Inconsistencies of Weighted Summation Voting Systems

Let α_N be any ordinal ordering of N ≥ 3 alternatives. Select some one alternative and let α_N−1 be some ordering of the remaining (N − 1) alternatives where α_N−1, need not have any relationship to α_n. It is shown for a voting system coming from a large class of weighted summation voting systems that there exist examples of voter profiles such that if the voters vote on N alternatives the aggregated result is α_N, but if they vote on (N − 1) alternatives, the aggregated result is α_N−1. This result holds even if the voting system changes with the number of alternatives.