III. Fair Voting Procedures (Social Choice)

Examples

• political election
• where to build a new road
• winner in an ice skating competition (figure skating)
• picking the "best" football team
• searching the Internet (Which are the "best" sites for a particular search?)

0. Easy Case: Only Two Alternatives

Here one just takes a vote and the choice with the majority of votes wins. This is an honest, though not very surprising, theorem.

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Thus, below we always assume there are at least three "candidates".

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1. Sequential Pairwise Voting

Each row in the following represents the result of one "election" between two candidates.

Alice 5	Anne 4
Alice 4	Tom 5
Anne 6	Tom 3

If the first "election" between Alice and Ann, then Alice wins but then looses the next election between herself and Tom.
Winner: Tom.

If the first "election" between Alice and Tom, then Tom wins but he then looses the next election between himself and Anne.
Winner: Anne.

If the first "election" between Anne and Tom, then Anne wins but she then looses the next election between herself and Alice.
Winner: Alice.

MORAL: In this sort of election the winner may depend on the order in which the elections are held.

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Examples 2 - 6 below (from The perplexing mathematics of presidential elections) all use the following hypothetical data from the USA Presidential Election held in 2000:

6 million	Bush	Nader	Gore
5 million	Gore	Nader	Bush
4 million	Nader	Gore	Bush

Note that these voters give their ordered list of all the candidates, not just their first choices.

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2. Plurality Voting

Here we only look at the first choice of each voter. We ignore the rest of their ordering of the candiates. Most political elections currently use this system.
Bush wins (6 million 1st place) -- yet 9 million voters ranked him last.

3. Single Transferable Voting (Hare)

This is used in Australia and Ireland

The first round eliminates Nader. All his votes go to Gore, so in the second round, Gore has 9 million votes and Bush has 6 million.

Winner: Gore, but 10 million prefer Nader to Gore.

4. Borda Count

Give first choice 3 points, second choice 2 points, third place 1 point.

• Bush: 6m*3 + 5m*1 + 4m*1 = 27m points
• Gore: 6m*1 + 5m*3 + 4m*2 = 29m points
• Nader: 6m*2 + 5m*2 + 4m*3 = 34 points

Winner: Nader

5. Approval

How many voters oppose each candidate?

Say Gore and Nader voters can accept either candidate, but will not accept Bush.

Then: Nader 15m votes, Gore 9m voters, and Bush 6m votes.

Winner: Nader

One related alternate system is to give each voter 5 points, say, to distribute among the candidates.

6. Condorcet (head-to-head)

Between Gore and Nader: 5m   10m, so Nader wins.
Between Bush and Nader: 6m   9m, so Nader Wins.

Winner: Nader

Remark: In this sort of election, it could be that there is no winner. See Example 1 above. But if there is a winner in a Condorcet election, perhaps that person should be declared the "winner."

Desirable Voting Rules:

• Transitivity: If a voter prefers A to B (A > B) and B > C, then this voter also prefers A to C, that is, A > C. This may be thought as measuring the consistency of each voter's preferences.

• Unanimity: If everyone prefers A to B, then so does society.

• Independence of Irrelevant Alternatives (IIA): If the voting system prefers A to B and someone changes their ranking of C, then society still ranks A over B.

• Pareto: If everyone prefers B to D, then D is not among the winners.

Example: Borda Count fails IIA

3 voters:	A	B	C
2 voters:	C	B	A

Then:
A:   2*3 + 0*2 = = 6 points
B:   1*3 + 1*2 = 5 points
C:   0*3 + 2*2 = 4 points
  Winner: A

Now say 2 voters change their vote, putting C between A and B. Then:

3 voters:	A	B	C
2 voters:	B	C	A

Thus:
A:   2*3 + 0*2 = = 6 points
B:   1*3 + 2*2 = 7 points
C:   0*3 + 1*2 = 2 points
  Winner: B

Example: Sequential pairwise fails Pareto

1 voter:	A	B	D	C
1 voter:	C	A	B	D
1 voter:	B	D	C	A

A vs. B:   2 > 1   so A wins
A vs. C:   1 < 2   so C wins
C vs. D:   2 > 1   so D wins
BUT everyone prefers B to D.

Arrow's Impossibility Theorem

Transitivity, Unanimity, and IIA is only possible in a dictatorship.

Moral: Using these "features", there cannot be any perfect voting system. Thus, we must change something.

A Practical Problem

A separate, but key ingredient in any voting system is that its results must be accepted by the voters. If the procedure is even modestly complicated or controversial, voters may not trust the results.

One can see this vividly in the BCS procedure used to select the best college football team in the USA. It combines rankings by both "experts" (sports writers) and by computers. See

• College Football Ratings
• Final 2001-2002 BCS Rankings
• Bibliography on Rankings

The election in 2000 for the Mayor of London allows voters to specify their first two choices. It seems to have been understood only vaguely by the New Yorker author. See http://www.prairienet.org/icpr/news/Hertzberg052900.html