From individual to collective ordering through multidimensional attribute space

Abstract

A method is proposed for producing a collective ordering of N options or candidates, given the individual orderings of each of M voters or judges. For the sake of concreteness, the work is expressed in terms of candidates and voters. The method assumes that the candidates can be located in ‘attribute space’. This aspect of the method has previously been used in voting theory and in the scaling theory developed by psychologists. The representative points of the candidates in attribute space are regarded as fixed and known to the voters; whereas for each voter there is a point in attribute space that he would regard as ideal for the location of a candidate. Although there are only a finite number of voters, we assume that the location of their ideal points has a multivariate normal distribution in the attribute space, whose mode is called O. For each voter the surface of points in attribute space of any given non-optimal utility (indifference surface) is assumed to be an ellipsoid whose centre is at his ideal point. It is shown that any assumption for the locations of all the candidates can be used to deduce a set of putative values p v ( v = 1, 2,..., N !) for the fractions of the voters that will place the candidates in each of the N ! possible orderings. The squared error associated with any such assumption for the locations of the candidates is then defined as Ʃ v ( p v – q v ) 2 where the q v are the observed fractions of the actual ballots, and this squared error is to be minimized for estimating the locations of the candidates. The collective ordering of the candidates is then taken as that of the distances of their N representative points from O. The analysis in this paper could be applied to marketing and medical research as well as to politics and sociology.