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MATH 110 OL – Quantitative Literacy

EXAM #3

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I. Suppose there are 3 voters A with 12 votes, B with 7 votes, C with 5 votes. Suppose that a simple majority is required to win.

1. What is the quota?

2. How many coalitions are there?

3. Make a coalition table and indicate the winning/losing coalitions

4. Find the Banzhaf index for each voter

5. Make a table listing all the permutations of the voters and the swing voter in each case.

6. Find the Shapley-Shubik index for each voter.

II. 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)

1. 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 52% of them had preferred Gore to Bush, who would have been the 43rd President?

2. 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 55% of them had preferred Gore to Bush, who would have been the 43rd President?

III. Consider the information in the table below:

1. Who is the plurality winner?

2. Who wins a top-two runoff?

3. Who is the Borda winner?

4. Is there a Condorcet winner? If so, whom?

Problem I is worth 40 points

Problem II is worth 20 points

Problem III is worth 40 points

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