There seems to be a connection between the Gini function discussed here recently and the Borda Count.
On all ballots, rankings are assumed complete and strict. Positional methods like the BC correspond bijectively (apart from a scaling factor) to the lists of n non-decreasing numbers of the form w_1=1,w_2,w_3,...,w_(n-1),0=w_n. Saari calls this a "w-vector". Of course, the correspondence is, for each ballot, assign w_1=1 to the top-ranked candidate w_2 to the second ranked ... w_(n-1) to the second-last w_n=0 to the last. Then sum over all ballots. Examples: for plurality, w_2 = 0 for Borda, the w_i are in arithmetic progression Now consider the w-vector w_G obtained by squaring all the entries in the w-vector for Borda. This is the one with a connection to the Gini function. The positional methods also correspond bijectively to the convex hull of the n-1 points p_i (for 1<=i<n-1) in R^n defined by p_i=(1,1,...,1,0,0,...0) where there are i ones followed by n-i zeroes. The points p_i even form a simplex of dimension n-2, within the hyperplane defined by "first coord=1, last coord=0" So plurality corresponds to the point p_1 and Borda corresponds to the barycentre/centroid of the simplex, namely the point obtained by taking the coordinate-wise mean of the p_i: (say) p_(BC) = sum_(i=1 to i=n-1) (p_i)/(n-1) Now this mention of a mean makes one think of other ways of taking an average, one of which is the Gini function recently mentioned in this forum. For the non-decreasing sequence of reals 0<=b_1<=b_2<=...<=b_r the value of the Gini function of these is f_Gini = (1//(r^2))*sum_(j:1<=j<=r) (2r+1-2j)*b_j The p_i above are not real numbers, but we can decree an ordering on them by saying p_i > p_j if and only if i > j. Then calculating f_Gini for the (n-1) p_i gives a positional method, and it turns out that the w-vector for this method is w_G, the one obtained by squaring all the entries in the Borda vector. Curious, no? Of course, this comes down to the fact that the sum of the first r positive odd numbers starting from 1 is r^2. [Choosing the opposite ordering on the p_i doesn´t seem to give anything interesting.] Is there any welfare/utility connection in all this? __________________________________________________ Correo Yahoo! Espacio para todos tus mensajes, antivirus y antispam ¡gratis! Regístrate ya - http://correo.yahoo.es ---- election-methods mailing list - see http://electorama.com/em for list info