DIFFERENT RULES,
DIFFERENT OUTCOMES
1
Different Rules, Different Outcomes
A. Summary
1.
2.
Up to this point we have examined various limitations of plurality
rule, pairwise majority rule, majority rule with a runoff, and instant
runoff voting.
Today we will emphasize a different point, that different voting
rules can select different candidates – even if voters vote the same
way.
a.
3.
This is similar to y = f(x). Change the function f and you will change the range, y.
If the voting rules select different outcomes, then the natural
question is which voting rule is best.
Different Rules, Different Outcomes
B. Illustrated with three voting rules
1.
2.
3.
Plurality rule.
Majority rule with a runoff (MRR).
Borda Count
a.
Each voter ranks the alternatives according to their preferences giving
greater numbers to their most preferred alternatives. These numbers
are then added and the alternative with the largest total wins.
b.
The Borda Count is used in Slovenia to elect the member of its National
Assembly who represents the ethnic Italians and the member who represents
the ethnic Hungarians. It is also used to nominate presidential candidates in
Kiribati (Reilly, 2002), to determine the Most Valuable Player in Major League
Baseball, and to nominate Heisman Trophy winners.
Different Rules, Different Outcomes
5 voters
A
C
B
3 voters
B
C
A
3 voters
C
A
B
Different Rules, Different Outcomes
5 voters
A (3)
C (2)
B (1)
3 voters
B (3)
C (2)
A (1)
3 voters
C (3)
A (2)
B (1)
Borda Count
A: 3(5) + 1(3) + 2(3) = 24
B: 1(5) + 3(3) + 1(3) = 17
C: 2(5) + 2(3) + 3(3) = 25
C wins because it has the largest count.
Different Rules, Different Outcomes
C.
1984 NY Democratic Primary
Jackson coalition
Hart coalition
Mondale coalition
25%
35%
40%
J H M
H J M
MJ H
Practice: who wins under plurality rule, MRR, and
Borda Count?
(note: preferences are left to right, not top to bottom)
Different Rules, Different Outcomes
C.
1984 NY Democratic Primary
Jackson coalition
Hart coalition
Mondale coalition
1)
Plurality:
25%
35%
40%
J H M
H J M
MJ H
Different Rules, Different Outcomes
C.
1984 NY Democratic Primary
Jackson coalition
Hart coalition
Mondale coalition
1)
Plurality:
1)
25%
35%
40%
Mondale Wins (M – 40%, H – 35%, J – 25%).
J H M
H J M
MJ H
Different Rules, Different Outcomes
C.
1984 NY Democratic Primary
Jackson coalition
Hart coalition
Mondale coalition
1)
2)
25%
35%
40%
J H M
H J M
MJ H
Plurality:
1)
Mondale Wins.
1)
no one has a majority in the first round so Hart and
Mondale go to the second round.
Majority with Runoff:
Different Rules, Different Outcomes
C.
1984 NY Democratic Primary
Jackson coalition
Hart coalition
Mondale coalition
1)
2)
25%
35%
40%
J H M
H J M
MJ H
Plurality:
1)
Mondale Wins
1)
no one has a majority in the first round so Hart and Mondale go to the
second round.
Hart wins by majority in the second round (60% to 40%). Hart wins.
Majority with Runoff:
2)
Different Rules, Different Outcomes
C.
1984 NY Democratic Primary
Jackson coalition
Hart coalition
Mondale coalition
1)
2)
3)
25%
35%
40%
J H M
H J M
MJ H
Plurality:
1)
Mondale Wins
1)
no one has a majority in the first round so Hart and Mondale go to the
second round.
Hart wins by majority in the second round. Hart wins.
Majority with Runoff:
2)
Borda Count:
Different Rules, Different Outcomes
C.
1984 NY Democratic Primary
Jackson coalition
Hart coalition
Mondale coalition
1)
2)
3)
Borda Count Points ->3 2 1
25%
J H M
35%
H J M
40%
MJ H
Plurality:
1)
Mondale Wins
1)
no one has a majority in the first round so Hart and Mondale go to the
second round.
Hart wins by majority in the second round. Hart wins.
Majority with Runoff:
2)
Borda Count:
Jackson:
3(.25) + 2(.35) + 2(.40) = 2.25
Different Rules, Different Outcomes
C.
1984 NY Democratic Primary
Jackson coalition
Hart coalition
Mondale coalition
1)
2)
3)
Borda Count Points ->3 2 1
25%
J H M
35%
H J M
40%
MJ H
Plurality:
1)
Mondale Wins
1)
no one has a majority in the first round so Hart and Mondale go to the
second round.
Hart wins by majority in the second round. Hart wins.
Majority with Runoff:
2)
Borda Count:
Jackson:
Hart:
3(.25) + 2(.35) + 2(.40) = 2.25
3(.35) + 2(.25) + 1(.40) = 1.95
Different Rules, Different Outcomes
C.
1984 NY Democratic Primary
Jackson coalition
Hart coalition
Mondale coalition
1)
2)
3)
Borda Count Points ->3 2 1
25%
J H M
35%
H J M
40%
MJ H
Plurality:
1)
Mondale Wins
1)
no one has a majority in the first round so Hart and Mondale go to the
second round.
Hart wins by majority in the second round. Hart wins.
Majority with Runoff:
2)
Borda Count:
Jackson:
Hart:
Mondale:
3(.25) + 2(.35) + 2(.40) = 2.25
3(.35) + 2(.25) + 1(.40) = 1.95
3(.40) + 1(.35) + 1(.25) = 1.8
Different Rules, Different Outcomes
C.
1984 NY Democratic Primary
Jackson coalition
Hart coalition
Mondale coalition
1)
2)
3)
25%
35%
40%
JHM
HJM
MJH
Plurality:
1)
Mondale Wins
1)
no one has a majority in the first round so Hart and Mondale go to the
second round.
Hart wins by majority in the second round. Hart wins.
Majority with Runoff:
2)
Borda Count:
Jackson:
Hart:
Mondale:
3(.25) + 2(.35) + 2(.40) = 2.25
3(.35) + 2(.25) + 1(.40) = 1.95
3(.40) + 1(.35) + 1(.25) = 1.8
Jackson wins.
In Class Election Results
Javier
Adam
Javier
Javier
Javier
Eli
Eli
Adam
Javier
Kingsley
Kingsley
Eli
Chris
Adam
Eli
Eli
Ovi
Eli
Adam
Kingsley
Javier
Kingsley
Ovi
Eli
Adam
Kingsley
Javier
Ovi
Ovi
Kingsley
Adam
Ovi
Ovi
Adam
Kingsley
Chris
Chris
Chris
Chris
Chris
Ovi
Chris
-----
-----
-----
-----
Ovi
Adam
Eli
Eli
Adam
Javier
Adam
Javier
Eli
Kingsley
Javier
Kingsley
Javier
Ovi
Kingsley
Adam
Kingsley
Eli
Ovi
Ovi
Chris
Chris
Chris
Chris
In Class Election Results
Adam
Ashley
Zack
Ricky
Rakyah
Jackson
Hallie
------Jackson
Rakyah
Hallie
Ricky
Ashley
Adam
Zack
------Hallie
Jackson
Ricky
Rakyah
Adam
Zack
Ashley
------Zack
Adam
Ashley
Hallie
Ricky
Jackson
Rakyah
Adam
Hallie
Zack
Ashley
Jackson
Ricky
Rakyah
------Rakyah
Ricky
Hallie
Adam
Ashley
Jackson
Zack
------Hallie
Zack
Ashley
Jackson
Adam
Rakyah
Ricky
------Zack
Adam
Rakyah
Ricky
Hallie
Ashley
Jackson
Ashley
Adam
Hallie
Ricky
Rakyah
Zack
Jackson
------Ricky
Ashley
Jackson
Adam
Rakyah
Hallie
Zack
------Jackson
Hallie
Zack
Ashley
Ricky
Adam
Rakyah
-------
Ashley
Adam
Zack
Ricky
Hallie
Rakyah
Jackson
------Ricky
Hallie
Ashley
Adam
Jackson
Zack
Rakyah
------Ricky
Adam
Ashley
Hallie
Jackson
Rakyah
Zack
-------
Ashley
Hallie
Rakyah
Ricky
Adam
Jackson
Zack
------Zack
Ricky
Adam
Jackson
Hallie
Rakyah
Ashley
------Ricky
Adam
Rakyah
Ashley
Jackson
Hallie
Zack
-------
Hallie
Rakyah
Adam
Zack
Ashley
Ricky
Jackson
------Adam
Ashley
Hallie
Jackson
Rakyah
Ricky
Zack
------Ricky
Adam
Rakyah
Jackson
Zack
Ashley
Hallie
-------
Hallie
Ricky
Ashley
Rakyah
Adam
Zack
Jackson
------Adam
Hallie
Zack
Ashley
Jackson
Ricky
Rakyah
------Ricky
Ashley
Rakyah
Jackson
Hallie
Adam
Zack
-------
In Class Election Results
Winner under each method (seven candidates at once)
Plurality: Ricky
MRR: tie
IRV: Ricky
Borda Count: Adam
Criteria:
Condorcet Winner: none
Condorcet Loser: Zack
Different Rules, Different Outcomes
D. Discussion
•
•
•
•
Notice that different candidates won depending upon
which voting rule we used.
When we talk about popular will, which voting procedure
are we referring to?
If you observe that a country elects say Jones, can you say
that the choice of Jones represents the interest of the
people?
If different voting rules select different candidates, then a
conniving politician can manipulate the outcome of the
vote by changing the voting rule.
Ex: Post-Cold War Poland.
Different Rules, Different Outcomes
•
•
Note: This should not suggest that different voting rules will
always select different candidates, but it should suggest
that they can (and by the way frequently do) select
different candidates.
Since we know that the voting rule matters, the natural
question is which voting rule is best.
Different Rules, Different Outcomes
E. More Practice
5 voters
D
A
C
B
3 voters
B
C
D
A
3 voters
C
D
A
B
Which candidate wins under plurality rule?
Which candidate wins under MRR?
Which candidate wins under IRV?
Which candidate wins under Borda Count?
Which candidate is the Condorcet winner?
Which candidate is the Condorcet loser?
Different Rules, Different Outcomes
movie: social choice the impossible dream