Mathematical Proofs: Table of Contents

The following mathematical proofs require a knowledge of algebra and geometry to General Certificate of Secondary Education Advanced Level (GCSE A-level) Mathematics standard only. This is an entry level qualification to a UK university. These proofs employ algebraic notation and election profile formats as defined on the notation and formats page.

Chapter: Evaluations (Ranked Ballot)

• Section: General Criteria
  1. GV (any r) fails the Condorcet Winner Criterion
  2. GV (any r) fails the Condorcet Loser Criterion
  3. GV (any r) fails the Reversal Symmetry Criterion

• Section: Majority and Related Criteria
  1. Majority Threshold as a function of Common Ratio
  2. Majority Threshold as a function of Common Ratio and Number of Candidates
  3. Pairwise Majority Thresholds as a function of Common Ratio
  4. Effect of Strong versus Broad Support as a function of Common Ratio

• Section: Clones, Teaming and Independence Criteria
  1. GV (any r) is not Independent of Irrelevant Alternatives

• Section: Teaming Thresholds
  1. GV Teaming Thresholds for Clone Set versus One Candidate
  2. GV Teaming Thresholds for Clone Set versus Two Candidates
  3. GV Teaming Thresholds for Clone Set versus Three Candidates
  4. GV Teaming Thresholds for Clone Set versus Many Candidates
  5. GV Teaming Effect and Success Thresholds for Clone Set versus One Candidate

Chapter: Evaluations (Party-List)

• Section: Optimality of CHPV
  1. Extreme Seat Share Domain Volume Ratios for OPV Maps
  2. Extreme Seat Share Domain Volume Ratios for CHPV Maps

• Section: Proportionality of CHPV
  1. Worst Case Resolution Least Squares Index for OPV
  2. Voting System Least Squares Index for CHPV

Chapter: Comparisons (Single-Winner)

• Section: Borda Count
  1. Candidate Preferences and Average Rank Positions

• Section: Geometric Voting
  1. Common Ratio 'Arc' on a Three-Preference Map
  2. Equivalent Positional Voting Systems to Three-Candidate GV Variants
  3. Identifying Equivalent Three-Candidate Voting Systems
  4. Iso-Tally Grid Lines on Three-Candidate Maps
  5. Minimum Winning Rank for a Consensus Candidate
  6. Rank of the Preference with the Mean Weighting
  7. Common Ratio for Balance between Vote Splitting and Teaming
  8. Example of Vulnerability to Tactical Voting

• Section: Positional Voting
  1. Consensus and Polarization Indices for Positional Voting Vectors
  2. Common Ratio for Anti-GV(r) Vectors
  3. Consensus and Polarization Indices for Geometric Voting
  4. Consensus Index Values for other Example Vectors
  5. Bias Indices and Truncation
  6. Consensus and Polarization Indices for Truncated GV(r) Vectors

Chapter: Comparisons (Multiple-Winner)

• Section: Droop Quota ~ Proportionality
  1. Voting System Least Squares Index for Droop Quota

• Section: D'Hondt Method ~ Proportionality
  1. Voting System Least Squares Index for D'Hondt Method

Refer to > Mathematical Proofs: Notation and Formats

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