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# mgf1107notes - MATHEMATICS FOR SOCIAL CHOICE AND DECISION...

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MATHEMATICS FOR SOCIAL CHOICE AND DECISION MAKING CHAPTER ELEVEN-UNWEIGHTED VOTING SYSTEMS A) ELECTIONS WITH ONLY TWO ALTERNATIVES: When there are just two alternatives in an election, the procedure is straightforward-We use “majority rules”. Question: If there are 87 voters, then how many are needed to make a majority? Answer: 44 Question: If there are 100 voters, how many are needed for a majority? Answer: 51. ______________________________________________ B) ELECTIONS WITH THREE OR MORE ALTERNATIVES: This is when it gets more “fun” (i.e., complicated.). There are five main systems we will look at and they are: Plurality Plurality With Runoff The Hare System The Borda Count Sequential Pairwise Voting

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We will examine how each one of these systems works along with their advantages and disadvantages. To do this, let’s use this hypothetical situation: Suppose we poll the class to see whether you prefer the pizza at Domino’s, Pizza Hut, Papa John’s, Godfather’s or Little Caesars. Suppose there are 55 students who vote as follows: Domino’s 18 Papa John’s 12 Pizza Hut 10 Little Caesars 9 Godfather’s 6 1) The Plurality method: Plurality is the method which says take the one with the most votes. So which pizza wins? The winner is Dominoes . 2) Plurality with Runoff : The problem with plurality is that many elections require a majority to favor the winner. There is no majority in this poll. In these situations, we would have a runoff between the top two finishers, namely Dominoes and Papa Johns. In this case, we need more information from the voters, namely how those who voted for the other three ranked the top two (Dominoes and Papa Johns).
Suppose we in fact gathered everybody’s rankings of all 5 chains and the results were summarized in this table: 18 votes 12 votes 10 votes 9 votes 4 vot 1 st Domino’s Papa John’s Pizza Hut Little Caesars Godf 2 nd Little Caesars Godfather’s Papa John’s Pizza Hut Papa John 3 rd Godfather’s Little Caesars Godfather’s Godfather’s Little Caes 4 th Pizza Hut Pizza Hut Little Caesars Papa John’s Pizza 5 th Papa John’s Domino’s Domino’s Domino’s Dom As it turned out, there were only six distinct rankings, and all voters except those of the first column ranked Papa Johns above Dominoes. By the way, each column in this table is called a preference list . The total number of voters preferring Papa Johns to Dominoes is 12+ 10 + 9 +4 +2 = 37 [we could subtract 18 (the Dominoes people) from 55 to also find that 37 people preferred Papa Johns to Dominoes.]

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So, with Plurality with Runoff , Papa Johns wins and those 18 who liked Dominoes better are not very happy. (WHAT IF Plurality with Runoff was decided beforehand as the method used for an election with more than two alternatives and an actual majority occurred? Then the majority alternative wins automatically without a runoff.) Let’s summarize how Plurality with Runoff works: Step One: All participants cast their votes. If there is a
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mgf1107notes - MATHEMATICS FOR SOCIAL CHOICE AND DECISION...

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