WHO do we elect?a look at voting methods

Robert Cutler ★ Purdue University ★ Jan 29, 2013

Definitions

Voting system or Electoral system

A method by which voters choose a winner from among one or more candidates

Algorithmic in nature

Evaluated using voting system criteria

Definitions

Election

The votes of all of the voters to determine a winner based on the rules of a voting system

Candidate

A choice option for the voter

Voter

A person choosing from among the candidates

Definitions

Winner

The overall choice of candidate determined by the voting system based on the preferences of the voters

Ballot

The choices of a voter in an election

Simple Majority

Each voter gets one vote

The winner is the candidate with more than 50% of the votes

Example:

20 voters

Candidates: Coke, Pepsi

Coke: 11 votes; Pepsi: 9 votes

Coke is the winner with 55% of the vote

Plurality

Each voter gets one vote

The winner is the candidate with most votes (even if not >50%)

Example:

20 voters

Candidates: Coke, Pepsi, Sprite

Coke: 5 votes; Pepsi: 8 votes; Sprite: 7 votes

Pepsi is the winner with only 40% of the vote

What if…?

Voters have preferences

What if they order their preferences?

Example:

20 voters; candidates: Coke, Pepsi, Sprite

7 voters like Pepsi better than Sprite and both better than Coke [P > S > C: 7]

And so on…

What if…?

P > S > C: 7P > C > S: 1C > S > P: 4C > P > S: 1S > C > P: 6S > P > C: 1

By plurality, Pepsi still wins by the same margin over Sprite and Coke (40% to 35% to 25%) as before

But…if Sprite were not in the mix, more people still prefer Coke to Pepsi (55% to 45%) as before

Yet…more people prefer Sprite to Pepsi (55% to 45%)

And way more people prefer Sprite to Coke (70% to 30%)

Who SHOULD WIN?

One option: RunoffP > S > C: 7P > C > S: 1C > S > P: 4C > P > S: 1S > C > P: 6S > P > C: 1

Eliminate all but top two vote getters. “Re-vote” based on preferences.

In the first round, Pepsi and Sprite win and Coke is eliminated

Of the 5 votes Coke received, 4 prefer Sprite; 1 prefers Pepsi

So in the second round, Sprite beats Pepsi by a 11 to 9 margin (55% to 45%)

But wait, it gets better!

Now let’s add Dr. Pepper to the mix…

We’ll simplify somewhat, but keep the same basic preferences:P > D > S > C: 7C > D > S > P: 4S > D > C > P: 6D > C > S > P: 3

Plurality

P > D > S > C: 7C > D > S > P: 4S > D > C > P: 6D > C > S > P: 3

Pepsi still wins in plurality voting with 35%

Runoff

P > D > S > C: 7C > D > S > P: 4S > D > C > P: 6D > C > S > P: 3

Coke and Dr. Pepper are eliminated in the first round.

Sprite gets all 7 of their votes to win in the second round 65% to 35%.

Another option: Instant Runoff

P > S > C: 7P > C > S: 1C > S > P: 4C > P > S: 1S > C > P: 6S > P > C: 1

Eliminate the lowest vote getter. “Re-vote” losing voters choices based on preferences. Continue until there is a winner.

With 3 candidates, exactly the same as Runoff

Instant Runoff

P > D > S > C: 7C > D > S > P: 4S > D > C > P: 6D > C > S > P: 3

Dr. Pepper is eliminated in the first round. All 3 votes go to Coke. Pepsi: 7 votes; Sprite: 6 votes; Coke: 7 votes.

Sprite is eliminated in the second round. All 6 votes go to Coke (Dr. Pepper already eliminated).

Coke wins with 13 votes (65% to 35%)!

One more option: Borda Count

P > S > C: 7P > C > S: 1C > S > P: 4C > P > S: 1S > C > P: 6S > P > C: 1

For each voter assign 1 point to the first place candidate; 2 points to the second place candidate; and so on…

Candidate with lowest score wins

Pepsi (8×1 + 2×2 + 10×3 = 42)Coke (5×1 + 7×2 + 8×3 = 43)Sprite (7×1 + 11×2 + 2×3 = 35) ➙ winner!

Borda Count

P > D > S > C: 7C > D > S > P: 4S > D > C > P: 6D > C > S > P: 3

Pepsi (7×1 + 0×2 + 0×3 + 13×4 = 59)Coke (4×1 + 3×2 + 6×3 + 7×4= 56)Sprite (6×1 + 0×2 + 14×3 + 0×4 = 48)Dr. Pepper (3×1 + 17×2 + 0×3 + 0×4 = 37) ➙ winner!

WTF?

Plurality: Pepsi wins

Runoff: Sprite wins

Instant Runoff: Coke wins

Borda Count: Dr. Pepper wins!

WTF?All methods seem “fair”

The plurality winner is preferred by more people than anyone else

The runoff winner is one of the top two preferred candidates and more preferred than the other when voters are asked to choose between them

The instant runoff winner is based on eliminating candidates one-by-one based on their preferences

The Borda count winner is based on the ranking of all candidates by all voters

Measures of “fairness”

We look at goals of an election and set criteria according to certain categories:

Absolute result criteria

Relative result criteria as voters change

Relative result criteria as nominees change

Administration criteria

Voter criteria

Absolute Result Criteria

Majority criterion: If one candidate is preferred by a majority of the voters, then that candidate must win.

Fails: Borda Count

Mutual majority criterion: If there is some subset of candidates such that the majority of voters prefer every candidate of the subset to every candidate outside the subset, then the winner must be in the subset.

Fails: Borda Count, Plurality

Absolute Result Criteria

Condorcet winner: The candidate who, when compared with every other candidate, is preferred by more voters.

Condorcet criterion: Chooses the Condorcet winner if one exists.

Satisfies: Majority vote

Fails: Borda Count, Plurality, Instant Runoff

Relative Result Criteria

Monotonicity criterion: A candidate should not be harmed if it is given higher preference by some voters.

In other words, if I change my ballot to rank winner x higher, x should not then lose the election

Satisfies: Plurality, Borda Count

Fails: Runoff, Instant Runoff

Relative Result Criteria

Participation criterion: The addition of a ballot where candidate A is strictly preferred to candidate B should not change the winner from candidate A to candidate B.

Satisfies: Plurality, Borda Count

Fails: Instant Runoff, Any Condorcet method!

Relative Result Criteria

Independence of irrelevant alternatives criterion: The addition of a candidate Y to an election where candidate X wins should not cause some third candidate Z to win.

Fails: Plurality

Administration Criteria

Can we find a winner in polynomial time?

Can we detect cheating (sum of tallies at polling stations) in polynomial time?

Voter Criteria

Ease of voting

Too many candidates

Alphabetical ordering of candidates

Understanding of tabulation method

Can we rank candidates equally?

Other Methods/Criteria

Methods

Approval

Copeland

Kemeny-Young

Minimax

Range voting

Ranked pairs

Schulze

Criteria

Condorcet loser

Reversal symmetry

Cloneproof

Equal rankings allowed

Arrow’s Impossibility Theorem

Can’t have a voting system that uses voters’ ranked preferences of candidates into a group winner while meeting three “fairness” criteria:

1) If every voter prefers X over Y, then the group prefers X over Y

2) If every voter’s preference between X and Y remains unchanged, then the group’s preference between X and Y also remains unchanged (even if other pairwise preferences change)

3) No single voter can always determine the group’s preference.

What does this mean?

Some say “No voting system is fair”

Trivially seen in “rock-paper-scissors” preferences

Fairness depends on:

Culture

History

Goals of election

What does this mean?

Bigger issues:

Voters who do not vote honestly (i.e., for candidates other than their preference

“I voted for Romney even though I prefer Gingrich because Romney has a better chance against Obama.”

“I voted for Santorum even though I prefer Gingrich because if Santorum drops out, then Romney will win in the two-person race.”

Next Thursday

• Prepare a 15 min presentation on one or two methods of tabulating votes.

• Make sure to share your method choices by emailing everyone so we don’t end up with the same methods!