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Published on August 27, 2007

Author: Amateur

Source: authorstream.com

§ 1.1 Preference Ballots and Preference Schedules:  § 1.1 Preference Ballots and Preference Schedules Example: :  Example: Starting in the early ‘90s, the Henson production company started to pay the Muppets with stock options rather than a straight salary. Quietly, the Muppets, as a group, gained a controlling interest in Henson productions. In a move that shocked the world, the Muppets decided to elect one of their own as the CEO of the company. Example: :  Example: Suppose the ballots broke down as follows: Ballot 1st Piggy 2nd Kermit 3rd Gonzo 4th Fozzie Ballot 1st Gonzo 2nd Kermit 3rd Fozzie 4th Piggy Ballot 1st Fozzie 2nd Gonzo 3rd Kermit 4th Piggy Ballot 1st Kermit 2nd Fozzie 3rd Gonzo 4th Piggy 21 15 12 7 We could also represent this information with a table: Example: (cont’d):  Example: (cont’d) This kind of ballot, in which the voters rank candidates in order of preference is called a preference ballot. If ties are disallowed then we have a linear ballot. The table we used is an example of a preference schedule for the election. Transitivity and Candidate Elimination:  Transitivity and Candidate Elimination Voter preferences are transitive--that is if a voter prefers candidate A to candidate B and prefers B over candidate C then the voter prefers A to C. This means that if we want to see which candidate someone would vote for in a two person election all we need to check is which candidate is placed higher on the ballot. Transitivity and Candidate Elimination:  Transitivity and Candidate Elimination Now suppose a candidate drops out of the race. In such a case, the relative preferences of a voter are not affected. § 1.2 The Plurality Method:  § 1.2 The Plurality Method Kent: Senator Dole, why should people vote for you instead of President Clinton? Kang/Dole: It makes no difference which one of us you vote for. Either way, your planet is doomed. DOOMED! Kent: Well, a refreshingly frank response there from senator Bob Dole. - The Simpsons, Treehouse of Horror VII Slide8:  The Plurality Method The plurality method says that the candidate (or candidates) with the most first-place votes wins a given election. This method is an extension of the concept of majority rule, which states that in an election between two candidates one with the majority of votes wins. Slide9:  The Majority Criterion If a choice receives a majority of first-place votes in an election, then that choice should be the winner of the election. Slide10:  The Majority Criterion If a choice receives a majority of first-place votes, but does not win then we have a violation of the majority criterion. Does the plurality method satisfy the Majority Criterion? Slide11:  The Majority Criterion If a choice receives a majority of first-place votes, but does not win then we have a violation of the majority criterion. Does the plurality method satisfy the Majority Criterion? Yes! (Since a candidate with a majority of first-place votes would automatically have a plurality as well.) Slide12:  The Plurality Method What’s wrong with the plurality method? Slide13:  Example: Let’s look at the Muppet example again. Slide14:  The Condorcet Criterion If there is a choice that in a head-to-head comparison is preferred by the voters over each of the other choices, then that choice should be the winner of the election. A candidate that wins in every head-to-head comparison against each of the other candidates is called the Condorcet candidate. Slide15:  The Plurality Method What’s wrong with the plurality method? It violates the Condorcet Criterion. Slide16:  Insincere voting occurs when a voter changes his or her true preferences on the ballot in an effort to influence the election against a certain candidate. Insincere Voting ``Don`t blame me - I voted for Kodos.`` - Homer Simpson, Treehouse of Horror VII § 1.3 The Borda Count:  § 1.3 The Borda Count The Borda Count Method:  The Borda Count Method The Idea: Assign points to each ranking on the ballot--the candidate with the highest total wins. This method produces the best compromise candidate. If we have an election with N candidates we will give 1 point for last place, 2 points for second to last, . . . , and N points for first place. The candidate with the highest total number of points is the winner. Slide19:  Example: Let’s look at the Muppet example again. If we tally up the points we find: Piggy gets 21(4) + 15(1) + 12(1) +7(1) = 118 Kermit gets 21(3) + 15(3) + 12(2) + 7(4) = 160 Gonzo gets 21(2) + 15(4) + 12(3) + 7(2) = 152 Fozzie gets 21(1) + 15(2) + 12(4) + 7(3) = 120 Slide20:  The Borda Count Method What’s wrong with this method? Slide21:  Example: The Springfield Republican primary Krusty the Clown, Sideshow Bob, Dracula and Mr. Burns are running in the primary to be Springfield’s congressional representative. Suppose the vote breaks down like this: Slide22:  The Borda Count Method What’s wrong with this method? It violates the Majority Criterion. . . Slide23:  The Borda Count Method What’s wrong with this method? It violates the Majority Criterion. . . . . .and the Condorcet Criterion.

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