Linear Algebra

146 Chapter Two. Vector Spacesdirection that we have chosen for the arrows, while the second voter’s spin isnegative.The fact that these opposite voters cancel each other is reflected in the factthat their vote vectors add to zero. This suggests an alternate way to tally anelection. We could first cancel as many opposite preference lists as possible, andthen determine the outcome by adding the remaining lists.The rows of the table below contain the three pairs of opposite preferencelists. The columns group those pairs by spin. For instance, the first row containsthe two voters just considered.positive spinDemocrat > Republican > ThirdD−11T R1DD1/31/3 −4/32/3=T R1/3+T R2/3Republican > Third > DemocratD1−1T R1=DD1/31/3 2/3−4/3T R1/3+T R2/3Third > Democrat > RepublicanD11T R−1=DD1/31/3 2/32/3T R1/3+T R−4/3negative spinThird > Republican > DemocratD1−1T R−1D−1T−1D−1/3=T R−1/3D−1/3 4/3−2/3+T R−2/3Democrat > Third > Republican1RDD−1/3−1/3−2/34/3=T R−1/3+T R−2/3Republican > Democrat > ThirdD−1−1T R1D−1/3=T R−1/3D−1/3−2/3−2/3+T R4/3If we conduct the election as just described then after the cancellation of asmany opposite pairs of voters as possible then there will be left three sets ofpreference lists: one set from the first row, one from the second row, and onefrom the third row. We will finish by proving that a voting paradox can happenonly if the spins of these three sets are in the same direction. That is, for avoting paradox to occur, the three remaining sets must all come from the left ofthe table or all come from the right (see Exercise 3). This shows that there issome connection between the majority cycle and the decomposition that we areusing — a voting paradox can happen only when the tendencies toward cyclicpreference reinforce each other.For the proof, assume that we have cancelled opposite preference orders andwe are left with one set of preference lists from each of the three rows. Considerthe sum of these three (here, the numbers a, b, and c could be positive, negative,or zero).−aTDRa+bTDR−b+cTDRc=−a + b + cTDa − b + cRab−ca + b − cA voting paradox occurs when the three numbers on the right, a − b + c anda + b − c and −a + b + c, are all nonnegative or all nonpositive. On the left,