The Majority Criterion (page 6) - If a choice receives a majority of the first place votes in an election, then that choice should be the winner of the election. - PowerPoint PPT Presentation

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The Majority Criterion (page 6) - If a choice receives a majority of the first place votes in an election, then that choice should be the winner of the election.

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Title: The Majority Criterion (page 6) - If a choice receives a majority of the first place votes in an election, then that choice should be the winner of the election.


1
1st criteria for a fair election
The Majority Criterion (page 6) - If a choice
receives a majority of the first place votes in
an election, then that choice should be the
winner of the election.
2nd criteria for a fair election
The Condorcet Criterion (page 8) - If there is a
choice that in a head-to-head comparison is
preferred by the voters over every other choice,
then that choice should be the winner of the
election.
2
3rd criteria for a fair election
The Monotonicity Criterion (page 15). If choice
X is a winner of an election and, in a
reelection, the only changes in the ballots are
changes that only favor X, then X should remain a
winner of the election.
4th criteria for a fair election
The Independence-of-Irrelevant-Alternatives
Criterion (page 18). If choice X is a
winner of an election and one (or more) of the
other choices is removed and the ballots
recounted, then X should still be a winner of the
election.
3
Arrows Impossibility Theorem (page 3) - A method
for determining election results that is
democratic and always fair is a mathematical
impossibility.
  • Methods used to find the winner of an election
  • Plurality Method
  • Borda Count Method
  • Plurality-with-Elimination Method
  • Method of Pairwise Comparison

4
Example 1.2 The Math Club Election (Page 6)
5
I. THE PLURALITY METHOD
plurality method (page 6) - the candidate (or
candidates) with the most first place votes wins.
A plurality does not imply a majority but a
majority does imply a plurality.
6
Whats wrong with the plurality method?
If we compare the Hula Bowl to any other bowl on
a head-to-head basis, the Hula Bowl is always the
preferred choice.
7
Y
N
8
(No Transcript)
9
II. THE BORDA COUNT METHOD
The Borda count method (page 10) - Each place on
a ballot is assigned points. In an election with
N candidates we give 1 point for last place, 2
points for next to last place, , and N points
for first place. The points are tallied for each
candidate separately and the candidate with the
most points is the winner.
4 points
56
40
32
16
4
42
30
24
12
3
3 points
28
20
16
8
2
2 points
14
10
8
4
1
1 point
10
Math Appreciation Society A gets 561084179
points B gets 423016162106 points C gets
28402484104 points D gets 14203212381
points
11
(No Transcript)
12
Whats wrong with the Borda count method?
Example 1.6 The School Board Principal Election
(page 11)
TABLE 1-5
13
TABLE 1-5
The School Board Principal Election A gets
242329 points B gets 188632 points C
gets 1261230 points D gets 64919 points
The Borda count method violates the majority
criterion.
14
Y
N
N
N
N
15
III. THE PLURALITY-WITH-ELIMINATION METHOD
16
TABLE 1-6
37 voters so a majority requires at least 19 votes
17
III. THE PLURALITY-WITH-ELIMINATION METHOD
18
TABLE 1-6
37 voters so a majority requires at least 19 votes
19
III. THE PLURALITY-WITH-ELIMINATION METHOD
20
TABLE 1-6
37 voters so a majority requires at least 19 votes
21
Example 1.9 (page 15)
TABLE 1-9
11 voters so a majority requires at least 6 votes
22
(No Transcript)
23
26 voters so a majority requires at least 14 votes
24
TABLE 1-9
Round 1
Candidate A B C
Number of first place votes 11 8 10
29 voters so a majority requires at least 15 votes
Round 2
Candidate A B C
Number of first place votes 11 18
25
29 voters so a majority requires at least 15 votes
Round 1
Candidate A B C
Number of first place votes 7 8 14
Round 2
Candidate A B C
Number of first place votes 15 14
26
Whats wrong with the plurality-with-elimination
method?
3th criteria for a fair election
The Monotonicity Criterion. If choice X is a
winner of an election and, in a reelection, the
only changes in the ballots are changes that only
favor X, then X should remain a winner of the
election.
The plurality-with-elimination method violates
the monotonicity criterion.
27
Y
N
N
N
N
N
N
Y
28
Homework
  • Read Pages 6 23
  • Page 30 18, 19, 24, 28, 29, 30, 33, 59, 60, 71
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