Comparisons: Summary (Single-Winner) 2

CHPV versus GV(r) continued

• The average candidate tally contribution awarded by a voter is equal to the mean weighting of the N preferences.
• The minimum winning rank of the preference awarded to a consensus candidate is always greater than the rank of the preference with the mean weighting.
• A candidate with a unanimous preference from every voter cannot win when the rank of this preference is lower than the mean-weighted one.
• For CHPV, the rank position (n) of the mean-weighted preference is approximately equal to the binary logarithm of the number of candidates (N) or n ≅ log2N.
• As r increases from 0 towards 1, the relative rank position of the mean-weighted preference decreases.
• As N increases, the relative rank position of the mean-weighted preference increases.
• As the relative rank position of the mean-weighted preference rises from middle (≡ Borda Count) to top (≡ FPTP), the GV variant increasingly swings from favouring consensus candidates to favouring polarized ones.
• CHPV is the central GV variant that is not biased in favour of either polarized or consensus candidates.
• With CHPV, a promotion of one place in rank for a preference is worth the same as receiving one additional preference of the original rank instead so candidates need to maximise both the quality (rank) and quantity of their above-mean-weighted preferences.
• CHPV is the only GV variant that achieves an equal balance between vote splitting and teaming in terms of both magnitude and potential scope.
• Also, where the proportion of opposition supporters adhering to the reverse slate is greater than the proportion of clone set supporters voting for the forward one, then attempts at teaming are guaranteed to fail when using CHPV.
• CHPV is the GV variant that is closest to the Borda Count and furthest from FPTP that minimises vote splitting while still retaining the capacity to thwart teaming.
• CHPV is less susceptible than GV(r<1/2) to the tactical voting option of compromise lifting.
• CHPV is less susceptible then GV(r>1/2) to the tactical voting option of burying.
• Like GV(r), CHPV does not staisfy the No-Favourite-Betrayal voting system criteria.

CHPV versus Positional Voting

• To qualify as Positional Voting (PV), each preference must be worth no more than the preceding one (w_n+1 ≤ w_n and so w_n+1/w_n ≤ 1) but the first one must be worth more than the last one (w₁ > w_N).
• For the lower of two adjacent preferences, its Consensus Index CI_L = w_n+1/w_n where 0 ≤ w_n ≤ 1.
• For the lower of two adjacent preferences, its Polarization Index PI_L = (w_n - w_n+1)/w_n where 0 ≤ w_n ≤ 1.
• For all PV preferences and vectors, their consensus and polarization indices sum to unity; namely CI + PI = 1
• The consensus or polarization index for a vector is the weighted average of all the individual CI_L or PI_L indices respectively where Σ is the sum of all the normalized preference weightings.
• The first preference of a vector is normalised at unity (w₁ = 1) and the last preference at zero (w_N = 0) before Σ, CI_V or PI_V is determined.
• The vector Consensus Index CI_V = (Σ - 1)/Σ.
• The vector Polarization Index PI_V = 1/Σ.
• With its PI_V = 1, Plurality is a wholly and consistently polarized voting system.
• With only two candidates, all vectors are wholly polarized as PI_V = 1 regardless of the common ratio r.
• For the Borda Count, its bias converges on CI_V = 1 as the number of candidates N becomes large and tends to infinity.
• The Borda Count is the most consensual vector for any given N (but the invalid indifference vector is the only wholly consensual vector).
• A normalized vector is one where 1 ≤ Σ ≤ N/2 and its anti-vector is where -1 ≥ Σ ≥ -N/2
• A conjugate vector is one where N/2 ≤ Σ ≤ N-1
• Use the format [1, w₂, w₃, ..., w_N-2, w_N-1, 0] to determine Σ and the bias indices for any vector
• Use the conjugate format [1, 1-w_N-1, 1-w_N-2, ..., 1-w₃, 1-w₂, 0] to determine Σ and the bias indices for any anti-vector
• For the Intermediate Indifference [1, y, ..., y, 0] and the Vote-for-X-out-of-N voting systems, their PI_V ranges from 1 where Σ = 1 then down to 2/N where Σ = N/2 and on to 1/(N-1) where Σ = N-1.
• The Intermediate Indifference vector is equivalent to Plurality when y = 0 and to Anti-Plurality when y = 1.
• The Vote-for-X-out-of-N vector is equivalent to Plurality when X = 1 and to Anti-Plurality when X = N-1.
• Any three-candidate positional voting vector (u₁,u₂,u₃) between Anti-Plurality and Anti-Borda-Count is equivalent to A-GV(r) or C-GV(r) with a common ratio r = (u₁-u₂)/(u₂-u₃).
• For GV(r), CI_V = r and PI_V = 1 - r where N becomes large and tends to infinity.
• CHPV, with CI_V = PI_V = 1/2, is exactly central between consensus (Borda Count) and polarization (Plurality) as N → ∞.
• The state of Nauru vector uses the formula w_n = 1/n for elections and its CI_V rises from 0 to 1 as 2 → N → ∞.
• The vector CI_V for the 'Square' formula w_n = 1/n² converges towards 0.392 from initially 0 as 2 → N → ∞.
• The vector CI_V for the 'Factorial' formula w_n = 1/n! rapidly converges on 0.418 from initially 0 as 2 → N → ∞.
• All GV(r) anti-vectors are inherently vulnerable to teaming when cloning occurs.
• When the last weighting in a GV(r) vector is truncated, its consensus index is increased.
• When further weightings in a GV(r) vector are also truncated, its polarization index then increases.
• CI_V for an untruncated GV(r) vector is never closer to its asymptote than it is for its truncated one.
• For balanced (unbiased) N-candidate positional voting, a CHPV vector with w_N truncated is optimal.

Proceed to next page > Comparisons: Summary (Single-Winner) 3

Return to previous page > Comparisons: Summary (Single-Winner) 1