Example: 4 A>B>C 5 B>C>A 2 C>A>B
We start with a matrix whose rows represent the ballots. It has four rows of [3,2,1], five rows of [1,3,2], and two rows of [2,1,3]. The average of all eleven rows is the center of gravity vector V0=[21,25,20]/11. We can see from this that B is the Borda winner and C the Borda loser. Subtracting this V0 from every row we get a matrix A with four rows of [12,-3,-9]/11, five rows of [-10,8,2]/11, and two rows of [1,-14,13]/11. The product of the transpose of A with A is a three by three matrix whose rows are [98,-52,-46]/11, [-52,68,-16]/11, and [-46,-16,62]/11. The normalized eigenvectors corresponding to positive eigenvalues are v1=[.814...,-.462...,-.3518...] and v2=[.06369...,.6731...,-.7367...], so the issue space is two dimensional. The vector A*v1 has four entries of approximately 1.3, five of -1.14, and two of .246, so the median is c1=.246 . The vector A*v2 has four entries of .489, five of .298, and two of -1.72, so the median is c2=.298. Then X=V0+c1*V1+c2*V2 is approximately [2.1,2.4,1.5]. Since 2.4 > 2.1 > 1.5 the winning order is B>A>C, in this case the same as as the Borda order, rather than the Ranked Pairs order of A>B>C. Forest On Wed, 8 Jan 2003, Forest Simmons wrote: > Suppose that (in a certain election) candidate X is preferred over any > other candidate Y on any issue Z by some majority (depending on Y and Z). > > Such a candidate would seem like a logical choice for winner of the > election, if there were such a candidate. > > How could we locate such a candidate if there happened to be one? > > Otherwise, how might one find the candidate closest to this ideal? > > Here's an idea along these lines: > > Form the matrix whose entry in row i and column j is the rank or rating of > candidate j by voter i. > > Let V0 be the average of all the row vectors of the resulting matrix. > Subtract V0 from all of the rows of this matrix, and call this new matrix > A. Let B be the transpose of A. > > Let V1, V2, etc. be normalized eigenvectors of the matrix product B*A, in > descending order of the magnitudes of the eigenvalues (until the > eigenvalues are too small to represent anything other than roundoff > error). > > V0 is the center of gravity vector, and the other V's are the > principal axes of rotation, i.e. those axes (through the center of > gravity) about which the rigid body (with unit masses positioned by the > row vectors) can rotate without wobble. > > Statistically, the eigenvectors give the uncorrelated directions in voter > space (the row space of A offset by the cg vector V0). They may be thought > of as the voter space shadows of the decorrelated issues, in order of > importance to the voters. > > Sort (by numerical value) the components of the vector given by the > product A*V1 to find the median value c1. > > Sort (by numerical value) the components of the vector given by the > product A*V2 to find the median value c2. > > Etc. > > > Let X be the vector V0+c1*V1+c2*V2+... > > Then X is the voter space position of the ideal candidate we are looking > for. In other words, if voter space were identical to issue space, then a > candidate positioned at X would be unbeaten on any of the (uncorrelated) > issues by any of the other candidates. > > What do we do in the almost sure case that no candidate vector (image in > voter space) points in the direction of X ? > > Here are two suggestions out of many possibilities: > > (1) Find the candidate vector (image) that has the largest dot product > with X. > > (2) Look at the components of X, and award the win to the candidate > corresponding to the largest component. In other words, let the > components of X order the candidates from winner to loser. > > After experimenting I prefer the second approach for reasons which I will > explain if this becomes a hot topic. > > Remark. I'm sure that this method doesn't satisfy the Local IIAC, because > each candidate, no matter how marginal in support contributes something to > the issue space profile. In other words, even the losing candidates help > to plumb the depths of issue space. > > Any method that makes intentional, intelligent use of the issue space > information inherent in the distribution of votes in voter space should > not be concerned with satisfying any version of the IIAC, because the > voter response to each candidate helps outline the shape of issue space. > > Sometimes I think that the best way to look at Arrow's Paradox is that the > IIAC was just one of those "good" ideas that didn't pan out. It was based > on a lack of imagination; nobody thought of using the profiles of the > losing candidates to help determine the relative positions of stronger > candidates in issue space. > > On the other hand, intelligent issue space methods should automatically > get rid of clone problems, because clones just give redundant information > about the shape of issue space. The singular value decomposition > decorrelates whatever residual information there might be when the > redundancy is factored out, just as it finds uncorrelated linear > combinations of random variables in taxonomy problems associated with > measuring and recording various dimensions and features of related plants, > for example. > > [In that context the SVD brings out the distinguishing features for > classification of related species. Similar uses are made in image > processing. If nostril width is directly proportional to ear lobe > thickness, the SVD will find that out automatically and count these two > measurements as only one dimension.] > > Thanks for your patience in wading through these rather technical musings. > > Forest > > ---- > For more information about this list (subscribe, unsubscribe, FAQ, etc), > please see http://www.eskimo.com/~robla/em > > ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em
