Comparisons: Borda Count 3

Borda Count as an Extreme Variant of GV

Up to this point, a Borda Count election and one using GV with a common ratio approaching one (r → 1) always appear to produce the same outcome in terms of candidate rankings. On reflection, this seems odd as the Borda Count is based on a linear progression of points yet by definition GV employs a geometric sequence of weights. Despite initial appearances, they are indeed equivalent voting systems.

Recall that neither the value of the common difference nor the absolute value of the first or last preference are important. Provided both are fixed beforehand, any Borda Count election will generate the same unique and complete ranking of the candidates despite the variation in their absolute tally values. So, let the first preference have a weighting of one and the common difference be δ where δ is extremely small and tends to zero. The weightings used by this Borda Count are therefore given below.

• w₁ = 1
• w₂ = 1 - δ
• w₃ = 1 - 2δ
• w_n = 1 - (n-1)δ
• w_N = 1 - (N-1)δ

The second weighting divided by the first defines the common ratio (r) for a geometric progression. This yields (1 - δ)/1 or just 1 - δ as the value of r. As δ tends to zero, then r tends to one but never equals it. However, the ratio between all adjacent weightings must be common (identical) for the progression to be truly geometric. Taking the ratio of the third to second weightings yields (1 - 2δ)/(1 - δ). By adding δ to both the numerator and denominator gives (1 - δ)/1 or just 1 - δ again. This addition can only be performed because δ is as close to zero as it is possible to be without actually equalling it. In effect, essentially nothing is added to the top or the bottom of the rational fraction. As mathematicians say, in the limit as δ approaches zero (δ → 0), then the common ratio of 1 - δ approaches one (r → 1).

For GV, the common ratio can never actually equal one since, for example, many GV equations involve division by 1 - r which for r = 1 would mean dividing by zero. The upper limit for r is therefore 1 - δ where δ tends to zero and r approaches one. Hence the Borda Count is directly equivalent to this extreme variant of GV.

• Borda Count ≡ GV (r = 1 - δ) ≡ GV (r → 1) provided δ → 0

It must be bourne in mind that this equivalence only holds when the common difference tends to zero. Hence care should always be taken when equating them. It is only the collective candidate rankings that are identical for the Borda Count (with a finite common difference) and for GV (with r → 1); the candidate tallies are different. At the other extreme, the common ratio is equal to zero (r = 0). This unconditionally equates to the FPTP system as both candidate rankings and tallies are identical. In the next section, both these GV extremes and all the in-between values of r are compared to its central variant where r = 1/2; namely CHPV.

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