Triple-consistent social choice and the majority rule

We define generalized (preference) domains \(\mathcal{D}\) 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 \(\mathcal{D}\) . We prove that, for any domain \(\mathcal{D}\) , the outcome of issue-wise majority voting φ _m belongs to \(\mathcal{D}\) at any profile where φ _m is well-defined if and only if this is true when φ _m is applied to any profile involving only 3 elements of \(\mathcal{D}\) . 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.