Question

Here are the preferences of the seven voters, choosing among options A, B, C, D, and E..

rank	2	2	1	1	1
first	c	e	c	d	a
second	e	b	a	e	e
third	d	d	d	a	c
fourth	a	c	e	c	d
fifth	b	a	b	b	b

Answer the following question about the possible outcomes of an election involving all 7 of these voters using various voting methods:

Who wins a Majority Rule election?
Who wins a Plurality Rule election?
Who wins a Borda Count election?
Who wins an Instant Runoff (Hare) election?

Who wins a Pairwise Comparison Election?

Answer 1

Given the preferences table

Rank/voters (Total = 7) ->	2	2	1	1	1
First priority	c	e	c	d	a
second priority	e	b	a	e	e
Third priority	d	d	d	a	c
Fourth priority	a	c	e	c	d
Fifth priority	b	a	b	b	b

(We will only look at first priority of candidates for majority and plurality, other priorities are used in Borda count.)

At first priority, votes are as follows

A - 1

B - 0

C-3

D-1

E-2

Majority Rule Election

This suggests that the candidate with more than 50% of votes wins the election.

Here most no. of votes are given to C but it is only 3/7*100 = 42.8%

Hence no candidate gets more than 50% votes. No one is the winner according to the majority rule election.

Plurality Rule Election

This suggests that candidates with a maximum number of votes win.

Clearly, C has the maximum number of votes. C is the winner according to plurality rule election.

Borda Count Election

In this method, the Last priority is given 1 point, last-second given 2 points, and so on.

Then each vote is multiplied/weighted by their priority and total points for each candidate is counted. One with maximum points win the election. The method is as follows

Rank/voters (Total = 7) ->	2	2	1	1	1
First priority (5 pts.)	c = 2*5 = 10	e = 2*5 = 10	c = 1*5 = 5	d = 1*5 = 5	a = 1*5 = 5
second priority (4 pts.)	e = 2*4 = 8	b = 2*4 = 8	a = 1*4 = 4	e = 1*4 = 4	e = 1*4 = 4
Third priority (3 pts.)	d = 2*3 = 6	d = 2*3 = 6	d = 1*3 = 3	a = 1*3 = 3	c = 1*3 = 3
Fourth priority (2 pts.)	a = 2*2 = 4	c = 2*2 = 4	e = 1*2 = 2	c = 1*2 = 2	d = 1*2 = 2
Fifth priority (1 pt.)	b = 2*1 = 2	a = 2*1 = 2	b = 1*1 = 1	b = 1*1 = 1	b = 1*1 = 1

Now count total points of each candidate.

A - 4+2+4+3+5 = 18

B - 2+8+1+1+1 = 13

C - 10+4+5+2+3 = 24

D - 6+6+3+5+2 = 22

E - 8+10+2+4+4 = 28

Cleary E has maximum number of points. Hence E wins according to Borda count election rule.

Instant Runoff Election

Also called Plurality with elimination. In this method, the candidate with least first-choice/first-priority votes is eliminated and the votes are transferred to the next candidate.This is done till one candidate have a majority. Let's look at the example for better understanding.

Rank/voters (Total = 7) ->	2	2	1	1	1
First priority	c	e	c	d	a
second priority	e	b	a	e	e
Third priority	d	d	d	a	c
Fourth priority	a	c	e	c	d
Fifth priority	b	a	b	b	b

A - 1

B - 0

C-3

D-1

E-2

Here B has the least first priority votes. So we eliminate B.

Now situation will look like

Rank/voters (Total = 7) ->	2	2	1	1	1
First priority	c	e	c	d	a
second priority	e		a	e	e
Third priority	d	d	d	a	c
Fourth priority	a	c	e	c	d
Fifth priority		a

A - 1

C-3

D-1

E-2

Now eliminate A and D. On doing this, look at 4th column, the first priority candidate D would be eliminated and the candidate below that is E. So D's votes will go to E. Similarly in fifth column, A's votes will go to E.

Now the situation is

Rank/voters (Total = 7) ->	2	2	1	1	1
First priority	c	e	c
second priority	e			e	e
Third priority					c
Fourth priority		c	e	c
Fifth priority

C-3

E-2+1+1 = 5

At this stage E has 5 votes while C has 3 votes. E has 62.5% of votes(more than 50%) So E has a majority hence E win.

Pairwise Comparison Election

In this method, we compare each pair of candidates and give 1 point for each pairwise win. For each voter, votes go to that candidate which the voter choose more priority over other.

For example in the table below, if we compare E and D then the first column voter has more priority towards E. So their votes will go to E.

Now let's look at the method.

We have 5 candidates, so total pairs that can be made are n*(n-1)/2 = 5*4/2 = 10

Rank/voters (Total = 7) ->	2	2	1	1	1
First priority	c	e	c	d	a
second priority	e	b	a	e	e
Third priority	d	d	d	a	c
Fourth priority	a	c	e	c	d
Fifth priority	b	a	b	b	b

A vs B - 5 vs 2 => A wins

A vs C - 2 vs 5 => C wins

A vs D - 2 vs 5 => D wins

A vs E - 2 vs 5 => E wins

B vs C - 2 vs 5 => C wins

B vs D - 2 vs 5 => D wins

B vs E - 0 vs 7 => E wins

C vs D - 4 vs 3 => C wins

C vs E - 3 vs 4 => E wins

D vs E - 2 vs 5 => E wins

Now final points will be A(1 pt.), B(0 pts.), C(3 pts.), D(2 pts.), E(4 pts.)

E is the winner according to the pairwise comparison rule.