# Math Mathematical Literacy in Today s World Lecture Fall First

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```							     Math 1000
Mathematical Literacy
in Today’s World

Lecture 19
Fall 2007
First a little review
Of Irrelevant Alternatives
And of the Hare system
A wins by Borda score, by
plurality and by Condorcet
Rank             V O T E S   A   B C
Count            3   2
First   2        A   C       6   0   4

Second 1         B   B       0   5   0

Third   0        C   A       0   0   0
Total   Points               6   5   4
Now change order of losers
(an irrelevant change)
Rank             V O T E S   A   B C
Count            3   2
First   2        A   B       6   4   0

Second 1         B   C       0   3   2

Third   0        C   A       0   0   0
Total   Points               6   7   2
So A won first, but now…
Changing the order of two losers on some
ballots would change the result
This is called “Failure of the Independence
of Irrelevant Alternatives”
Because the order of B and C should be
irrelevant to A’s winning
A new condition for voting
systems
The condition “Independence of Irrelevant
Alternatives” says the only way a non-
winner can become a winner is by jumping
over the winner on at least one ballot (a
relevant change in the ballots)
The Borda count voting system can fail
this condition
Hare System, Round One

Rank         V O T E S         A has 6
Count    5    4   3      1     B has 3
First    A    C   B      A     C has 4

Second   B    B   C      B
So eliminate B
Third    C    A   A      C    and move up
preferences
Total
After B eliminated, Round Two

Rank          V O T E S       Now A has 6
Count     5    4   3      1    first-place
First     A    C   B      A
C has 7
Second    B    B   C      B   (counting 3
from B’s
spot)
Third     C    A   A      C
Total
Now to manipulation of voting
systems
In which voters may cast ballots which do
not reflect their true preferences, but
instead try to manipulate the voting system
Plurality voting allows
manipulation
In the case of the 2000 election, in which Bush,
Some people who preferred Buchanan over
Bush instead voted for Bush because of
electability
So people chose to vote differently from their
actual opinion because of the electoral system
Manipulability
A voter may choose to vote in a manner
that misrepresents his own true
preferences (voting insincerely)
This usually occurs because of the
particular voting system involved
The voters are then choosing to
manipulate the system
Avoiding manipulability
We may try to choose a voting system that
does not allow manipulation (or is
“strategy-proof”) if possible
Or to reduce the ease with which
manipulation can be done
Manipulating the Borda count
Rank         1   2   A has 4
B has 5
First    3   A   B   C has 3
Second 2     B   C   D has 0
Third    1   C   A
Fourth   0   D   D
Manipulating the Borda count
Rank         #1   #2        A has 4
B has 5
First    3   A    B         C has 3

Second 2     B    C         D has 0

Third    1   C    A    If voter #1 knows
Fourth   0   D    D         voter #2’s
preferences, he
can change his
list
Manipulating the Borda count
Rank         #1   #2   If voter #1 knows
voter #2’s
preferences, he
First    3   A    B      can change his
list by moving B
Second 2     B    C           down
Third    1   C    A
Fourth   0   D    D
Manipulated Borda count
Rank         #1   #2    Voter #1 has
moved B down

First    3   A    B
Second 2     C    C     Now A has 4
Third    1   D    A       B has 3
Fourth   0   B    D       C has 4
D has 1
Re-manipulated Borda count
Rank         #1   #2    Voter #1 has
moved B down
AND C down
First    3   A    B
Second 2     D    C
Now A has 4
Third    1   C    A
B has 3
Fourth   0   B    D
C has 3
D has 2
Two elections to compare

Election #1   Election #2

#1     #2      #1      #2

A      B       A       B
B      C       D       C
C      A       C       A
D      D       B       D
This was a unilateral change
Only one voter changed to an insincere (or
disingenuous) ballot in an attempt to
manipulate the election
A case of “single-voter manipulation”,
rather than a group of voters
We could also tell what Voter #1
wanted
The original preferences for #1 were A-B-
C-D, so we could tell that he would prefer
a victory by A to one by B
If he could only arrange a tie by
manipulation, we cannot tell if he would
prefer a tie between A and D, or instead a
tie between B and C.
When manipulable?
A voting system is manipulable if there a
scenario in which some voter can achieve
a preferable result by changing to an
insincere ballot
Or, more carefully:
If there are two sequences of preference list
ballots (elections #1 and #2) so that
Neither election ends in a tie
Only one ballot is changed between the two
elections
The one changing his ballot prefers the result in
#2 to the result in #1 (according to the sincere
preference list ballot he gave in #1)
Manipulability in two-candidate
elections
If a voter could manipulate a two-
candidate election, that would mean by
changing his vote from his original
preference to a different one, he could
change the result
If that voter preferred candidate A, he
would have to change his vote to B
Manipulability in two-candidate
elections
If that voter preferred candidate A, he
would have to change his vote to B
But that would mean that B won and the
voter is trying to change that
But adding a vote for B won’t change the
result (monotone condition)
So majority rule is
nonmanipulable
You can’t change the result in your favor
by changing from your actual preference
to an insincere preference
As before, it is the only nonmanipulable
system that treats voters and candidates
equally
Now for three or more
candidates
Our first method was Condorcet’s, which
matchup
A winner was declared if he beat all others
How could you try to manipulate that?
Condorcet manipulability
Suppose you prefer candidate A but
Candidate B won by Condorcet’s method
You can lower B in your preferences, but
you can’t keep him from beating A, since
you can only change your ballot
So you can’t make A win
Condorcet manipulability
So Condorcet’s method is nonmanipulable
in the sense that a voter can never change
an election result from one candidate to
another candidate that the voter prefers
But he could turn the election into a
situation where Condorcet doesn’t provide
a winner (e.g., the paradox example)
Turning Condorcet winner into a
tie
Election #1   Election #2

#1 #2 #3      #1 #2 #3

A    B    C   A    B    C

C    C    A   B    C    A
B    A    B   C    A    B
A voting system is manipulable:
If there are two sequences of preference list
ballots (elections #1 and #2) so that
Neither election ends in a tie
Only one ballot is changed between the two
elections
The one changing his ballot prefers the result in
#2 to the result in #1 (according to the sincere
preference list ballot he gave in #1)
Manipulability of the Borda
count
Election #1   Election #2

#1     #2      #1      #2

A      B       A       B
B      C       D       C
C      A       C       A
D      D       B       D
Manipulability of the Borda
count
So Borda count can be manipulated with
four candidates
Three candidates?
The Borda count with three
candidates
Not manipulable to change the winner
To see this, suppose B wins with highest
Borda score but your choice was not B
How much can you change B’s Borda
score?
The Borda count with three
candidates
If your preference ranked B last, you
cannot decrease B’s score and you can
only increase A or C’s score by 1 (no
better than a tie)
If your preference ranked B second, you
cannot raise the Borda score of your
preferred candidate and you can only
lower B’s score by 1 (no better than a tie)
The Borda count with three
candidates
So a unilateral change in preference list
ballots cannot change the winner to a
winner preferred by the single voter
So Borda count with three candidates is
nonmanipulable
Borda count is manipulable for
more candidates
For any number of candidates more than
three, the Borda count is manipulable
We can show this by adding candidates
and voters to the four-candidate example
that don’t change which scores are higher

Election        Election with 2 more

#1       #2       #1   #2    #3      #4

A        B        A    B     A       D
B        C        B    C     B       C
C        A        C    A     C       B
D        D        D    D     D       A

Election        Election with 2 more

#1       #2       #1   #2    #3      #4

A        B        A    B     A       D
B        C        B    C     B       C
C        A        C    A     C       B
D        D        D    D     D       A

Election        Election with 2 more

#1       #2       #1   #2    #3      #4

A        B        A    B     A       D
B        C        B    C     B       C
C        A        C    A     C       B
D        D        D    D     D       A

Election        Election with 2 more

#1       #2       #1   #2    #3      #4

A        B        A    B     A       D
B        C        B    C     B       C
C        A        C    A     C       B
D        D        D    D     D       A
Election #1     Election #2
#1     #2        #1     #2

A      B         A      B
B      C         D      C
C      A         C      A
D      D         B      D
E      E         E      E
For n candidates and even
number of voters
Use the same two ballots with four
More candidates, all ranked at the end
Enough irrelevant voters to make the right
number
For n candidates and odd
number of voters
Use the three ballots from Exercise 9 with
More candidates, all ranked at the end
Enough irrelevant voters to make the right
number
Manipulability with three voters
of the Borda count
Election #1    Election #2

#1 #2 #3       #1 #2 #3

A    B B       ?    B B
B    A A       ?    A A
C    C C       ?    C C
D    D D       ?    D D
The End for Today

```
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