The ultimate of chaos resulting from weighted voting systems

When a group is to rank order n alternatives, it is common for them to adopt some sort of weighted voting system. Namely, n numbers w(l), W), * . ., w(n) are selected. Next, each individual of the group assigns to each alternative one of these numbers. The number of points cast for each alternative is then totaled. The n alternatives are ordered according to their total point values, where the group ordinal ranking of the n altematives is given either by this ordering or by the reversal of this ranking. For example, for a plurality vote the weights are w(1) = 1 while w(i) = 0; and the more votes an alternative receives, the higher is its group ranking. There are voting systems, notably in some athletic events, where lower values are assigned to higher-ranked choices, and the smaller the total assigned to an alternative the higher is its group ranking. For example, consider a track meet where the teams are to be ranked. Each “team” becomes an “altemative”, and an “event” is a “voter”. The way a “voter” marks the ballot corresponds to how the competitors from the different teams placed in the event. Perhaps most people instinctively realize that the choice of the weights can alter the outcome of the election, For example, for four alternatives, should one advocate the use of the weights 5,3,2,1 or the weights 4,3,2,1; how will the choice affect the outcome? (Several years ago several countries employed different weighting systems to determine which country was the “real” victor of the Olympics. As one might expect, the “conclusions” depended upon the choice of the adopted system.) On the other hand, unless there is a drastic difference between two weighted voting systems, it would seem reasonable to expect only slight differences between the two resulting rankings. This need not be the case. In a recent paper [2], P. Fishbum established the following surprising result. Suppose for n > 2 alternatives there are two different monotone-weighted voting systems. (This