Chapter 12
Social Choice: The Impossible Dream
Two Alternatives
Majority Rule - candidate with the most votes wins.
Majority Rule has 3 desirable properties:
1. All voters are treated equally.
2. Both candidates are treated equally.
3. If a new election were held and a single voter changes his/her vote from the loser to the winner (and no one else changes a vote) then the outcome of the election will be the same.
May’s Theorem
If the number of voters is odd, and we are interested only in voting systems that never result in a tie, then majority rule is the only voting system for two alternatives that satisfies the three conditions just listed.
Three or More Alternatives
Plurality voting - Only 1st place votes are considered. The candidate with the most votes wins.
A Condorcet winner defeats each of the other candidates in a head-to-head election.
In the 2000 presidential election if Al Gore had been pitted against any of the other three candidates, based on preference lists, Gore would have won the election (both in Florida and the presidential election). Thus, Gore was a Condorcet winner.
Condorcet Winner Criterion (CWC) A voting procedure satisfies the Condorcet winner criterion provided that, for every possible sequence of preference lists either
1) There is no Condorcet winner, or
2) There is a Condorcet winner and it is the unique winner of the election.
Shortcomings of plurality voting-
Does not satisfy the Condorcet Winner Criterion
Ballots do not provide an opportunity for a voter to express any preferences except for naming his or her top choice.
It is subject to manipulability. That is, there are elections in which it is to a voter’s advantage to submit a ballot that misrepresents his or her true preferences.
Borda Count - example of a rank method.
A rank method assigns points in a nonincreasing manner to the ordered candidates on each voter’s preference list and then sums these points to arrive at a groups final ranking. The special case where there are n alternatives with each first-place vote worth n-1 points, each second-place vote worth n-2 points, and so on down to each last-place vote worth zero points is known as the Borda Count.
There are other rank methods (See examples)
Example of Borda count method.
RANK NUMBER OF VOTERS (5) POINTS
First A A A B B 2
Second B B B C C 1
Third C C C A A 0
A 6
B 7 (Winner)
C 2
A voting system satisfies independence of irrelevant alternatives if it is impossible for an alternative B to move from nonwinner status to winner status unless at least one voter reverses the order in which he or she had B and the winning alternative ranked.
RANK NUMBER OF VOTERS (5) POINTS
First A A A C C 2
Second B B B B B 1
Third C C C A A 0
A 6 (Winner)
B 5
C 4
RANK NUMBER OF VOTERS (5) POINTS
First A A A B B 2
Second B B B C C 1
Third C C C A A 0
A 6
B 7 (Winner)
C 2
The above shows that the Borda Count does not satisfy independence of irrelevant alternatives.
Also the Borda Count is subject to manipulability.
Sequential Pairwise Voting - starts with an agenda and pits the first alternative against the second in a one-on-one contest. The winner (or both, if they tie) then moves on th confront the 3rd alternative in the list, one-on-one. Losers are deleted. This process continues throughout the entire agenda, and those remaining at the end are winners.
For a given sequence of individual preference lists, the particular agenda chosen can greatly affect the outcome of the election.
Agenda: A, B, C, D
RANK NUMBER OF VOTERS (3)
First A C B
Second B A D
Third D B C
Fourth C D A
A vs B A wins
A vs C C wins
C vs D D wins
Note- Everyone prefers B to D yet D wins.
Pareto Condition: If everyone prefers one alternative to another alternative, then that second alternative is not among the winners.
Sequential pairwise voting fails to satisfy the Pareto condition.
Hare System (single transferable vote system) - arrives at a winner by repeatedly deleting alternatives that are “least preferred” in the sense of being at the top of the fewest preferences lists. If a single alternative remains after all others have been eliminated, it alone is the winner. If two or more alternatives remain and all of theses remaining alternatives would be eliminated in the next round, then these alternatives are declared to be tied for the win.
NUMBER OF VOTERS
RANK 5 4 3 1
First A C B B
Second B B C A
Third C A A C
A wins.
NUMBER OF VOTERS
RANK 5 4 3 1
First A C B A
Second B B C B
Third C A A C
C wins
Monotonicity- if an alternative is a winner, and a new election is held in which the only ballot change made is for some voter to move the former winning alternative higher on his or her preference list, then the original winner should remain a winner.
The Hare system does not satisfy monotonicity.
The Hare system can also be mainpulated.
Voting Paradox of Condorcet
RANK NUMBER OF VOTERS (3)
First A B C
Second B C A
Third C A B
Arrow’s Impossibility Theorem
There does not exist, and never will exist, any social choice procedure that satisfies both the Condorcet Winner Criterion and Independence of Irrelevant Alternatives.
Approval Voting