Triple-consistent Social Choice and the Majority Rule

We define generalized (preference) domains D as subsets of the hypercube {−1, 1}, 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 D. We prove that the outcome of issue-wise majority voting φm belongs to D at any profile where φm is well-defined if and only if this is true when φm is applied to any profile involving only 3 elements of D. 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.