Chapter 1: Methods
of Voting
Arrow’s Impossibility Theorem

For elections with three or more
candidates “ a method for determining
election results that is democratic and
always fair is a mathematical impossibility
1.1 Preference Ballots &
Preference Schedules
Preference Ballot- a ballot where the
voters are asked to rank the candidates in
order of preference
 Linear Ballot- a ballot in which ties are not
allowed

1.2 The Plurality Method
Plurality Method – all we care about is the
first place votes, the candidate with the
most votes wins. We don’t need the
voters to rank the candidate.
 Majority candidate – the candidate with a
majority (more than half) of the first place
votes.

1.2 The Plurality Method
with 3 or more candidates there is no
guarantee that there is going to be a
majority candidate
 The Majority criterion – If candidate X has
a majority of the first place votes, the
candidate X should be the winner of the
election.

1.2 The Plurality Method


Condorcet candidate – a candidate preferred by
a majority of voters over every other candidate
when the candidates are compared in head-tohead comparisons. Not every election has one.
The Condorcet criterion – If candidate X is
preferred by the voters over each of the other
candidates in a head-to-head comparison, then
candidate X should be the winner of the election.
1.2 Flaws of the Plurality Method
It sometimes leads us to pick a choice that
is loved or hated by the voters instead of
one that is preferred by most
 The ease with which election results can
be manipulated by a voter or block of
voters through insincere voting.

1.2 Flaws of the Plurality Method

Insincere Voting (Strategic Voting) – If we
know the candidate we really want doesn’t
have a chance of winning, then rather than
“waste our vote” on our favorite candidate
we can cast it for a lesser choice who has
a better chance of winning.
The Borda Count Method

Each place on the ballot is assigned
points. For an election with N candidates,
last place = 1 point, second to last place =
2 points, and so on until first place = N
points. The points are then added up and
the candidate with the most points wins.
The Borda Count Method

This method is widely used in many
different settings such as sports, awards,
and companies. For example Heisman
Trophy winner, NBA Rookie of the Year,
NFL MVP, college football polls, music
awards, and hiring processes.
Flaws of the Borda Count Method
This method can violate the 2 basic
criterion for fairness, the majority criterion
and the condorcet criterion.
 These violations are RARE especially if
there are a lot of candidates.

1.4 The Plurality-with-Elimination
Method (Instant Runoff Voting)

Instant Runoff Voting – use the preference
schedule to eliminate the candidates with
the fewest first place votes until 1
candidate has the majority of the votes.
The election must use a preference ballot.
1.4 The Plurality-with-Elimination
Method (Instant Runoff Voting)

Plurality-with-Elimination
 Round
1: Count the first place votes. If there is a
majority candidate, then that candidate is the winner.
If not eliminate the candidates with the fewest first
place votes.
 Round 2: Cross out the candidate(s) that were
eliminated and recount the first place votes. If there is
a majority, declare the winner. If not continue the
process until you have a majority candidate.
1.4 The Plurality-with-Elimination
Method (Instant Runoff Voting)

The Monotonicity criterion – If candidate X
is a winner of an election and in a
reelection, the only changes in the ballot
are changes that favor X (and only X),
then X should remain the winner of the
election.
1.4 The Plurality-with-Elimination
Method (Instant Runoff Voting)

This method is becoming more popular
because you don’t have to hold a run-off
election. This method is currently used
with the Olympic Committee, and some
local elections in California, Vermont,
Michigan, and Australia.
1.5 The Method of Pairwise
Comparisons
Pairwise comparison – every candidate is
matched head-to-head against every other
candidate
 The one with the most votes in the
comparison wins. If they are split equally,
the comparison ends in a tie.

1.5 The Method of Pairwise
Comparisons

The winner of the comparison gets 1 point,
the loser gets none; if there was a tie,
each candidate gets ½ point. The winner
of the election is the one with the most
points after all the comparisons are done.
1.5 The Method of Pairwise
Comparisons
This method satisfies all 3 of the fairness
criteria discussed so far.
 The Independence-of-Irrelevant criterion
(IIA) – If candidate X is a winner of an
election and in a recount one of the
nonwinning candidates withdraws or is
disqualified, then X should still be a winner
of the election.

1.5 The Method of Pairwise
Comparisons
This method can be quite indecisive, it is
not unusual to have multiple ties for first
place.
 As the number of candidates grow, the
number of comparisons grows even
quicker.

1.5 The Method of Pairwise
Comparisons

Sum of consecutive
LL  1
integers formula – 1  2  3    L 
2

Number of Pairwise
Comparisons with N
Candidates -
N  1N
2
1.6 Rankings
Ranking of candidates – who comes first,
second, third, etc.
 With the plurality method - count the
number of first place votes, then rank in
order from greatest to least.
 With the borda method – find the total
points for each candidate and use that to
rank them.

1.6 Rankings
With the plurality-with-elimination method
– the first one eliminated is in last place &
so on until you get to first place.
 With the pairwise comparisons – rank the
candidates using the total points from the
comparisons.

1.6 Rankings

Recursive Ranking – Find the winner,
remove the winners name from the
preference schedule and make a new one.
Continue the process until all places are
filled. You can do this using any method.