Voting Methods and Fairness Criteria

Preference Schedules and Permutations of Candidates

• Preference Schedule Definition: A table that summarizes voter preferences by displaying the number of voters who cast ballots for each specific ranked ordering of candidates.
• Permutations of Candidates: For an election with $n$ candidates, the total number of possible ranked orderings (permutations) is given by $n!$ (n factorial).
  • 3 Candidates: $3 \times 2 \times 1 = 6$ possible ranked orderings.
  • 4 Candidates: $4 \times 3 \times 2 \times 1 = 24$ possible ranked orderings (4 choices for 1st place, 3 choices for 2nd place, 2 choices for 3rd place, and 1 choice for 4th place).
• Columns representing candidate orderings with zero votes are customarily omitted from preference schedules for conciseness.

Majority vs. Plurality Standards

• Total Voters Calculation: Found by summing the number of voters across all columns in the preference schedule.
  • Example calculation: $10 + 11 + 20 + 3 + 4 + 8 + 2 = 58$
• Majority: Requiring strictly more than $50\%$ of the total votes cast.
  • Even Number of Voters ($N$): $\text{Majority Threshold} = \frac{N}{2} + 1$
  • Example: For $N = 58$ voters, $\frac{58}{2} + 1 = 29 + 1 = 30$ votes are required for a majority (since $29$ votes is exactly $50\%$).
  • Odd Number of Voters ($N$): Round up $\frac{N}{2}$ to the nearest whole integer.
  • Example: For $N = 59$ voters, $\frac{59}{2} = 29.5$, which rounds up to $30$ votes.
• Plurality: Receiving the most votes relative to all other candidates, regardless of whether that total exceeds $50\%$.
• Theoretical Minimum Votes for a Plurality: The smallest possible number of votes a candidate could possess and still win under the plurality method.
  • Calculation Method: Divide total voters $N$ by the total number of candidates $k$, then round up to the next integer: $\lceil \frac{N}{k} \rceil$
  • Example: For $N = 58$ total voters and $k = 4$ candidates:
  • $\frac{58}{4} = 14.5$
  • Rounding up gives $15$ votes as the smallest possible plurality threshold (occurring in a close scenario such as $15 - 14 - 14 - 14$).

Analyzing First-Place and Last-Place Vote Counts

• Consider an election with $N = 58$ total voters and candidates $A$, $D$, $X$, and $K$:
  • First-Place Vote Breakdown:
  • Candidate $A$: $10$ votes
  • Candidate $D$: $11$ votes
  • Candidate $X$: $20 + 4 + 8 = 32$ votes
  • Candidate $K$: $5$ votes
  • Result: Candidate $X$ receives $32$ votes, satisfying both the plurality requirement and exceeding the $30$-vote majority threshold.
  • Fewest First-Place Votes: Candidate $K$ received the fewest first-place votes ($5$ votes).
  • Most Last-Place Votes:
  • Candidate $A$: $7$ last-place votes
  • Candidate $D$: $28$ last-place votes
  • Candidate $X$: $13$ last-place votes
  • Candidate $K$: $10$ last-place votes
  • Candidate $D$ received the greatest number of last-place votes ($28$ votes).

The Condorcet Candidate and Condorcet Criterion

• Flaw of the Plurality Method: The plurality method considers only first-place choices and ignores lower-rank voter preferences, which can lead to unfair outcomes.
• Condorcet Candidate: A candidate who beats every other candidate in individual head-to-head (one-on-one) matchups.
  • Uniqueness: An election can have at most one Condorcet candidate. If candidate $X$ beats candidate $Y$ head-to-head, $Y$ cannot simultaneously beat $X$.
  • Existence: An election is not guaranteed to have a Condorcet candidate (e.g., non-transitive voter preferences where $A$ beats $B$, $B$ beats $C$, and $C$ beats $A$).
• Head-to-Head Comparison Example:
  • Election with 12 total voters ($5 + 4 + 3 = 12$) and candidates $A$, $B$, and $C$:
  • $5$ voters rank: $A > B > C$
  • $4$ voters rank: $B > A > C$
  • $3$ voters rank: $C > B > A$
  • Plurality Method Outcome: Candidate $A$ gets $5$ votes, $B$ gets $4$ votes, $C$ gets $3$ votes. Plurality winner = Candidate $A$.
  • Head-to-Head Comparisons:
  • $A$ vs. $B$: Column 1 ($5$ voters) prefers $A$. Columns 2 and 3 ($4 + 3 = 7$ voters) prefer $B$ over $A$. Result: Candidate $B$ beats Candidate $A$ by a score of $7$ to 5$.\n - **B$vs.$C$**: Columns 1 and 2 ($5 + 4 = 9$voters) prefer$B$over$C$. Column 3 ($3$voters) prefers$C$. Result: Candidate$B$beats Candidate$C$by a score of$9$to$3$.
  • Candidate $B$ is the Condorcet Candidate because Candidate $B$ wins head-to-head matchups against both Candidate $A$ and Candidate C$.\n- **Condorcet Criterion**: A fairness criterion stating that if a Condorcet candidate exists, that candidate should win the election.\n- **Criterion Violation**: The plurality method can violate the Condorcet Criterion because it elected Candidate A$even though Candidate$B beat all opponents head-to-head.\n\n# Real-World Case Study: The 1992 U.S. Presidential Election\n\n- **Context**: A three-way election between George H. W. Bush (incumbent Republican), Bill Clinton (Democratic governor of Arkansas), and Ross Perot (Independent Texas billionaire focused on the national debt).\n- **Outcome**: Bill Clinton won the election with a plurality of approximately 47.5\% of the popular vote, without obtaining an absolute majority.\n- **Analysis**: Polling data indicated that in head-to-head matchups, George H. W. Bush would likely have defeated Bill Clinton and comfortably defeated Ross Perot. Ross Perot drew disproportionately from voters who otherwise preferred George H. W. Bush.\n- **Implication**: The 1992 election is a real-world example of the plurality method violating the Condorcet Criterion.\n\n# Arrow's Impossibility Theorem\n\n- **Fairness Criterion Definition**: A mathematical standard or rule reflecting ideal principles of a fair democratic election.\n- **Mathematical Reality**: In the 1940s, mathematicians proved (Arrow's Impossibility Theorem) that **no voting method exists** that can satisfy all fairness criteria simultaneously for elections involving three or more candidates.\n- Every voting method carries systemic flaws and potential violations of specific fairness criteria.\n\n# Instant Runoff Voting (IRV)\n\n- **Usage**: Used in political jurisdictions such as Alaska and New York City mayoral elections.\n- **Algorithm Procedure**:\n 1. Voters rank all candidates from first choice to last choice.\n 2. If a candidate receives an absolute **majority** ($> 50\%$) of first-place votes, that candidate is declared the winner.\n 3. If no candidate achieves a majority, the candidate with the **fewest first-place votes** is eliminated.\n 4. Ballots belonging to the eliminated candidate are transferred to those voters' next highest ranked surviving candidate.\n 5. Recalculate the preference schedule and first-place vote totals.\n 6. Repeat the elimination and vote reallocation process iteratively until one candidate achieves an absolute majority.\n\n# Step-by-Step Execution of Instant Runoff Voting\n\n- **Initial Setup**:\n - Candidates: A$,$B$,$C$,$D\n - Voter columns: 12, 15, 8, 13, 10, 15\n - Total voters: 12 + 15 + 8 + 13 + 10 + 15 = 73\n - Majority threshold: \frac{73}{2} = 36.5 \rightarrow 37 votes required.\n - Theoretical minimum plurality threshold: \frac{73}{4} = 18.25 \rightarrow 19 votes.\n\n- **Round 1 Counts**:\n - Candidate A$:$12 + 15 = 27 votes\n - Candidate B$:$8 + 13 = 21 votes\n - Candidate C$:$10 votes\n - Candidate D$:$15 votes\n - *Plurality Winner*: Candidate A$($27 votes).\n - *Majority Check*: No candidate achieved 37 votes.\n - *Elimination*: Candidate C$received the fewest first-place votes ($10 votes) and is eliminated.\n\n- **Round 2 (Reallocated preferences after dropping Candidate C)**:\n - Candidate A$:$27 votes\n - Candidate B$:$21 + 10 = 31$votes (gains$10$votes from column where$C$was 1st and$B was 2nd)\n - Candidate D$:$15 votes\n - *Majority Check*: No candidate achieved 37$votes ($B$leads with$31 votes).\n - *Elimination*: Candidate D$received the fewest first-place votes ($15 votes) and is eliminated.\n\n- **Round 3 (Reallocated preferences after dropping Candidate D)**:\n - Candidate A$:$27 votes\n - Candidate B$:$31 + 15 = 46 votes\n - *Majority Check*: Candidate B$achieves$46$votes, exceeding the$37-vote majority threshold.\n - *Winner*: **Candidate B** wins the election under Instant Runoff Voting.\n\n- **Comparative Conclusion**: The Plurality Method selects Candidate A$as the winner, whereas Instant Runoff Voting selects Candidate$B as the winner from the exact same ballot set.\n\n# Introduction to Alternative Voting Methods: Borda Count\n\n- **Monotonicity Criterion**: A fairness criterion that Instant Runoff Voting can potentially fail.\n- **Borda Count Method**: A point-based positional voting system.\n - Points are assigned based on rank order.\n - In a 4-candidate election: 4$points for 1st place,$3$points for 2nd place,$2$points for 3rd place, and$1$$ point for 4th place.
  • Points are summed across all ballots; the candidate with the highest total points wins.