 Harder than you might think  There are many examples in history where the results were disputed

 Al Gore vs. George W. Bush  There were two other candidates: Ralph Nader (Green Party) and Pat Buchanan (Reform Party)  The result of the popular vote was:  Gore 48.4%  Bush 47.9%  Nader 2.7%  Buchanan 0.4%

 Even though Al Gore won the popular vote, George W. Bush won the electoral college  The result of Florida was in dispute for several weeks, but eventually Florida’s electoral votes were given to Bush after the Supreme Court ordered a stop to recounts  The official margin of victory for Bush in Florida was 537 votes, out of 5.8 million votes cast

 Three main candidates:  Norm Coleman (R)  Hubert Humphrey (D)  Jesse Ventura (Reform Party)  The results of the vote were:  Ventura (37%)  Coleman (34%)  Humphrey (28%)

 Few people expected the former professional wrestler to win the election  Most of the people who voted for Coleman or Humphrey probably had Ventura as their last choice  That means that Ventura was elected governor even though 63% of the voters would have ranked him last!

 There are many different systems, as we will learn  The most common system used in US elections is the plurality system  The candidate who gets more votes than any other candidate is said to receive a “plurality”  A candidate receives a “majority” if they earn more than half of the total number of votes  Al Gore won a plurality of the popular vote in 2000  Jesse Ventura won a plurality of the vote in 1998  Neither of these candidates won a majority

 In most US elections, voters can only cast a single ballot for the candidate he or she likes the best  However, most voters will have “preference lists”: a ranking of the candidates in order of most preferred to least preferred  For example, many (but not all) of the people who voted for Ralph Nader in 2000 would have had Al Gore as their second choice

 Suppose a class of children is trying to decide what drink to have with their lunch  The choices are milk, soda, and juice  Each child votes for their top choice  The results are:  Milk 6  Soda 5  Juice 4  Milk wins a plurality of the votes

 What if we examine the full voting preferences of the children?  6 have the preference Milk > Soda > Juice  5 have the preference Soda > Juice > Milk  4 have the preference Juice > Soda > Milk  Is the outcome fair? If we choose Milk as the winner of this election, 9 of the 15 students are stuck with their last choice

 We will not allow ties on individual preference lists, though some methods will result in a tie  All candidates must be listed in a specific order  We will sometimes assume that the number of voters is odd to avoid ties (remember we will think about applying these methods to situations where we have thousands or millions of voters)

 We’ll start off simple and only consider the case where we have two candidates  There are only two preferences: A > B and B > A  Voters with preference A > B vote for A  Voters with preference B > A vote for B  The candidate with the most votes wins  This method is called majority rule

 Notice that one of the two candidates will definitely get a majority (they can’t both get less than half of the votes)  Majority rule has three desirable properties  All voters are treated equally  Both candidates are treated equally  Monotone

 If any two voters exchange (marked) ballots before submitting them, the outcome of the election does not change

 If a new election were held and every voter reversed their vote (people who voted for A now vote for B, and vice versa), then the outcome of the election is also reversed

 If a new election were held and a single voter were to change his or her ballot from being a vote for the loser of the previous election to being a vote for the winner of the previous election, and everyone else voted exactly as before, then the outcome of the new election would be the same as the outcome of the previous election.

 Majority rule is not the only way to determine the winner of an election with two candidates  May’s Theorem states that majority rule is the only method for determining the winner of an election with two candidates that treats all voters equally, treats both candidates equally, and is monotone