Geometric Voting Vector Analysis of Strategic Candidate Nomination and Voter Retaliation in Positional Voting

Positional voting fails both the Independence of Irrelevant Alternatives and the Independence of Clones criteria. It is therefore vulnerable to the strategic nomination of insincere candidates that may adversely affect election outcomes. By introducing identical clones, other candidates may be either promoted or demoted in the resultant collective ranking. The analysis of the ramifications of cloning is restricted here to ‘geometric voting’ vectors where preference weightings form a geometric progression. With the common ratio of the weightings as the sole variable, a full continuous spectrum of vectors from plurality through the Borda count to anti-plurality can be simultaneously analyzed. This universal vector may handle any given number of candidates. Starting with an example three-candidate election, the effect of varying the common ratio and the margin between the clone and non-clone candidates is analyzed. It is then extended to include additional such candidates. Cloning maps that display all possible election outcomes for the example elections are introduced. They exhibit regions where teaming attempts either succeed or fail; the latter due to vote-splitting. The geometric voting vector that represents all those from the Borda count to anti-plurality is highly vulnerable to teaming while the one equivalent to plurality is instead vulnerable to vote-splitting. The intermediate vector with a common ratio of one-half – called the consecutively halved positional voting vector – is a balanced one with no net bias towards either polarizing or consensus candidates. The research here establishes that this vector exactly counterbalances the possibilities for teaming against those for vote-splitting when a non-cloned candidate and a pre-cloned one are tied. Also, it inherently thwarts teaming attempts where identical clones are undifferentiated, and is the most consensual vector to exhibit this property. Further, it enables opposition voters to retaliate successfully using a tit-for-tat clone-reversal strategy even when one clone is consistently promoted by clone supporters over the others. Unlike the Borda count, it possesses such disincentives to clone candidates. This balanced vector hence offers an optimal compromise between maximizing the consensus index of a geometric voting vector and minimizing its vulnerability to teaming attempts through strategic nominations.