: It is mathematically impossible for a democratic voting method to satisfy all of the fairness criteria was proven in 1949.

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1 Chapter 1 Notes from Voting Theory: the mathematics of the intricacies and subtleties of how voting is done and the votes are counted. In the early 20 th century, social scientists and mathematicians working on social choice theory examined the methods of democratic voting and their related advantages and disadvantages. : It is mathematically impossible for a democratic voting method to satisfy all of the fairness criteria was proven in Fairness Criteria (we will study more about these as we go through the chapter): Criterion: if there is a choice that has a majority of the first-place votes, then that choice should be the winner of the election Criterion: If there is a choice that is preferred by the voters over each of the other choices, then that choice should be the winner of the election. Criterion: If choice X is a winner of an election and, in a reelection, all the changes in the ballots are changes favorable only to X, then X should still be a winner of the election. Criterion: If choice X is a winner of an election, and one (or more) of the other choices is disqualified and the ballots recounted, then X should still be a winner of the election. Voting Methods you are Responsible From this Chapter: Plurality Method Borda Count Instant Runoff Voting (Plurality-with-Elimination) Pairwise Comparison (Copeland s Method) Extended Ranking Methods Recursive Ranking Methods Majority Rule: In a democratic election between two candidates, the one with more than half of the votes wins. Plurality Method: The winner is the one with the most first-place votes. Frequently when there are more than 2 candidates, no one candidate receives more than half of the vote, and so the winner, with the most votes, wins by plurality rather than majority. The student organization MAS holds an election for president. There are four candidates: Alisha (A), Boris (B), Carmen (C), and Dave (D). All 37 members of the club vote by means of a ballot indicating his or her first, second, third, and fourth choice. There are 4! = = 24 different ways a voter could rank the four candidates when ties are not allowed. 50

2 It turns out in the MAS election there are only 5 ways that actually occurred in this election: When the ballots are sorted, these are the frequencies of the 5 ballot rankings that occurred. There should be 37. Are there? In summary, we can make a Preference Schedule for the MAS Election. This is the simplest and most compact way to summarize the voting in an election based on preference ballots. Ex. 1 a) How many voters submitted ballots that ranked the candidates in the order B, D, C, A? b) How many voters ranked candidate C in first place? c) Is there a majority candidate? Explain why or why not. d) Is there a plurality candidate? Explain why or why not. 51

3 Ex. 2 The Latin club holds an election to choose its president. There are three candidates: Arsenio (A), Beatrice (B) and Carlos (C). All 11 members voted. Their ballots are listed below: a) Make a preference schedule for this election: b) How many first-place votes are needed for a majority? c) Which candidate has the most first-place votes? Is it a majority or a plurality? Plurality Method: The candidate with the most first-place votes wins. Notice that the only information we use from the ballots are the votes for first place nothing else matters. 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. This is a criteria of a fair and democratic election. The majority criteria is violated only when there IS a candidate with the majority of the vote, but that candidate does not win. If a candidate has plurality, does that mean the candidate also has majority? No. When there are more than 2 candidates, it is possible for a candidate to have plurality, but none of the candidates have more than ½ of all the votes. If a candidate has majority, does that mean the candidate also has plurality? Yes. If the candidate has more than ½ of all the votes, that candidate has more of the votes than any other candidate. If a candidate wins by plurality, can the election have violated the majority criterion? No. Because the majority criterion is only invoked IF there is a candidate that has received more than ½ of the votes. What is wrong with the plurality method? It fails to take into consideration the voters preferences other than first choice It can therefore lead to some very bad election results. It can violate the Condorcet Criterion It can be easily manipulated by insincere voting (next topic) 52

4 Insincere Voting: When a voter or block of voters change the true order of their preferences in an effort to influence the outcome of the election by AGAINST a certain candidate. Often used when the voters actually favor a third party candidate who has no chance of winning. Condorcet Candidate: A candidate who wins in every head-to-head comparison against each of the other candidates is called the Condorcet Candidate. The Condorcet criterion says that when there is a Condorcet candidate, then that candidate should be the winner. When there is no Condorcet candidate, the Condorcet criterion does not apply. Ex. 3 The band must choose which of the five bowl games to march at: Rose Bowl (R), Hula Bowl (H), Cotton Bowl (C), Orange Bowl (O), and Sugar Bowl (S). a) Which Bowl game wins by the plurality method? b) Which Bowl game wins head-to-head comparison Hula Bowl to Rose Bowl? c) Which Bowl game wins head-to-head comparison of Hula Bowl to Cotton Bowl? d) Which Bowl game wins head-to-head comparison of Hula Bowl to Orange Bowl? e) Which Bowl game wins head-to-head comparison of Hula Bowl to Sugar Bowl? f) Is there a Condorcet candidate in this election? g) By the Condorcet Criterion, which Bowl Game should the Band march at? What does this represent in terms of the Band s Preferences? h) Suppose the three band members who voted C, H, S, O, R above are extremely allergic to lots of kinds of flowers and could not go if the Rose Bowl is selected. IF these three band members had done an informal poll ahead of time and discovered that the Cotton Bowl had no chance of winning, how might they manipulate the election against the Rose Bowl? 53

5 Ex 4. An election with 4 candidates (A, B, C, D) and 150 voters is to be decided using the plurality method. After 120 ballots have been recorded, A has 26 votes, B has 18 votes, C has 42 votes, and D has 34 votes. (a) What is the smallest number of remaining 30 votes that B must receive to guarantee a win for B? Explain? (b) What is the smallest number of the remaining 30 votes that D must receive to guarantee a win for D? Explain? Assignment Due 10/17 Read Chapter 1 pp Do #1, 4, 6, 13, 19 on pp Make Test 2 Corrections on a separate sheet by Oct