Triple-consistent social choice and the majority rule

We define generalized (preference) domains <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal{D}$</EquationSource> </InlineEquation> as subsets of the hypercube {−1,1}<Superscript> D </Superscript>, 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal{D}$</EquationSource> </InlineEquation>. We prove that, for any domain <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal{D}$</EquationSource> </InlineEquation>, the outcome of issue-wise majority voting φ <Subscript> m </Subscript> belongs to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal{D}$</EquationSource> </InlineEquation> at any profile where φ <Subscript> m </Subscript> is well-defined if and only if this is true when φ <Subscript> m </Subscript> is applied to any profile involving only 3 elements of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal{D}$</EquationSource> </InlineEquation>. 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. Copyright Sociedad de Estadística e Investigación Operativa 2014