Math Mathematical Literacy in Today s World Lecture Fall First

							     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
                                  votes
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,
 Gore, Nader, and Buchanan ran
 Some people who preferred Buchanan over
 Bush instead voted for Bush because of
 electability
 Similarly for Gore over Nader
 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
 applied majority rule to each head-to-head
 matchup
 A winner was declared if he beat all others
 in the head-to-head matchups
 How could you try to manipulate that?
Condorcet manipulability
 Suppose you prefer candidate A but
 Candidate B won by Condorcet’s method
 That means your sincere ballot had A first
 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
 What about five 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
Adding irrelevant voters

   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
Adds 3 to A’s score

   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
Adds 3 to B’s score

   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
Adds 3 to C’s score

   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
Adding candidates
    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
 candidates, but add:
 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
 four candidates, but add:
 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