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 Preferences Number of people with this choice 1st2nd3rd ACheese CakeCrepesApple Pie6 B CrepesCheese Cake5 CCrepesApple PieCheese Cake4 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: = 10 Apple pie: 5 Crepes is preferred to apple pie by more people! The Chair should have chosen crepes! Outcomes Ranked Preferences Number of people with this choice 1st2nd3rd A CrepesApple Pie6 B Crepes 5 C 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 = 9 Cheese cake 6 Crepes was actually the first choice! What is going on! Outcomes Ranked Preferences Number of people with this choice 1st2nd3rd ACheese CakeCrepes 6 B Cheese Cake5 CCrepes Cheese Cake4

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 votes B voters prefer 2. to 1. : 5 votes Resolution 1. wins first round. Second round: 1. vs 3. A voters prefer 1. to 3. : 5 votes B and C voters prefer 3. to 1.: 10 votes Resolution 3. Wins by a 10:5 margin. The chair is replaced! Outcomes Ranked Preferences Number of people with this choice 1st2nd3rd Ares. 1res. 2res. 35 B res. 2res. 3res. 15 Cres. 3res. 1res. 25

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 votes Resolution 2. wins first round. Second round 1. vs 2. A and C voters prefer 1. to 2. : 10 votes B voters prefer 2. to 1.: 5 votes Resolution 1. Wins by a 10:5 margin! The chair has a vote of confidence! Outcomes Ranked Preferences Number of people with this choice 1st2nd3rd Ares. 1res. 2res. 35 B res. 2res. 3res. 15 Cres. 3res. 1res. 25

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 Preferences Number of people with this choice 1st2nd3rd AAnnBarbCarol4 B BarbAnnCarol3 C AnnBarb3 DCarolBarbAnn3 Runoff Method. First round: Ann 4, Barb 3, Carol 6: Barb is eliminated Second round: Ann 4+3=7, Carol 6: Ann wins.

12 Second Committee’s Preferences Outcomes Ranked Preferences Number of people with this choice 1st2nd3rd AAnnBarbCarol4 B BarbAnnCarol3 C AnnBarb3 D CarolAnn3 Runoff Method. First round: Ann 4, Barb 6, Carol 3: Carol is eliminated Second round: Ann 4+3=7, Barb 6: Ann wins.

13 Combined Committee’s Preferences Outcomes Both Committees :Ranked Preferences Number of people with this choice 1st2nd3rd AAnnBarbCarol8 BBarbAnnCarol6 CBarbCarolAnn3 DCarolAnnBarb6 ECarolBarbAnn3 Runoff Method. First round: Ann 8, Barb 9, Carol 9: Ann is eliminated Second 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 Preferences Number of people with this choice 1st2nd3rd AAbbottBoyceChair7 B BoyceAbbottChair7 C AbbottBoyce1

15 Election of the New Chair Vote Tally: Abbott: (7x2)+(8x1)=14+8=22 Boyce: (7x2)+(7x1) =14+7=21 Chair: 2 Predicted Winner: Abbott Outcomes Ranked Preferences Number of people with this choice 1st2nd3rd AAbbottBoyceChair7 B BoyceAbbottChair7 C AbbottBoyce1

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 Preferences Number of people with this choice 1st2nd3rd AAbbottChairBoyce7 B ChairBoyce7 CChairAbbottBoyce1 Vote Tally: Abbott: (7x2)+1=14+1 =15 Boyce: (7x2 ) =14 Chair: (1x2)+(14x1) =16 The Chair is relected!

17 Analysis of the Department Dessert Problem Election MethodWinning Drink PluralityCheese Cake Pairwise Comparisoncrepes RunoffApple 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 AgendaWinning Resolutions 1. vs 2. then vs 3. then 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.