How Voting Systems Affect Democracy and Math Affects Voting Systems

56 Pages·2010·0.46 MB·English

by  Matthew Smedberg

Early and Often
How Voting Systems Affect Democracy
and
Math Affects Voting Systems
Matthew Smedberg
VanderbiltUniversity
DepartmentofMathematics
2 November 2010
M.Smedberg (VanderbiltUniv.) VotingSystems 2November2010 1/20
What is an electoral system?
The phrase “electoral system” refers to the rules governing how voters
express their preferences as to who governs, together with rules for how
those votes determine who is seated in office.
Example: In the United States, the electoral system is quite simple. The
nation is broken into states, which are the electoral districts for Senate
elections, and states into smaller districts for House elections. At each
election, voters in each district select exactly one candidate for office; the
candidate with the most votes cast for him/her is seated.
M.Smedberg (VanderbiltUniv.) VotingSystems 2November2010 2/20
A motivating example
In the 2000 U.S. Presidential election in Florida, 5,963,110 votes were
cast, distributed
(cid:18) (cid:19)
Bush Gore Nader Other
2,912,790 2,912,253 97,488 40,579
Mr. Bush received the state’s electoral votes and went on to win the
Presidency. What made this result interesting is that most Nader voters
actually preferred Gore to Bush, while not many Bush voters preferred
Nader to Gore: that is, if voters were presented with a series of two-way
choices, the outcomes would probably have been approximately
(cid:18) (cid:19)
Bush/Gore Bush/Nader Gore/Nader
2,912,790 / 3,009,741 Bush (cid:29) Nader Gore (cid:29) Nader
Thus Gore wins every individual match, but loses the tournament!
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2 What different electoral systems are possible?
3 What effects, if any, would changes have on U.S. public life?
Three questions
1 How might a different electoral system have avoided this
“paradoxical” outcome?
M.Smedberg (VanderbiltUniv.) VotingSystems 2November2010 4/20
3 What effects, if any, would changes have on U.S. public life?
Three questions
1 How might a different electoral system have avoided this
“paradoxical” outcome?
2 What different electoral systems are possible?
M.Smedberg (VanderbiltUniv.) VotingSystems 2November2010 4/20
Three questions
1 How might a different electoral system have avoided this
“paradoxical” outcome?
2 What different electoral systems are possible?
3 What effects, if any, would changes have on U.S. public life?
M.Smedberg (VanderbiltUniv.) VotingSystems 2November2010 4/20
Advantage: Borda count (n = 3) resolves Bush v. Gore v. Nader without
paradoxes.
Disadvantage?
Borda count
A first attempt (Jean-Charles de Borda, ca. 1800):
Instead of simply picking one favorite, the voter ranks their top n
candidates. A voter’s most favored candidate receives n−1 weighted
votes, their next favored n−2, etc.
M.Smedberg (VanderbiltUniv.) VotingSystems 2November2010 5/20
Borda count
A first attempt (Jean-Charles de Borda, ca. 1800):
Instead of simply picking one favorite, the voter ranks their top n
candidates. A voter’s most favored candidate receives n−1 weighted
votes, their next favored n−2, etc.
Advantage: Borda count (n = 3) resolves Bush v. Gore v. Nader without
paradoxes.
Disadvantage?
M.Smedberg (VanderbiltUniv.) VotingSystems 2November2010 5/20
40 30 0
130 130 130
125 135 165
100 90 50
R L! L by a landslide!
K > L 50
R 130
L 115
K 110
Winner R
Borda’s shortcomings
Example: Suppose we have one right-wing candidate R and two left-wing
candidates L and K. 65 right-wing voters prefer R > L > K, but a total of
50 left-wing voters may prefer either K > L > R or L > K > R.
Predict: With all 50 voting hard-left (K > L), who wins the election?
What happens as voters move towards the center (preferring L over K)?
M.Smedberg (VanderbiltUniv.) VotingSystems 2November2010 6/20
40 30 0
130 130 130
125 135 165
100 90 50
R L! L by a landslide!
Borda’s shortcomings
Example: Suppose we have one right-wing candidate R and two left-wing
candidates L and K. 65 right-wing voters prefer R > L > K, but a total of
50 left-wing voters may prefer either K > L > R or L > K > R.
Predict: With all 50 voting hard-left (K > L), who wins the election?
What happens as voters move towards the center (preferring L over K)?
K > L 50
R 130
L 115
K 110
Winner R
M.Smedberg (VanderbiltUniv.) VotingSystems 2November2010 6/20