Mathematical Proofs: Positional Voting

Proof CV1: Consensus and Polarization Indices for Positional Voting Vectors

The consensus index (CI_V) for a positional voting vector is defined as the weighted average of all the individual consensus indices (CI_L) for the lower preference in each adjacent pair of preferences. The higher preference in each pair is employed to weight these indices in every case. The CI_V is therefore calculated as follows.

The polarization index (PI_V) is similarly calculated using all the individual polarization indices (PI_L) instead; as follows.

As CI_V and PI_V should sum to one, the following check confirms this relationship.

For vectors normalized such that w₁ = 1 and w_N = 0, the two vector bias indices may be simplified further as shown below.

The polarization index is simply the reciprocal of the sum of all the normalized preference weightings.

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