Near the end of the 2000 U.S. Presidential Election, if either Bush or Gore won the Elector votes of Florida, they would become the 43rd President of the United States. Votes were cast and the final results were (after being recounted several times and the Supreme Court voting 5-4 to stop the recounting)

Candidate	Number of Votes
George W. Bush	2,912,790
Al Gore	2,912,253
Ralph Nader	97,488
Other Candidates	40,579

Notice that no candidate received the majority of the votes (had the voting system that decides which candidate receives the electoral votes been majority rules, there would have been a tie). Rather George W. Bush won the State of Florida and became the 43rd President of the U.S., since he receive more votes than any other candidate. This voting system is know as the Plurality Method. Note that in two candidate elections plurality and majority mean the same thing. However in multi-candidate elections we have the possibility of having spoiler candidates, meaning a candidate that doesn't stand a realistic chance of winning the election, but has an effect on the outcome just by being on the ballot. Most political scientist say that in the State of Florida Ralph Nader was a spoiler candidate. Since, it would make since that more than 50% of those that voted for Nader would prefer Gore to Bush.

Suppose that every voter that voted for Nader realized that he had no chance of winning and instead of voting for Nader they had voted for their second choice candidate. If only 52% of them had prefered Gore to Bush, who would have been the 43rd President?

This brings us to the idea of preference orders, where each voter ranks the candidates according to there personal preferences for them. For those 52%, or more, that prefered Nader to Gore and Gore to Bush, we could write there preference order as N > G > B. Voting systems take as input each voters preference order and output the societal preference order, which is the ranking of the candidates, that according to the voting system being used, best represents the will of the people. We can keep track of every individual voters preference order with a preference schedule, which is just a table that counts how many voters had which preference orders. To show an example of this, suppose that Peter, Sergio, Jason, and Liz are all running for the office of Graduate Student Representative and that there are 27 commitee members which have a vote, each of which must submit there preference order. Votes are in and they are as follows

Number of Voters

Rank	12	7	5	3
1	P	S	J	L
2	S	J	L	J
3	J	L	P	S
4	L	P	S	P

Meaning that 12 voters had the preference order P>S>J>L, and so on.

Problem Set 3 :Under majority rules, what would be the outcome of this election? Under plurality method? What societal preference order would be produced if the plurality method was used? Does plurality best represent the will of the voters? Explain your answer.

Notice that in this election only four preference orders were voted for. How many are actually possible with four candidates? What about with n candidates?

A critic of the plurality method claims that:

Under plurality, it is possible for the winner of an election to ranked first by an arbitrarily small percentage of the electorate and to be ranked last by an arbitrarily large percentage of the electorate.

Construct an argument in favor or opposition of this claim using preference schedules.

End of Problem Set 3

Borda count system for determining the societal preference order for n candidates works as follows:

• Each voter submits there entire preference order.
• For each ballot cast, points are awarded to each candidate as follows
  • A first-place ranking is worth n-1 points.
  • A second-place ranking is worth n-2 points.
  • A third-place ranking is worth n-2 points.
  • ...
  • A last place ranking is worth n-n=0 points.
• The candidates are then ranked in the societal preference order by how many points they have accumulated (any ties are viewed equally in the societal preference order).

Note: For those of you that are familiar with matrix algebra, to get the Borda count for each candidate you can replace the candidate's name in the preference schedule by a 1 and the others with 0 and multiply on the right by some simply (appropiate matrices) to quickly compute the Borda count for each candidate.

A voting system is said to satisfy the majority criterion, if whenever a candidate is ranked first by a majority of the voters that candidate will be ranked first in the resulting societal preference order.

For the next activity we need only adjust the definitions of anonymous, neutral, and monotone for multi-candidate elections.

• A voting system is said to be anonymous if it treats all of the voters equally, meaning that if any two voters traded preference orders, the resulting societal preference order would remain unchanged.
• A voting system is said to be neutral if it treats all of the candidates equally, meaning that if every voter switched the positions of two particular candidates on their individual preference orders, the positions of these two candidates would be switched on the resulting societal preference order as well.
• A voting system is said to be monotone if changes favorable to a candidate on individual preference ballots do not cause that candidate to experience a decrease in rank on the resulting societal preference order.

Problem Set 4 :Who would have won the race for the Graduate Student Representative in the previous example had the Borda count been used? What societal preference order would be produced?

Construct a preference schedule that shows the Borda count system violates the majority criterion. Hint: You can do this in an election with three candidates A,B,C and 25 voters.

Do you think that the majority criterion is reasonable? Should the Borda count be discarded because it does not satisfy the majority criterion?

Which of the properties of anonymity, neutrality, and monotonicity are satisfied by plurality? Which of the properties of anonymity, neutrality, and monotonicity are satisfied by the Borda count? Give an argument to justify each of your answers.

With what you know now, is plurality a quote system? Give an argument to support your answer. Does this contradict the Lemma from the previous section?

End of Problem Set 4

References and Further Reading:
[1]Jonathan K. Hodge and Richard E. Klima.The Mathematics of Voting and Elections: A Hands-On Approach. American Mathematical Society, Providence, R.I., 2005.
[2]CNN.com Election 2000 http://www.cnn.com/ELECTION/2000/results/index.president.html
[3]Wikipedia's entry on Borda count http://www.cnn.com/ELECTION/2000/results/index.president.html