1

Voting: Perception and Reality

• Do voters vote in their best interest?

• When they do, does the outcome of the vote reflect those interests?

• What type of political system will give representatives of the people power to act in the interests of population?

2

What Makes a Vote Fair?

• Who gets to vote?

• How are the votes counted?

• The ballot should be easy to understand.

• The process should be transparent.

3

Criteria for a Fair Voting System

• The candidate who most people put as their first choice should win. (plurality)

• If a candidate would beat all other candidates in pairwise comparisons they should win the election.

• If most voters prefer A to B then the addition of a third candidate C into the election should not result in B winning the election.

• If A is the winner of an election, then in a re-election if voters change their votes only in ways that are favorable to candidate A, then A should still win the election.

4

An Electoral Fable

Departmental Politics: The Annual Department Party

Outcomes

Ranked PreferencesNumber of people with this choice1st 2nd 3rd

A Cheese Cake Crepes Apple Pie 6

B Apple Pie Crepes Cheese Cake 5

C Crepes Apple Pie Cheese Cake 4

What Dessert to Provide? Cheese Cake, Crepes Suzette , or Apple Pie?

• Conclusion: Counting the voters’ first preferences (plurality) the winner is Cheese Cake, the second place goes to Apple Pie and the third is Crepes.

• The department Chair chooses Cheese Cake for the party.

5

Bad Milk

Unfortunately there was no cheese cake, so the Chair went with the second choice: apple pie

Is apple pie really the second choice? If cheese cake was not available what would people have chosen?

Crepes: 6 + 4 = 10 Apple pie: 5Crepes is preferred to apple pie by more people! The Chair should have chosen crepes!

Outcomes

Ranked PreferencesNumber of people with this choice1st 2nd 3rd

A Crepes Apple Pie 6

B Apple Pie Crepes 5

C Crepes Apple Pie 4

6

It Gets Worse

What games is the Chair playing. Should cheese cake have been the winner?

Clearly apple-pie is the last choice. What are the preferences between cheese cake and crepes?

Crepes 6 + 5 = 9 Cheese cake 6Crepes was actually the first choice! What is going on!

Outcomes

Ranked PreferencesNumber of people with this choice1st 2nd 3rd

A Cheese Cake Crepes 6

B Crepes Cheese Cake 5

C Crepes Cheese Cake 4

7

Department Unrest

The Chair had the order of preferences exactly reversed! Was the Chair’s known preference for cheese cake a factor?

The department are up in arms. What other votes had he rigged? They call a special meeting to discuss the issue and decide to vote for one of the following resolutions:

Resolutions:1. Commend the Chair for his honest efforts to be

fair.2. Recommend that the Chair teach remedial math

to improve his vote tallying skills.3. Ask the Dean to find a new Chair.

In order to avoid accusations of bias, the Chair decides to have the professors first vote between resolutions 1. and 2. Then the winner would be matched against the dreaded resolution 3.

8

Successive Elimination

Results:

First round: 1. vs 2.A and C voters prefer 1. to 2. : 10 votesB voters prefer 2. to 1. : 5 votes

Resolution 1. wins first round.Second round: 1. vs 3.

A voters prefer 1. to 3. : 5 votesB and C voters prefer 3. to 1. : 10 votes

Resolution 3. Wins by a 10:5 margin. The chair is replaced!

Outcomes

Ranked PreferencesNumber of people with this choice1st 2nd 3rd

A res. 1 res. 2 res. 3 5

B res. 2 res. 3 res. 1 5

C res. 3 res. 1 res. 2 5

9

Rounds of Elimination

But wait, what if he had asked people to vote on 2. vs 3. first, and then had the winner go up against 1.?

First round: 2. vs 3.A and B voters prefer 2. to 3. : 10 votes

C voters prefer 3. to 2. : 5 votesResolution 2. wins first round.Second round 1. vs 2.

A and C voters prefer 1. to 2. : 10 votesB voters prefer 2. to 1. : 5 votes

Resolution 1. Wins by a 10:5 margin! The chair has a vote of confidence!

Outcomes

Ranked PreferencesNumber of people with this choice1st 2nd 3rd

A res. 1 res. 2 res. 3 5

B res. 2 res. 3 res. 1 5

C res. 3 res. 1 res. 2 5

10

The Dean Steps in

In his attempt to be fair and balanced the Chair made a major blunder. Now he has to face the Dean.

Before replacing the Chair, the Dean considered some other evidence.

Selection of student representative to Dean’s council. Evidence: • Three candidates stood for election. Ann, Barb and Carol. • The Chair had polled two student committees (13 in each committee). • Both selected Ann as the preferred candidate. • The Chair appointed Barb!

Was the Chair playing favorites? Did he have a soft spot for Barb?

Is there an explanation?

11

First Committee’s Preferences

Outcomes

Ranked PreferencesNumber of people with this choice1st 2nd 3rd

A Ann Barb Carol 4

B Barb Ann Carol 3

C Carol Ann Barb 3

D Carol Barb Ann 3

Runoff Method. First round: Ann 4, Barb 3, Carol 6 : Barb is eliminatedSecond round: Ann 4+3=7, Carol 6 : Ann wins.

12

Second Committee’s Preferences

Outcomes

Ranked PreferencesNumber of people with this choice1st 2nd 3rd

A Ann Barb Carol 4

B Barb Ann Carol 3

C Carol Ann Barb 3

D Barb Carol Ann 3

Runoff Method. First round: Ann 4, Barb 6, Carol 3 : Carol is eliminatedSecond round: Ann 4+3=7, Barb 6 : Ann wins.

13

Combined Committee’s Preferences

Outcomes

Both Committees :Ranked PreferencesNumber of people with this choice1st 2nd 3rd

A Ann Barb Carol 8

B Barb Ann Carol 6

C Barb Carol Ann 3

D Carol Ann Barb 6

E Carol Barb Ann 3

Runoff Method. First round: Ann 8, Barb 9, Carol 9 : Ann is eliminatedSecond round: Barb 8+9=17, Carol 9 : Barb wins easily!

14

Election of the New Chair

The Dean calls for an election for a new Chair. Three candidates stand: Abbott, Boyce and the disgraced Chair.

The Dean decides to use a new method for voting called the Borda count: Each candidate gets a certain number of points: 2 points for each first place vote, 1 point for each second place vote and 0 points for each last place vote. The candidate with the most points wins.

Based on department gossip this is what was known about the preferences before the actual vote:

Outcomes

Ranked PreferencesNumber of people with this choice1st 2nd 3rd

A Abbott Boyce Chair 7

B Boyce Abbott Chair 7

C Chair Abbott Boyce 1

15

Election of the New Chair

Vote Tally: Abbott: (7x2)+(8x1)=14+8=22Boyce: (7x2)+(7x1) =14+7=21Chair: 2Predicted Winner: Abbott

Outcomes

Ranked PreferencesNumber of people with this choice1st 2nd 3rd

A Abbott Boyce Chair 7

B Boyce Abbott Chair 7

C Chair Abbott Boyce 1

16

Strategic Voting

Boyce supporters decided to switch their second and third choices to reduce the points for Abbott – thinking this will help their candidate will win.

Abbott supporters are worried about such a ploy so preempt it by switching their second and third choices.

Outcomes

Ranked Strategic PreferencesNumber of people with this choice1st 2nd 3rd

A Abbott Chair Boyce 7

B Boyce Chair Boyce 7

C Chair Abbott Boyce 1

Vote Tally: Abbott: (7x2)+1=14+1 =15Boyce: (7x2 ) =14Chair: (1x2)+(14x1) =16 The Chair is relected!

17

Analysis of the Department Dessert Problem

Election Method Winning Drink

Plurality Cheese Cake

Pairwise Comparison crepes

Runoff Apple pie

• Each of the methods seems to be fair – yet they each yield a different outcome.

• The winning choice depends more on the decision procedure then the voters’ preferences.

• What is the fair choice?

• Lessons for political elections?

18

Analysis of the Resolution Vote

Agenda Winning Resolutions

1. vs 2. then 3. 3.

2. vs 3. then 1. 1.

3. vs. 1. then 2. 2.

•The winning resolution depends on the order in the agenda.

•The person setting the agenda has a lot of power to influence the outcome.

•Are there objective criteria to determine the right/fair agenda or is it entirely strategic?

•Political relevance: voting on bills in the Senate and House.

19

Lessons of the Student Representative

• Dividing up a population into groups and polling the groups separately may not yield a result that reflects the will of the population as a whole.

• What does this say about how we elect presidents by combining the results from state wide elections through the Electoral College?

• Are primaries the best way to select candidates? Might the most preferred candidate be eliminated?

20

Conclusion

• There is no voting system that meets all the criteria for the fair voting.

• Some voting systems have more problems than others.

• The simplest and probably the worst is the plurality system. Favors candidates who are strong first choices, but often yields candidate who are not broadly supported.

• The most fair by some objective (geometric) criteria is the Borda count. Approval voting is reasonably simple and fair by most criteria. These two are best at producing consensus candidates.

Voting systems considered:• Plurality• Pairwise comparison• Successive Elimination• Plurality with Runoff/Instant Runoff• Borda Count

• Others: Approval Voting: Give one vote to each of the candidates that you could approve.