Upload
arlene-rogers
View
221
Download
2
Embed Size (px)
Citation preview
The Difference a Different Voting System Makes
Alex Tabarrok
Different Voting Systems
Individual Rankings
(Inputs)B D
AC C
CA B
DD A
B
Voting System
(Aggregation Mechanism)
Election Outcome
(Global Ranking)BDAC
What is the Will of the Voters?
A Simple Election Example
Number of Voters → 39 24 37
1st A C B
2nd C B C
3rd B A A
• Plurality Rule: A>B>C
• Borda Count: C>B>A
• “7,1,0” Count: B>A>C
Mathematics of Point Score Voting Systems 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].
The standardized plurality rule system is s= The standardized Borda Count is s=
1
0
0
2
1
0
Plurality Rule
BordaCount
1-s
s
0
Standardized Point Score System
01
3
?
?
Mathematics of Point Score Voting Systems
A voter may rank three candidates in any one of six possible ways.
The vote matrix can be read in two ways. 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.
ABC ACB CAB CBA BCA BACA 1-s 1-s s 0 0 sB s 0 0 s 1-s 1-sC 0 s 1-s 1-s s 0
p₁ (ABC)p₂
(ACB)p₃
(CAB)p₄
(CBA)p₅
(BCA)p₆
(BAC)
1-s 1-s s 0 0 ss 0 0 s 1-s 1-s0 s 1-s 1-s S 0
=
A’s tallyB’s
tallyC’s
tally
Familiar Examples
0390
24370
1 1 0 0 0 00 0 0 0 1 10 0 1 1 0 0 =
393724
0390
24370
2/3 2/3 1/3 0 0 1/31/3 0 0 1/3 2/3 2/30 1/3 2/3 2/3 1/3 0
=26
32.6641.33
Plurality Rule
Borda Count
A Simple Election Example
39 24 37
1st A C B
2nd C B C
3rd B A A
p₁ =0(ABC)
p₂=39(ACB)
p₃=0(CAB)
p₄=24(CBA)
p₅=37(BCA)
p₆=0(BAC)
The Representation Triangle and the Procedure Line
p₁+p₂+(-p₁-p₂+p₃+p₆) s∗p₆+p₅+(p₄-p₅+p₁-p₆) s∗p₃+p₄+(p₂-p₃-p₄+p₅) s∗
1-s 1-s s 0 0 ss 0 0 s 1-s 1-s0 s 1-s 1-s s 0 =
0.5 1A
0.5
1B
1) ABC
2)ACB 3)CAB
5)BCA 6)BAC
C
4)CBA
p₁p₂p₃p₄p₅p₆
• Interpret p₁…p6 as shares of each type of voter then the tallies are vote shares.
• Vote shares must sum to 100% so one of the equations is redundant. Thus we can graph in 2-dimensions.
A Simple Election Example
39 24 37
1st A C B
2nd C B C
3rd B A A
Plurality Rule: A>B>C
Borda Count: C>B>A
“7,1,0” Count: B>A>C
0390
24370
39-39 s∗37-13 s∗24+52 s∗
One Ranking Many Outcomes
Maximum Outcomes from One Ranking
Candidates OutcomesMax. Outcomes from 1 Ranking
Max. Outcomes from 1 Ranking_____________Total Outcomes
2 2 1 0.53 6 4 0.664 24 18 0.755 120 96 0.86 720 600 0.8337 5040 4320 0.8578 40,320 35,280 0.8759 362,880 322,560 0.888
10 3,628,800 3,265,920 0.9n n! n!-(n-1)! 1-(1/n)
Source: Saari (1992)
0.5 1A
0.5
1B
1) ABC
2)ACB 3)CAB
5)BCA 6)BAC
C
4)CBA
Intuition for these results comes from the geometry of the procedure line extended to higher dimensions.
Many Outcomes is the Norm
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.
The 1992 Election
0.5 1 Clinton
0.5
1
Bush
1
23
4
5 6
Perot
Pluralityy Rule
Borda Count
Anti-Plurality Rule
• Multiple outcomes from the same profile 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!
President Perot and Approval Voting
0.5 1 Clinton
0.5
1
Bush
1
23
4
5 6
Perot
Pluralityy Rule
Borda Count
Anti-Plurality Rule
• Approval voting is an increasing 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!
Voting
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.
In Defense of Democracy
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.”
Democracy and Famines
The Democratic and Capitalist Peace
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.
Economic Freedom, Democracy and Living Standards
The Democratic/Capitalist Peace
Democracies
Free Economies
10
20
30
40
50
60
Fre
e E
cono
mie
s
0
10
20
30
40
50
60
70
Num
ber
of D
emo
crac
ies
1900
1910
1920
1930
1940
1950
1960
1970
1980
1990
2000
2010
Year
Source: Polity IV, Democracy measured as democ>=8.Economic Freedom of the World 2010, measured as chained summary index>7
1900-2009The Growth of Democracy and Economic Freedom
International Conflict is Down
So What is Democracy Good For?
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.