Majority decision with “phantom” voters can be used to establish Arrow consistency when the number of people is even. Let there be \(2n\) people, and let each profile in the domain be single peaked with respect to one and the same ordering of the alternatives. Let \(R_{2n+1}\) be an ordering that also is single peaked with respect to this common ordering.  \(R_{2n+1}\) represents the preferences of a “phantom” voter. Now take each profile \(\langle R_{1},\ldots, R_{2n}\rangle\) in the domain and expand it into \(\langle R_{1},\ldots, R_{2n}, R_{2n+1}\rangle\), by adding \(R_{2n+1}\). The set of all the expanded profiles is a single peaked domain and, because the real voters together with the phantom are odd in number, Black’s result applies to it. Let \(g\) be pairwise majority decision for the expanded domain. We obtain a social welfare function \(f\) for the original domain by assigning to each profile the ordering that \(g\) assigns to its expansion. That is, we put: