1
Plurality reads only the top
Each ballot counts once, for its first choice. The candidate with the most first places wins, with a majority or without one, and the rest of every ballot is ignored.
Lesson · Justice and Collective Choice · Collective choice
Plurality, Runoff and Borda
Plurality counts first places. The two-round runoff counts first places and then holds a head-to-head contest, instant runoff removes one candidate at a time, and the Borda count adds points for every position on every ballot. Run on one profile they can elect different candidates, and each answers a different question about the voters. The wrong model is that the candidate with the most first choices is the people's choice.
plurality counts first places
runoff: the top two meet head to head
Borda: 2, 1, 0 points down each ballot
same ballots, different rule, new winner
One profile, four counts
Core ideas
What this lesson establishes
Every rule in this lesson takes the same input, a profile of rankings, and returns a winner. What differs is what the rule reads off the ballots.
2
Runoff rules count first places more than once
The two-round runoff keeps the two candidates with the most first places and lets the voters choose between them. Instant runoff removes the candidate with the fewest first places, passes those ballots to their next choice, and repeats until someone holds a majority.
3
Borda reads every position
With three candidates a ballot gives 2 points to its first choice, 1 to its second and 0 to its last, and the points are added over all ballots. A candidate who is rarely ranked low can win without being ranked first often.
The material
Four counts of one profile
A nine-voter profile in which most rules agree, one in which they split three ways, and one in which plurality elects the candidate most voters rank last.
The previous lesson asked whether majorities can be assembled into a ranking and found that sometimes they cannot. Real elections need a winner anyway, and the rules in use answer in different ways. The way to compare them is to hold the ballots fixed and change only the rule.
Take nine voters and three candidates. Four voters rank them A B C, three rank them B C A, and two rank them C B A.
voters first second third
4 A B C
3 B C A
2 C B A
Plurality reads the first column. A has 4 first places, B has 3 and C has 2, so A wins with fewer than half the voters. The two-round runoff keeps A and B, the two with the most first places, and asks the voters to choose between them using their whole ballots: 4 rank A above B and 5 rank B above A, so B wins. Instant runoff reaches the same place by another road. C has the fewest first places and is removed, its two ballots pass to their next choice, which is B, and B holds 5 of the 9 ballots.
The Borda count does not eliminate anyone. Each ballot gives 2, 1 and 0 points down the ranking, and the voters' points are summed. The sums below are the arithmetic the lab leaves to you; it prints the winner.
A B C
4 voters 8 4 0
3 voters 0 6 3
2 voters 0 2 4
total 8 12 7
B wins the Borda count with 12 points against 8 and 7. B is also the Condorcet winner: it beats A by 5 to 4 and C by 7 to 2. So three of the four counts elect B, and plurality is the one that elects A, a candidate who loses the head-to-head contest with B.
Three rules, three winners
Change one ballot group. Let four voters rank the candidates A C B, three B C A and two C B A. Plurality still elects A, with 4 first places against 3 and 2. The runoff and instant runoff elect B: A meets B and loses 4 to 5, and C is removed first. The Borda count elects C, with 11 points against 8 and 8. C is also the Condorcet winner, beating A by 5 to 4 and B by 6 to 3.
Three different counts, three different winners, from nine ballots. C is never ranked last by anyone, and it is eliminated in the first round of both runoff rules because only two voters put it first. Nothing in the ballots has changed between the rules; what changed is which part of the ballots each rule looks at.
What each rule is counting
- Plurality counts enthusiasm: how many voters would be delighted. It cannot see how the others feel.
- The runoffs count strength against the strongest rival, but only after first places have chosen who the rivals are.
- Borda counts average standing: who is placed high by most voters, whether or not first.
- The Condorcet rule counts victories: who beats every other candidate by majority, when anyone does.
The lab runs these on the profile you type, with the winner under the chosen rule, the Condorcet result and the table of pairwise counts. It reports a tie as a tie. It scores Borda as 2, 1, 0 for three candidates and with the matching run of points for more, and it takes every ballot as the voter's real ranking; whether voters would rank the same way under a different rule is the question of “Strategic Voting and Manipulation”.
Interactive
Collective choice
six voting rules, the majority relation, judgment aggregation and the jury theorem
Method
Running four rules on one profile
Five moves, in this order, for any profile of three candidates. Keep the profile table in front of you throughout.
1
Write the profile as a table of rankings with counts
One row per ranking, best to worst, with the number of voters who hold it. Check that the counts add up to the electorate.
2
Plurality: count the first column
Add the counts of the rows that begin with each candidate. The largest total wins, however small a share of the voters it is.
3
Runoff: take the top two, then count whole ballots
The two candidates with the most first places meet. For each row, see which of the two it ranks higher, and add the counts. Instant runoff instead removes the candidate with the fewest first places and moves those rows to their next choice.
4
Borda: give each row 2, 1, 0 and sum
Multiply a row's points by its count for each candidate, then add down the column. The largest total wins.
5
Compare with the Condorcet winner
Count each pair of candidates. If one candidate beats both others it is the Condorcet winner, and each rule can now be described by whether it elected that candidate.
Counting under four rules
Score 0 / 0Choose the answer you can justify.
Nine ballots, four counts
Four voters rank A B C, three rank B C A, two rank C B A.
plurality A 4, B 3, C 2 -> A
runoff A against B: 4 to 5 -> B
instant drop C, its 2 go to B: 5-4 -> B
Borda A 8, B 12, C 7 -> B
Condorcet B beats A 5-4, C 7-2 -> B
Plurality elects a candidate who loses a head-to-head contest with the Condorcet winner. The first column of a ballot is a small part of what the voter thinks, and a rule that reads only that part can disagree with every rule that reads more.
Common mistakes
1
Treating the most first choices as the people's choiceWith three voters at A B C, two at B A C and four at C A B, C has the most first places, 4 against 3 and 2, and five of the nine voters rank C last. A beats C by 5 to 4 and B by 7 to 2. The first-place count measures how many voters are enthusiastic, and it says nothing about how the rest rank the winner, so a lead of 4 in 9 can be a majority's last choice.
2
Speaking of the winner without naming the ruleWith four voters at A C B, three at B C A and two at C B A, plurality elects A, the runoff elects B and Borda elects C. “The winner” is a statement about a profile and a rule together, and the same nine ballots give three answers to it.
3
Expecting a runoff to rescue the Condorcet winnerIn that same profile C is the Condorcet winner, beating A by 5 to 4 and B by 6 to 3, and both runoff rules eliminate it in the first round because only two voters put it first. A runoff lets first places decide who may reach the contest, so a candidate everyone finds acceptable but few adore never gets to it.
Check your understanding
✓
Finish when you can run plurality, runoff, instant runoff and Borda on a three-candidate profile and say what each counts.
Given a profile with counts, compute each rule's winner by hand, find the Condorcet winner, and name a profile on which three of the rules elect three different candidates.
Note: Change the rule below the profile and watch the winner move while the table of pairwise counts stays put: the counts belong to the ballots and the winner belongs to the rule. Then take the spoiler profile and move one voter from C A B to B A C, and the plurality count becomes a three-way tie.