The Mathematics of Voting Voting Methods Summary Last time, we considered elections for Math Club President from among four candidates: Alisha (A), Boris (B), Carmen (C), and Dave (D). All 37 voters submitted their ranking of these four candidates using a linear preference ballot. The results are summarized in the following Preference Schedule: Preference Schedule for the MAS Election Number of voters 14 10 8 4 1 1 st choice A C D B C 2 nd choice B B C D D 3 rd choice C D B C B 4 th choice D A A A A You explored a number of different ways to decide in a fair manner which candidate should win the election. You noted that it seemed unfair for Alisha to win: even though she received more first-place votes than any of the other candidates, all the other votes (more than half) put her into last place. Borda Count Method One method some of you suggested was to assign various points to each of the candidates, based on their placement on the preference list: 1 st place gives 4 points, 2 nd place gives 3 points, etc. In this way, you take into account all the information the voters have provided. This is also called the Borda Count Method 1. Great! Yet, there is a problem with the Borda Count Method. Consider the following election at George Washington Elementary School. Four finalists for the job are Mrs. Amaro, Mr. Burr, Mr. Castro, and Mrs. Dunbar (A, B, C, D for short). These are the summary results of the eleven members of the school board: Preference Schedule for George Washington Elementary Number of voters 6 2 3 1 st choice A B C 2 nd choice B C D 3 rd choice C D B 4 th choice D A A 1 Named after the Frenchman Jean-Charles de Borda (1733-1799). Borda was a military man a cavalry officer and naval captain who wrote on such diverse subjects as mathematics, physics, the design of scientific instruments, and voting theory.
Investigations: 1. Who would be the winner of this election based on the Borda Count Method? 2. Do you think that s a fair choice? Discuss with your fellow students. 3. How many first place votes did Mrs. Amaro get? Out of how many total? 4. Do you want to reconsider the fairness question in this light? We can explicitly state one criterion for fairness in an election: Majority Criterion: If a choice receives a majority of the first-place votes in an election, then that choice should be the winner of the election. It appears that the Borda Count Method may violate the Majority Fairness Criterion, and result in an election result we would consider unfair. Hmmm, perhaps some other method would be better.
Method of Pairwise Comparisons Another voting method some of you explored was based on comparing each of the candidates head-to-head. Preference Schedule for the MAS Election Number of voters 14 10 8 4 1 1 st choice A C D B C 2 nd choice B B C D D 3 rd choice C D B C B 4 th choice D A A A A Let s compare A to B: 1) A is placed ahead of B by 14 voters, 2) B is placed ahead of A by all the remaining 10+8+4+1=23 voters. Therefore, B would win this particular head-to-head comparison of A:B we assign one point to B. Let s keep a tally: Candidate A B C D Head-to-head tally 1+ Investigations: 1. Make a list of all other head-to-head comparisons. (Do you have all of them? How many should there be?) 2. Who would win each of those? Keep a tally in the above table. (What do you suggest to do in case of a tie?) 3. One your tally is complete, who should emerge as the winner? This method is designed to satisfy the following fairness criterion: Condorcet Criterion: If there is a choice that in head-to-head comparisons is preferred by the voters over each of the other choices, then that choice should be the winner of the election
The Mathematics of Voting Instant-Runoff Voting (Plurality-With-Elimination Method) This counting method is also known as the Hare method. Examples include local or municipal election (San Francisco, for instance). Let s consider again our familiar example: Preference Schedule for the MAS Election Number of voters 14 10 8 4 1 1 st choice A C D B C 2 nd choice B B C D D 3 rd choice C D B C B 4 th choice D A A A A Investigations: 1. Among the four candidates, who has the least number of first place votes? 2. How many people ranked that candidate as their first choice? 3. We will eliminate this candidate from the election. But what will happen with the voices of the people who voted for that candidate. Who should their support now go to? Discuss you suggestions with your fellow students.
4. The idea is to continue with this elimination procedure until there s only one candidate left. That person will be declared the winner. In order to keep track of how many votes each of the candidates has as the elimination process proceeds, we want to keep a tally. 1 st round of eliminations: Candidate A B C D 1 st place votes Eliminate candidate Transfer votes Total support Notice that in the next round, we only have three candidates left. Enter their initials. 2 nd round of eliminations: Candidate 2nd place votes Eliminate candidate Transfer votes Total support then there were only two 5. Now you can choose a winner by comparing the total support each of the two remaining candidates has. Who wins? 6. How does that compare to our previous results for who should win this election?
What s wrong with Instant Runoff Voting? The main problem is quite subtle and is illustrated by the next example: Investigation: Three cities, Athens (A), Babylon (B), and Carthage (C), are competing to host the next Summer Olympic Games. The final decision is made by a secret vote of the 29 members of the Executive Council using Instant Runoff Voting. Two days before the vote, a straw poll is taken: Straw Vote Results (Two days before the election) # of voters 7 8 10 4 1 st choice A B C A 2 nd choice B C A C 3 rd choice C A B B 1. Use Instant Runoff Voting to decide who would come out the winner?
2. While results are supposed to remain secret, they are leaked. Since everybody loves a winner, all the people who chose A before C before B (4 people in the last column above) change their votes to C before A before B (ie. they join the 10 people in the third column). Election Results # of voters 7 8 14 1 st choice A B C 2 nd choice B C A 3 rd choice C A B Using the Instant Runoff Election method, who comes out as the winner now? We notice that Instant Runoff Voting violates the The Monotonicity Criterion: If choice X is a winner of an election and, in a reelection, the only changes in the ballots are changes that only favor X, then X should remain a winner of the election. In spite of its flaws, Instant Runoff Voting is used in many real-world situations, usually in elections in which there are a few candidates (typically three to four, rarely more than six). The International Olympic Committee uses the Hare method to choose the hosts for the Olympic Games (details on how Sydney was chosen to host the 2000 Summer Games are contained in one of the projects).
Summary From all the investigations about, we see that the answer to the question Who is the winner of the election? depends as much on the counting as it does on the voting: Voting Method Plurality Borda Count Instant Runoff (Hare) Pairwise comparisons Winner Alisha Boris Dave Carmen Even more, we have seen an illustration of the fact that no single counting method can satisfy all fairness criteria (this is known as Arrow s Impossibility Theorem).