of the result, each being related to a specific collective choice problem.