01:640:103 Chap.1 Preference Ballots and Voting Methods
Preference Ballots and Voting Methods
What a preference ballot records
Voters in this part of the course do not just mark a favourite. Each one ranks every candidate, with no ties, and identical ballots are grouped into the columns of a preference table. A voting method is a rule that reads that table and names a winner.
The chapter works four such rules on one set of ballots so that you can see exactly which part of the table each rule pays attention to. With N candidates there are N factorial possible ballots, so four candidates already allow 24 different rankings.
Four rules, four readings of the same table
Plurality counts only first choices, and the candidate with more of them than any single rival wins.
Instant Runoff Voting also starts from first choices but checks for a majority, more than half of all ballots; if nobody has one, the candidate with the fewest first choices is eliminated and those ballots move to their next surviving choice, round after round. The Borda Count gives N points for a first choice down to 1 point for a last choice and adds them up.
Pairwise Comparisons runs every head-to-head contest between two candidates and awards a point to each contest winner, half a point each for a tie. In the guide's 43-voter mascot election the four rules produce four different winners: Comet, Anchor, Bison and Drake.
Checks that catch arithmetic slips
Each method has a built-in check. Borda totals must equal the points on one ballot times the number of ballots.
Instant Runoff transfers must leave the ballot total unchanged from round to round. A head-to-head count must use every column, since a voter who ranks both candidates low still prefers one of them.
Extended versions of each method rank all the candidates rather than naming only a winner, and the course asks for those rankings as often as for the winner itself.
Reading the four winners in the mascot election
The guide's worked table has 43 ballots and four candidates. Comet has the most first choices, 14, so it wins under Plurality.
Instant Runoff eliminates Bison and then Drake, and Anchor collects their transferred ballots to finish with 29. Bison, last on first choices, has 23 second places and wins the Borda Count with 121 of the 430 points. Drake beats every rival head to head and wins Pairwise Comparisons with three points. Nothing about the voters changed between these four answers; only the rule did.
That is why an exam question always names the method, and why a complete answer shows the tally that decided it rather than just the winner's name.
What this chapter covers
- 01
Preference ballots and preference tables
- 02
Plurality and the extended ranking
- 03
Instant Runoff Voting round by round
- 04
The Borda Count and its points check
- 05
Head-to-head competitions and Pairwise Comparisons
Worked example · free
One table, three winners
- 2Plurality: first choices are A 8, B 6, C 5, so A wins without a majority.
- 2Instant Runoff: C has the fewest and is eliminated; its 5 ballots go to B, which then leads 11 to 8.
- 2Borda: A 35, B 44, C 35; the total 114 equals 6 points times 19 ballots.
Key terms
- Preference ballot
- A preference ballot is one voter's full ordering of the candidates from first to last, each in its own position.
- Plurality candidate
- A plurality candidate leads every rival on first places, which need not amount to a majority.
- Instant Runoff Voting
- Instant Runoff Voting eliminates the candidate with the fewest first choices, round by round, until someone holds a majority.
- Borda Count
- The Borda Count scores N points for each first choice down to 1 point for each last choice and elects the highest total.
- Head-to-head competition
- A head-to-head competition compares two candidates alone, counting the ballots that rank each one higher.
Preference Ballots and Voting Methods FAQ
Why can different voting methods pick different winners from the same ballots?
Each method reads a different part of the preference table. Plurality sees only the top row, Instant Runoff sees the top row repeatedly as candidates drop out, Borda weighs every position, and Pairwise Comparisons compares candidates two at a time, so a candidate strong in one reading can be weak in another.
How do I check a Borda Count total quickly?
Multiply the points on a single ballot by the number of ballots. With four candidates a ballot carries 4 + 3 + 2 + 1 = 10 points, so 43 ballots must hand out 430 points in total; any other sum signals a slip in a column.
Is Instant Runoff Voting the same as Ranked Choice Voting?
Yes. The course text notes that Instant Runoff Voting is also known as Ranked Choice Voting. Voters rank the candidates once, and the rounds of elimination and transfer are carried out from those rankings without a second trip to the polls.
Exam move
Build one small preference table of your own and run all four methods on it, writing the extended ranking for each. Then change one column and predict which winners move before recounting.
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