Comprehensive Study Notes on Voting Methods and Social Choice Theory

The Plurality Voting Method

• Core Mechanics and Principles:

  • Under the plurality method, the choice or candidate receiving the greatest number of first-preference (first-choice) votes is declared the winner.
  • Only the top row of a preference schedule (the first-choice votes) is evaluated; rankings in second, third, or lower positions are disregarded entirely.
  • Ties are possible under this method; when a tie occurs, it must be resolved through a runoff vote.
  • Winning by plurality does not require obtaining a majority of the total votes cast.

• Case Study Example and Preference Schedule:

  • Context: A college class ranks four dessert options from favorite to least favorite:
  • Cake, denoted by $c$
  • Cookies, denoted by $k$
  • Pie, denoted by $p$
  • Ice Cream, denoted by $I$
  • Preference Schedule Distribution across six voter groups:
  • Group 1 ($5$ voters): 1st choice Cake ($c$), 2nd choice Ice Cream ($I$), 3rd choice Pie ($p$), 4th choice Cookies ($k$).
  • Group 2 ($9$ voters): 1st choice Cookies ($k$), 2nd choice Cake ($c$), 3rd choice Ice Cream ($I$), 4th choice Pie ($p$).
  • Group 3 ($7$ voters): 1st choice Cookies ($k$), 2nd choice Ice Cream ($I$), 3rd choice Pie ($p$), 4th choice Cake ($c$).
  • Group 4 ($8$ voters): 1st choice Pie ($p$), 2nd choice Cake ($c$), 3rd choice Cookies ($k$), 4th choice Ice Cream ($I$).
  • Group 5 ($6$ voters): 1st choice Cake ($c$), 2nd choice Cookies ($k$), 3rd choice Ice Cream ($I$), 4th choice Pie ($p$).
  • Group 6 ($2$ voters): 1st choice Ice Cream ($I$), 2nd choice Cake ($c$), 3rd choice Pie ($p$), 4th choice Cookies ($k$).

• First-Choice Vote Calculations:

  • Total first-choice votes for Cake ($c$):     $5 + 6 = 11$
  • Total first-choice votes for Cookies ($k$):     $9 + 7 = 16$
  • Total first-choice votes for Pie ($p$):     $8$
  • Total first-choice votes for Ice Cream ($I$):     $2$

• Plurality Winner Determination:

  • Cookies received the highest number of first-choice votes ($16$ votes) and is declared the winner under the plurality method.

• Percentage and Majority Calculations:

  • Total number of votes cast across all preference columns:     $5 + 9 + 7 + 8 + 6 + 2 = 37$
  • Percentage of votes obtained by the winning choice (Cookies):     $\frac{16}{37} \approx 0.4324$
  • Converting the decimal to a percentage rounded to two decimal places yields:     $43.24\%$
  • Majority Criterion Assessment:
  • A majority requires securing strictly greater than $50\%$ of the total vote.
  • Because $43.24\% < 50\%$, Cookies won the election via plurality without achieving a majority of votes.

Condorcet Winner Criterion and Pairwise Comparisons

• Definition and Core Concept:

  • A candidate or choice that defeats every other candidate in a head-to-head (pairwise) comparison is designated as the Condorcet winner.
  • In head-to-head comparisons, all candidates appear on the ballot, but for each specific comparison, voter preferences are isolated between just those two options.

• Pairwise Head-to-Head Analysis (3 Options: Cake $c$, Cookies $k$, Pie $p$):

  • Preference schedule structure for $37$ total voters:
  • Group 1 ($5$ voters): Cake ($c$) > Pie ($p$) > Cookies ($k$)
  • Group 2 ($9$ voters): Cookies ($k$) > Cake ($c$) > Pie ($p$)
  • Group 3 ($7$ voters): Cookies ($k$) > Pie ($p$) > Cake ($c$)
  • Group 4 ($8$ voters): Pie ($p$) > Cake ($c$) > Cookies ($k$)
  • Group 5 ($6$ voters): Cake ($c$) > Cookies ($k$) > Pie ($p$)
  • Group 6 ($2$ voters): Cookies ($k$) > Pie ($p$) > Cake ($c$)

• Matchup 1: Cake ($c$) vs. Cookies ($k$):

  • Group 1 ($5$ voters): Cake is ranked above Cookies ($5$ votes for Cake).
  • Group 2 ($9$ voters): Cookies is ranked above Cake ($9$ votes for Cookies).
  • Group 3 ($7$ voters): Cookies is ranked above Cake ($7$ votes for Cookies).
  • Group 4 ($8$ voters): Cake is ranked above Cookies ($8$ votes for Cake).
  • Group 5 ($6$ voters): Cake is ranked above Cookies ($6$ votes for Cake).
  • Group 6 ($2$ voters): Cookies is ranked above Cake ($2$ votes for Cookies).
  • Cake total:     $5 + 8 + 6 = 19$
  • Cookies total:     $9 + 7 + 2 = 18$
  • Winner of Matchup 1: Cake ($19$ to $18$).

• Matchup 2: Cookies ($k$) vs. Pie ($p$):

  • Group 1 ($5$ voters): Pie is ranked above Cookies ($5$ votes for Pie).
  • Group 2 ($9$ voters): Cookies is ranked above Pie ($9$ votes for Cookies).
  • Group 3 ($7$ voters): Cookies is ranked above Pie ($7$ votes for Cookies).
  • Group 4 ($8$ voters): Pie is ranked above Cookies ($8$ votes for Pie).
  • Group 5 ($6$ voters): Cookies is ranked above Pie ($6$ votes for Cookies).
  • Group 6 ($2$ voters): Pie is ranked above Cookies ($2$ votes for Pie).
  • Cookies total:     $9 + 7 + 6 = 22$
  • Pie total:     $5 + 8 + 2 = 15$
  • Winner of Matchup 2: Cookies ($22$ to $15$).

• Matchup 3: Cake ($c$) vs. Pie ($p$):

  • Cake total across all columns:     $20$
  • Pie total across all columns:     $17$
  • Winner of Matchup 3: Cake ($20$ to $17$).

• Condorcet Winner Conclusion:

  • Cake defeated Cookies ($19$ to $18$) and defeated Pie ($20$ to $17$).
  • Because Cake won every pairwise head-to-head matchup in which it participated, Cake is the Condorcet winner.

Instant Runoff Voting (IRV) Method

• Step-by-Step Procedure for Instant Runoff Voting:

  • Step 1: Count the first-choice votes for every candidate.
  • Step 2: Determine if any candidate has achieved a majority. A majority requires strictly more than half of the total votes cast.
  • Step 3: If no candidate has a majority, eliminate the candidate with the fewest (least) first-place votes.
  • Step 4: Redistribute the votes of the eliminated candidate to each voter's next choice among the remaining candidates.
  • Step 5: Repeat this process until a single candidate reaches a majority.

• Round 1 Evaluation:

  • Total voters:     $5 + 9 + 7 + 8 + 6 + 2 = 37$
  • Majority threshold calculation:     $\frac{37}{2} = 18.5$
  • Exactly $19$ votes are required to secure a majority.
  • Initial First-Place Vote Counts:
  • Cake ($c$): $5 + 6 = 11$ votes
  • Cookies ($k$): $9 + 7 = 16$ votes
  • Pie ($p$): $8$ votes
  • Ice Cream ($I$): $2$ votes
  • Majority Check: The highest vote total is $16$ (Cookies), which is below the required $19$ votes. No majority exists.

• Round 1 Elimination and Preference Adjustment:

  • Eliminate Ice Cream ($I$) because it has the fewest first-place votes ($2$ votes).
  • Remove Ice Cream ($I$) from all ballots, leaving three choices (Cake, Cookies, Pie):
  • Column 1 ($5$ voters): Cake, Pie, Cookies
  • Column 2 ($9$ voters): Cookies, Cake, Pie
  • Column 3 ($7$ voters): Cookies, Pie, Cake
  • Column 4 ($8$ voters): Pie, Cake, Cookies
  • Column 5 ($6$ voters): Cake, Cookies, Pie
  • Column 6 ($2$ voters): Pie, Cookies, Cake
  • Vote Totals After Round 1 Elimination:
  • Pie ($p$): $8 + 2 = 10$ votes
  • Cake ($c$): $5 + 6 = 11$ votes
  • Cookies ($k$): $9 + 7 = 16$ votes

• Round 2 Elimination and Column Consolidation:

  • Majority requirement remains at least $19$ votes. No candidate currently has $19$ votes.
  • Eliminate Pie ($p$) because it has the fewest first-place votes ($10$ votes).
  • Remove Pie ($p$) from all ballots, leaving only Cake ($c$) and Cookies ($k$):
  • Column 1 ($5$ voters): Cake, Cookies
  • Column 2 ($9$ voters): Cookies, Cake
  • Column 3 ($7$ voters): Cookies, Cake
  • Column 4 ($8$ voters): Cake, Cookies
  • Column 5 ($6$ voters): Cake, Cookies
  • Column 6 ($2$ voters): Cookies, Cake
  • Combining Like Columns:
  • Voters ranking Cake over Cookies (Columns 1, 4, and 5):       $5 + 8 + 6 = 19$
  • Voters ranking Cookies over Cake (Columns 2, 3, and 6):       $9 + 7 + 2 = 18$

• Final IRV Outcome:

  • Cake secures $19$ votes, meeting the threshold of at least $19$ votes for a majority.
  • Cookies receives $18$ votes.
  • Cake is declared the overall winner using the Instant Runoff Voting method.

The Borda Count Method

• Point System and Allocation Rules:

  • Points are assigned to candidates based on their ranking position on each ballot.
  • Last choice receives $1$ point, second-to-last choice receives $2$ points, and point values increase by $1$ for each higher ranking position.
  • For an election with $3$ candidates:
  • 1st Choice: $3$ points per vote
  • 2nd Choice: $2$ points per vote
  • 3rd Choice: $1$ point per vote
  • Total point counts across all ballots are calculated for each candidate, and the candidate with the highest point total wins.

• Complete Borda Count Computation (3 Candidates: Cake $c$, Cookies $k$, Pie $p$):

  • Preference Schedule ($37$ total voters):
  • Column 1 ($5$ voters): 1st Cake ($c$), 2nd Pie ($p$), 3rd Cookies ($k$)
  • Column 2 ($9$ voters): 1st Cookies ($k$), 2nd Cake ($c$), 3rd Pie ($p$)
  • Column 3 ($7$ voters): 1st Cookies ($k$), 2nd Pie ($p$), 3rd Cake ($c$)
  • Column 4 ($8$ voters): 1st Pie ($p$), 2nd Cake ($c$), 3rd Cookies ($k$)
  • Column 5 ($6$ voters): 1st Cake ($c$), 2nd Cookies ($k$), 3rd Pie ($p$)
  • Column 6 ($2$ voters): 1st Cookies ($k$), 2nd Pie ($p$), 3rd Cake ($c$)

• Point Breakdown for Cake ($c$):

  • Column 1 (1st Place): $3 \times 5 = 15$ points
  • Column 2 (2nd Place): $2 \times 9 = 18$ points
  • Column 3 (3rd Place): $1 \times 7 = 7$ points
  • Column 4 (2nd Place): $2 \times 8 = 16$ points
  • Column 5 (1st Place): $3 \times 6 = 18$ points
  • Column 6 (3rd Place): $1 \times 2 = 2$ points
  • Total Points for Cake:     $15 + 18 + 7 + 16 + 18 + 2 = 76$

• Point Breakdown for Cookies ($k$):

  • Column 1 (3rd Place): $1 \times 5 = 5$ points
  • Column 2 (1st Place): $3 \times 9 = 27$ points
  • Column 3 (1st Place): $3 \times 7 = 21$ points
  • Column 4 (3rd Place): $1 \times 8 = 8$ points
  • Column 5 (2nd Place): $2 \times 6 = 12$ points
  • Column 6 (1st Place): $3 \times 2 = 6$ points
  • Total Points for Cookies:     $5 + 27 + 21 + 8 + 12 + 6 = 77$

• Point Breakdown for Pie ($p$):

  • Column 1 (2nd Place): $2 \times 5 = 10$ points
  • Column 2 (3rd Place): $1 \times 9 = 9$ points
  • Column 3 (2nd Place): $2 \times 7 = 14$ points
  • Column 4 (1st Place): $3 \times 8 = 24$ points
  • Column 5 (3rd Place): $1 \times 6 = 6$ points
  • Column 6 (2nd Place): $2 \times 2 = 4$ points (summed in calculation as $6$ points)
  • Total Points for Pie:     $10 + 9 + 14 + 24 + 6 + 6 = 69$

• Borda Count Summary and Final Result:

  • Cake point total: $76$
  • Cookies point total: $77$
  • Pie point total: $69$
  • Cookies achieves the highest point total ($77$ points) and is declared the winner under the Borda Count method.