The Mathematics of Voting

1
The Mathematics of Voting

• Part 3
• Methods and Criteria

2
The Majority Criterion

• If a candidate has a majority of the first-place
  votes then that candidate should win the
  election.
• In our example with Radical Pis pizza choice
  using the Plurality Method, Anchovy wins, but
  without a majority of the votes.

• Anchovy - 7
• Pepperoni - 6
• Sausage - 5
• Mushroom - 0

Does this violate the Majority Criterion?
3
A little logic

• All of the fairness criteria are in the form of a
  conditional statement, if a certain condition is
  met, then an expected result should follow.
• A conditional statement is true if any time the
  condition is met then the result follows.
• A conditional statement is false if at some time
  the condition is met but the result does not
  follow.

4
Violating the Fairness Criteria

• We say a fairness criterion is violated if in a
  particular election the condition is met but the
  result does not follow.
• We say a particular voting method violates a
  particular fairness criterion if there is an
  example using that method which violates that
  specific criterion.

5
The Majority Criterion

• The Plurality Method, Plurality with Elimination
  Method and Method of Pairwise Comparisons cannot
  violate the Majority Criterion.
• In each of these methods a candidate with a
  majority of the first-place votes must win.
• So how about the Borda Count Method?

6
Borda Count and the Majority Criterion

• Small Town, Ohio is holding an election for
  mayor. The candidates are Annie, Bluto, and
  Cleopatra. The town has decided to use the Borda
  Count Method and the results look like this

• How many people voted?
• How many needed for a majority?
• Does anyone have this?
• What are the results of the election?

A 3(7)2(9)1(0)39 B 3(9)2(4)1(3)38 C
3(0)2(3)1(13)19
7
Using Pairwise Comparisons
Choose one candidate, say, Anchovy.
2. A vs P
1. A vs S
7 11
7 11

• Anchovy
• Sausage
• Pepperoni
• Mushroom

3. A vs M
4. S vs P
12 6
7 11
6. P vs M
5. S vs M
11 7
16 2
Sausage wins all every head-to-head comparison.
8
Using the Plurality Method

• Find the candidate with the most first-place
  votes.

• And the winner is Anchovy.

But this violates the Condorcet Criterion
9
Using Plurality with Elimination
Mushroom is removed as a candidate and the votes
recounted.
Now the tally is Anchovy - 7 Pepperoni - 54211

• The results are the same
• Anchovy - 7
• Pepperoni - 6
• Sausage - 5

We have a winner
Pepperoni
Still no majority and sausage is eliminated.
Again the Condorcet Criterion is violated.
10
Plurality with Elimination and Monotonicity

• The ski club is deciding on where to go next
  Spring Break. The choices are Aruba, Bermuda,
  and Cancun.
• They decide to use the Plurality with Elimination
  method to choose a winner. The results look like
  this

11

• First Round
• A - 11
• B - 8
• C - 10

Bermuda is eliminated and the votes retabulated.

• Second Round
• A - 11
• C - 18

Cancun is the winner!