Fair Voting Procedures (Social Choice)

Examples

• When to have an exam in a course.
• Political elections
• Winner in a figure skating competition
• Picking the best college football teams
• Ranking tennis players
• Rookie of the Year (baseball)
• Academy Awards (movies)
• Time magazine's Person of the Year
• Choosing an architect for a big national museum
• Where to build a new road
• Which house should I buy? Which job should I take?
• Searching the Internet (Which are the "best" sites for a particular search?)
• Is Democracy possible?

Some Issues

• How many candidates are there?
• How many voters are there?
• Primary elections?

  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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  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.

  Another Version

  • Nondictatorship: The preferences of an individual should not become the group ranking without considering the preferences of others.

  • Individual Sovereignty: each individual should be able to order the choices in any way and indicate ties

  • Unanimity: If every individual prefers one choice to another, then the group ranking should do the same

  • Freedom From Irrelevant Alternatives: If a choice is removed, then the others' order should not change

  • Uniqueness of Group Rank: The method should yield the same result whenever applied to a set of preferences. The group ranking should be transitive.

    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.

    Kenneth Arrow, Wikipedia

  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.

  Voting Strategies: Gaming the system. If a voter sincerely prefers A to B to C, will (should?) she vote that way? How should she respond to a poll of her preference?
  In tennis, if player A generally wins over player B and B generally wins over player C, it is often not true that A will generally win over C

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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. Ranked Choice "Single Transferable Voting" (Hare)

  This is used in Maine, Australia, and Ireland.

  Under this system, a voter ranks all candidates in order of preference. If no one is ranked first by more than 50 percent of voters, the candidate least often ranked first is eliminated from the ballots and the second choice of those voters gets added to the tallies of the remaining candidates.
  The process then repeats until a candidate does achieve 50 percent of the top ranking. In that sense, that candidate has majority support -- and wins.
  Applied to our example, 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.
  Remark: If there are more than 3 candidates (choices), a ranked choice election asks voters to make many more decisions than they would need to make using a simpler election procedure. As a result, many voters might not rank the maximum number of candidates. This creates the possibility that the election outcome might be different than if every voter had filled out a full ballot.
  In the 2011 San Francisco mayoral race, 27 percent of ballots did not rank either of the two candidates who reached the final round. See the perecptive May 2021 New York Times article

  4. Borda 3,2,1 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 = 34m points
  Winner: Nader
  • What if you assign different weights?

  • What if there are more than 3 candidates?

  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.
  As was noted above, in tennis, if player A generally wins over player B and B generally wins over player C, it is often not true that A will generally win over C. Thus, in some types of elections it would be inappropriate to impose transitivity on the voting procedure.

• Unanimity: If all the voters prefer 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 3,2,1 Count fails IIA

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

Then:
A:   3*3 + 1*2 = 11 points
B:   2*3 + 2*2 = 10 points
C:   1*3 + 3*2 = 9 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:   3*3 + 1*2 = 11 points
B:   2*3 + 3*2 = 12 points
C:   1*3 + 2*2 = 7 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.

Additional illuminating examples and discussion

See:
Ellenberg, Jordan, How Not To Be Wrong, Penguin Press, New York, 2014, pages 376--392,   418--420.

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