Transcript The Difference a Different Voting System Makes

Alex Tabarrok
Individual
Rankings
Voting System
(Inputs)
Election
Outcome
(Global Ranking)
B
D
A
C
C
C
A
B
D
A
D
A
B
C
(Aggregation
Mechanism)
B
D
A Simple Election Example
Number of
Voters →
39
1st
A
C
B
2nd
C
B
C
3rd
B
A
A
24
37
• Plurality Rule:
A>B>C
• Borda Count:
C>B>A
• “7,1,0” Count:
B>A>C





Assume n=3 candidates then we can write the
plurality rule system as (1,0,0) and the Borda
Count as (2,1,0).
Clearly, (10,0,0) is equivalent to plurality rule.
It's also true although a little bit more difficult
to see that (5,3,1) is equivalent to the Borda
Count.
Any point score voting system can be
converted into a standardized point score
system denoted (1-s,s,0), where s∈[0, 1/2]. ?0
1
? is s=
The standardized plurality rule system
3
The standardized Borda Count is s=
Plurality
Rule
Borda
Count
1
2
0
1
0
0
Standardized Point
Score System
1-s
s
0
A voter may rank three candidates in
any one of six possible ways.
 The vote matrix can be read in two
ways.

A
B
C
ABC
1-s
s
0
ACB
1-s
0
s
CAB
s
0
1-s
CBA
0
s
1-s
BCA
0
1-s
s
BAC
s
1-s
0
 Reading down a particular column we
see the number of points given to each
candidate from a voter with the
ranking indicated by that column.
 Reading across the rows we see where
a candidate's votes come from.

We will write the number of voters
with ranking (1) ABC as p₁ the number
of voters with ranking (2) ACB as p₂
and so forth up until p₆. We can place
all this information in matrix form by
multiplying the vote matrix with the
voter type matrix.
p₁
(ABC)
p₂
(ACB)
1-s
s
0
1-s
0
s
s
0
1-s
0
s
1-s
0
1-s
S
s
1-s
0
p₃
(CAB)
p₄
(CBA)
p₅
(BCA)
p₆
(BAC)
=
A’s
tally
B’s
tally
C’s
tally
p₁ =0
(ABC)
A Simple Election Example
p₂=39
(ACB)
39
24
37
p₃=0
A
C
B
C
B
A
B
C
A
p₄=24
(CAB)
1st
2nd
3rd
(CBA)
p₅=37
(BCA)
p₆=0
0
1
0
0
39
0
24
37
0
1/3
2/3
0
0
39
0
24
37
0
(BAC)
Plurality Rule
Borda Count
1
0
0
2/3
1/3
0
1
0
0
2/3
0
1/3
0
0
1
1/3
0
2/3
0
0
1
0
1/3
2/3
0
1
0
0
2/3
1/3
=
39
37
24
=
26
32.66
41.33
1-s
s
0
1-s
0
s
s
0
1-s
0
s
1-s
0
1-s
s
s
1-s
0
• Interpret p₁…p6 as shares of each type
of voter then the tallies are vote shares.
p₁
0
39
p₂
p₃
0
24
p₄
37
p₅
p₆
0
=
39-39∗s
p₁+p₂+(-p₁-p₂+p₃+p₆)∗s
p₆+p₅+(p₄-p₅+p₁-p₆)∗s
37-13∗s
p₃+p₄+(p₂-p₃-p₄+p₅)∗s
24+52∗s
1
B
• Vote shares must sum to 100% so one
of the equations is redundant. Thus we
can graph in 2-dimensions.
5)BCA
6)BAC
0.5
A Simple Election Example
1st
2nd
3rd
39
24
37
A
C
B
C
B
A
B
C
A
1) ABC
Plurality Rule: A>B>C
4)CBA
Borda Count: C>B>A
“7,1,0” Count: B>A>C
3)CAB
C
2)ACB
0.5
1
A
Maximum Outcomes from One Ranking
Max. Outcomes
from 1 Ranking
Max. Outcomes _____________
Candidates Outcomes from 1 Ranking Total Outcomes
2
2
1
0.5
3
6
4
0.66
4
24
18
0.75
5
120
96
0.8
6
720
600
0.833
7
5040
4320
0.857
8
40,320
35,280
0.875
9
362,880
322,560
0.888
10
3,628,800
3,265,920
0.9
n
n!
n!-(n-1)!
1-(1/n)
Intuition for these results
comes from the geometry
of the procedure line
extended to higher
dimensions.
B
1
5)BCA
6)BAC
0.5
1) ABC
4)CBA
3)CAB
C
2)ACB
0.5
Source: Saari (1992)
1
A

For even small electorates (say 50 or more) and 3 candidates a
single profile generates:






7 different rankings (including ties) about 6.7 percent of the time
5 different rankings 18.6 percent of the time,
3 different rankings 41.3 percent of the time
a single ranking 33.3 percent of the time.
A single profile, therefore, generates more than one ranking 66
percent of the time.
As the number of candidates increases the probability that all
positional voting systems agree on the winner (K=1) quickly goes to
zero.
• Multiple outcomes from the same profile
Bush
1
5
6
0.5
Plurality
y Rule
Borda Count
Anti -Plurality Rule
1
4
3
Perot
2
0.5
1
Clinton
are not always the case.
• In 1992, conservative commentators
emphasized President Clinton's failure to
receive more than 50% of the vote and thus
his failure, in their minds, to achieve a
"mandate.“
• An analysis of voter preferences, however,
reveals the surprising fact that Clinton
would have won under any point-score
voting system!
• Approval voting is an increasing
Bush
1
5
6
0.5
Plurality
y Rule
Borda Count
Anti -Plurality Rule
1
4
3
Perot
2
0.5
1
Clinton
popular system where each voter can
approve of as many candidates as he or
she likes.
• e.g. if there are 5 candidates the voter
could approve 1,2,3, or 4 of them.
• Approval voting vastly increases what
can happen. Note, for example, that
with candidates under approval voting
each voter has the option of using
plurality rule or anti-plurality rule!
• What could have happened in 1992?
Anything!
 Group choice is not at all like individual choice.
 Groups will always choose in ways that would
appear irrational if chosen by an individual.
 The voting system determines the outcome of an
election at least as much as do preferences.
 Voting does not represent the “will of the voters.”
 The idea of a group will is incoherent.

Nobel prize winner Amartya
Sen has argued that:
 “No famine has ever taken place in
the history of the world in a
functioning democracy.”
 “Democracies have to win elections
and face public criticism, and have
strong incentive to undertake
measures to avert famines and
other catastrophes.”



Democratic Peace – democracies rarely go to war against one
another.
Capitalist Peace – trading countries, countries with private
property and capitalist economies rarely go to war against
one another.
Democratic and capitalist peace are strongly supported in the
data and a consensus has developed in the International
Relations literature but less consensus on why.
The Growth of Democracy and Economic Freedom
60
1900-2009
40
30
50
30
50
40
Free Economies
Democracies
Free Economies
20
20
60
10
10
Year
Source: Polity IV, Democracy measured as democ>=8.
Economic Freedom of the World 2010, measured as chained summary index>7
20
10
20
00
19
90
19
80
19
70
19
60
19
50
19
40
19
30
19
20
19
10
0
19
00
Number of Democracies
70







Democracies don’t kill their own citizens or let them starve.
Democracy is compatible with economic freedom ->
democratic/capitalist peace.
Democracies avoid some very bad possibilities.
The threat of throwing politicians out of office is a constraint on what can
happen in a democracy.
Dictatorships and oligarchies need only not abuse a minority – in a
democracy the standard is higher.
Democracy, however, is not good at representing the will of the voters
and in general we should not expect democracy to be a good way of
making decisions.
Democracy should be seen as a way of limiting or constraining
government.