Transcript Math and Elections

Math and Elections
A Lesson in the “Math + Fun!” Series
Nov. 2004
Math and Elections
Slide 1
About This Presentation
This presentation is part of the “Math + Fun!” series devised
by Behrooz Parhami, Professor of Computer Engineering at
University of California, Santa Barbara. It was first prepared
for special lessons in mathematics at Goleta Family School
during the 2003-04 and 2004-05 school years. The slides can
be used freely in teaching and in other educational settings.
Unauthorized uses are strictly prohibited. © Behrooz Parhami
Nov. 2004
Edition
Released
First
Nov. 2004
Revised
Math and Elections
Revised
Slide 2
We Vote to Choose Our Leaders
or Indicate Our Preferences
Who would you like
to be our president
(senator, school
board member) for
the next few years?
George
Bush
John
Kerry
What type of drink
should the cafeteria
serve with school
lunches this year?
Apple
juice
Nov. 2004
Math and Elections
Orange
juice
Ralph
Nader
Grape
juice
David
Cobb
Milk
Slide 3
We Use Different Voting Methods
Punched-card or
punched-paper
ballot processed
by special
reader devices
Write-in ballot with manual counting
Marked ballot with
optical reading
Nov. 2004
Computerized
touch-screen voting
Math and Elections
Slide 4
Isn’t Counting All There Is to Voting?
M
O
A
A
O
O
G
O
A
A
4 prefer
apple juice
O
5 prefer
orange juice
M
M
G
G
3 prefer
grape juice
M
M
M
M
M
8 prefer
milk
20 kids voting
Nov. 2004
Math and Elections
Slide 5
True, When We Have Only 2 Choices
Apple
juice
George
Bush
John
Kerry
Orange
juice
Proposition 71:
□ Yes □ No
Blank or doubly marked
votes do not count
Only one possible
complication: tie votes
(no winner, prop fails)
Things get tricky as soon as we go to three or more choices
In 1952, mathematical economist Kenneth Arrow proved that there is no
consistent method of making a fair choice among 3 or more candidates
All examples to follow will assume
three choices; you can imagine that
problems can only get worse if
there are more than three choices
Nov. 2004
A A
A A
Math and Elections
O O
O
O
O
M M M M
M M MM
Slide 6
Majority and Plurality Voting
M
O
A
A
A
O
M
M
O
M
M
O
M
A
O
M
M
Juice or Milk?
AJ, OJ, or Milk?
Juice gets a majority of votes
(majority means more than half)
Milk gets a plurality of votes
(plurality means more than others)
17 kids voting
Nov. 2004
Math and Elections
Slide 7
Meaning of Fairness in Voting
A
A
A
A
O O
O
O
O
M
M M
M M
In a 3-way race:
A gets 4 votes
O gets 5 votes
M gets 8 votes
So, M wins!
M
M
M
Results of 2-way races:
O or A?
O or M?
A or M?
9 to 8
9 to 8
9 to 8
O should win!
A
Juice people
always prefer juice
to milk; milk people
are equally divided
among A and O as
second choice
A solution: Run-off between
the top two vote getters
Nov. 2004
4
5
8
O
Math and Elections
M
Slide 8
Activity 1: Polling
1. On small pieces
of paper, vote for:
A
O
M
A
A
O O
O
O
O
A
A
Apple juice
Orange juice
Milk
Nov. 2004
M M
M M
M
M
M
A
2. Collect and tally the votes;
enter results in this triangle.
3. Suppose after the vote has
been tallied, you are informed
that the top choice is no longer
available. Can you make a fair
choice without voting again?
M
O
Math and Elections
M
Slide 9
Indicating Two-Way Preferences
Orange
juice
Apple
juice
Alice’s preferences:
A over O
O over M
M over A
Milk
Does this make sense?
No it does not!
if
and
then
5
3
5
2
Nov. 2004
>
>
>
3
A voter who prefers A > O,
O > M, and M > A is “confused”
3
2
2
5
Nonconfused voters can
order their choices from
most to least desirable:
e.g., A > O > M
Math and Elections
Slide 10
Indicating First and Second Choices
A
A
O O
O
O
O
A
A
M
M M
M M
M
M
M
Second choices:
A’s prefer O over M
O’s prefer A over M
M’s half are A > O,
the other half O > A
A
A
Number of A
kids who prefer
O over M
Number of A
kids who prefer
M over O
4
5
O
Nov. 2004
4 0
5
8
M
O
Math and Elections
4
0
4
M
Slide 11
Vote Tallying in Rounds
A
A
A
A
O O
O
O
O
M M
M
M
M M
M
M
A
4 0
95
Collect the ordered choices of voters
Remove the lowest vote getter (A)
O
4
0
4
M
Adjust the voter choices to account for the removed candidate
Repeat the process with the remaining choices until only two
candidates remain; then tally the votes as usual
Nov. 2004
Math and Elections
Slide 12
Borda Voting
A
A
A
A
O O
O
O
O
M
M M
M M
M
2 points for 1st choice
1 point for 2nd choice
0 point for 3rd choice
M
M
A
Number of A
kids who prefer
O over M
A points: 4 × 2 + 9 × 1 = 17
O points: 5 × 2 + 8 × 1 = 18
M points: 8 × 2 + 0 × 1 = 16
4 0
Is this outcome fair?
5
No, M has the most first-place votes
Yes, O would win against A or M
Nov. 2004
Number of A
kids who prefer
M over O
O
Math and Elections
4
0
4
M
Slide 13
Activity 2: Ordered Preferences
1. On small pieces
of paper, vote for
your first and
second choices
among A, O, M
A
A
O O
O
O
O
A
A
M
M M
M M
M
M
M
A
2. Collect and tally the votes;
enter results in this triangle.
3. Tally the votes in rounds
4. Tally the votes according to
Borda voting rules
5. Are the results fair? Why?
Nov. 2004
O
Math and Elections
M
Slide 14
Activity 3: A Variant of Borda Voting
A
A
A
A
O O
O
O
O
M
M M
M M
M
M
M
What happens if we
change the points to:
3 for 1st choice
2 for 2nd choice
1 for 3rd choice
A
A points: __ × 3 + __ × 2 + __ × 1 = ___ (was 17)
O points: __ × 3 + __ × 2 + __ × 1 = ___ (was 18)
M points: __ × 3 + __ × 2 + __ × 1 = ___ (was 16)
Is the outcome fair?
5
____________________________
____________________________
O
Nov. 2004
4 0
Math and Elections
4
0
4
M
Slide 15
Activity 4: Borda Voting
A
A
A
A
O O
O
O
O
M
M M
M M
M
M
M
A
A points: __ × 2 + __ × 1 = __
O points: __ × 2 + __ × 1 = __
M points: __ × 2 + __ × 1 = __
Number of A
kids who prefer
O over M
Number of A
kids who prefer
M over O
4 0
Is the outcome fair?
5
____________________________
____________________________
O
Nov. 2004
Show that if one of
the M voters
changes his/her 2nd
choice, A can win
Math and Elections
4
0
4
M
Slide 16
Borda Voting: Conspiracy
A
A
A
A
O O
O
O
O
M
M M
M M
M
M
M
Number of A
kids who prefer
O over M
A: 4 × 2 + 10 × 1 = 18 points
O: 5 × 2 + 6 × 1 = 16 points
M: 8 × 2 + 1 × 1 = 17 points
Number of A
kids who prefer
M over O
4 01
3
Is this outcome fair?
5
Yes, M has the most 1st place votes
No, M would not win against A or O
Nov. 2004
Suppose one A > O > M
and one M > O > A voter
conspire to change their
votes to A > M > O and M
> A > O (i.e., each tries to
help the other)
A
O
Math and Elections
0
45
43
M
Slide 17
Approval Voting
Each voter
lists all the
choices
that are
acceptable
to him/her
A
A
AGAG
A
A
AOAO
AO
O O
O
O
O
GO
GG
G
A,G
A
Votes are tallied and the total
for each choice is found
G
Approval voting makes
majority vote more likely
2
4
A = 9, G = 8, O = 11 (wins)
Nov. 2004
GO GO
3
0
A,O
Math and Elections
3
3
G,O
5
O
Slide 18
Activity 5: Approval Voting
A
1. On small pieces
of paper, vote for
all your approved
juice choices
among A, O, G
A
AGAG
A
A
AOAO
AO
O O
O
O
O
GO GO
GO
GG
G
A,G
A
G
2. Collect and tally
the votes; enter results
in this diagram.
3. Tally the approval votes
and choose a winner.
4. Are the results fair? Why?
Nov. 2004
Math and Elections
A,O
G,O
O
Slide 19
Conclusions
A
A
A
A
O O
O
O
O
M
M M
M M
M
M
M
When there are three
or more choices,
no voting method
guarantees a fair
outcome in all cases.
Choosing the candidate or option with the most votes (plurality)
is not a good idea, unless he/she/it has a majority of the votes.
Run-off election among the top two vote getters solves some,
but not all, of the problems.
Ordering all the candidates, and not just voting for the top one,
combined with Borda voting (point system) is usually the best.
Nov. 2004
Math and Elections
Slide 20
Next Lesson
Thursday, December 2, 2004
Nov. 2004
Math and Elections
Slide 21