Conclusions: Ranked Ballot CHPV 2
A Balanced Voting System
The consensus and polarization indices for a positional voting system enable a comparison to be made between two rival vectors. With only two candidates, all methods produce the same (polarized) election outcome. With three or more candidates however, the degree of bias in promoting consensus or polarization then emerges in the particular system. Inherent system bias should not be confused with voter support for a consensual or polarizing candidate.
Plurality is always a wholly polarized vector where any consensus candidate is severely disadvantaged. The Borda Count is the most consensual one of all the positional voting vectors as its bias against polarized candidates converges on extreme consensus as the number of candidates increases. Anti-Plurality, like all anti-vectors, is more consensual than the Borda Count. The Nauru voting system also becomes increasingly biased towards consensus albeit less extreme as more candidates enter an election.
For a GV vector, its consensus index equates to its common ratio when the number of candidates is large and tends to infinity. So, as its common ratio and consensus index vary in unison from zero towards one here, the vector ranges from Plurality to the Borda Count with CHPV balanced mid-way between them. CHPV is hence an unbiased vector with no net tendency towards either polarization or consensus. Where few candidates stand, the consensus index for any GV vector is somewhat reduced but it converges towards its common ratio asymptote rapidly as more candidates compete. Where the last preference weighting of the standard GV voting vector is truncated to zero, this index becomes significantly closer to its asymptote.
Truncation is the only form of voter choice that affects the inherent bias of a vector. Where permitted, truncating the lowest-ranked preference in a vector slightly shifts this bias towards greater consensus. However, additionally truncating other low-ranked preferences reverses this shift back towards greater polarization. Where the lowest preference weighting is small, the magnitude of any shift is also small.
An electoral method is generally established prior to the number of the candidates choosing to fight the election becomes known. If a balanced positional voting system with little or no bias towards either consensus or polarization is desired, then a GV vector with a common ratio of about one half is required. A slightly higher ratio is needed when a small number of candidates stand but adopting CHPV clearly offers the best practical solution. With or without truncation being permitted, its consensus and polarization indices will be very closely matched even if they are not exactly equal.
Optimising a Candidate Tally
Both FPTP and AV satisfy the majority criteria so any candidate with the full support of more than one half of the voters wins outright. With the Borda Count, a candidate with the vast majority of first preferences may still be beaten by another candidate with a greater number of second preferences. CHPV is intermediate between these systems and a candidate requires over two thirds of first preferences to be guaranteed victory. In all these systems, a candidate with a plurality but minority of first preferences may nevertheless still win the election.
Most single-winner voting systems either favour polarized candidates over consensus ones or the reverse. Polarized candidates are the more likely to win in FPTP and AV elections while consensus candidates are the more likely to win in Condorcet and Borda Count elections. Ranked ballot CHPV does not advantage one type of candidate over the other. CHPV is the threshold variant of GV for a consensus candidate with unanimous second preference support to tie with two polarized candidates who share all the first preferences equally between them.
For any consensus candidate with unanimous nth preference support to stand any chance of winning, the rank of this preference must be greater than that of the mean-weighted preference. The rank (n) of the preference with the mean weighting is approximately equal to the binary logarithm of the number of candidates (n ≈ log2N). CHPV preferences with weightings above the mean are of high rank while those below it are of low rank. Large proportions of the high preferences are necessary for victory in CHPV elections while any low preferences are a relative hindrance.
FPTP and GV vectors with low common ratios favour polarized candidates over consensus ones. The reverse is true for vectors with high common ratios and for the Borda Count. Ranked ballot CHPV is the central GV vector that is not biased in favour of either polarized or consensus candidates. For CHPV with its common ratio of one half, a preference at one rank position is worth the same as two preferences ranked one place lower. Alternatively stated, a promotion of one place in rank for a preference is worth the same as receiving one additional preference of the original rank instead.
For AV or GV with a low common ratio, candidates should predominantly focus on maximising the rank of their anticipated high preferences. For the Borda Count, Anti-Plurality, any Condorcet method, the Nauru voting system or GV with a high common ratio, they should instead focus on maximising the number of their expected high preferences. Uniquely, for ranked ballot CHPV, the focus on quality (rank) and quantity is evenly balanced. In seeking to win a single-winner CHPV election, each candidate must try to broaden their support as much as to strengthen it. Using CHPV, a powerful candidate may simultaneously beat both a polarized candidate and a consensus one to gain victory.
Summary Conclusion
Ranked ballot CHPV is a simple, transparent, efficient, reliable and balanced voting system for use in single-winner elections where there is no bias in favour of polarized or consensus candidates and where the capacity for voters to thwart teaming is retained while the effects of vote splitting are minimised.
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