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Voting: Does the Majority Always Rule?

Explore the concept of majority rule in voting and its variations, such as super-majorities. Analyze case examples to understand the outcomes in different voting scenarios.

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Voting: Does the Majority Always Rule?

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  1. Mathematics and Politics 12 Voting: Does the Majority Always Rule?

  2. Majority Rule The simplest type of voting involves only two choices. With majority rule, the choice receiving more than 50% of the vote is the winner. Some properties of majority rule are listed below. • Every vote has the same weight. • There is symmetry between the candidates. • If a vote for the loser were changed to a vote for the winner, the outcome of the election would not be changed.

  3. Example The table shows the 2000 presidential election official results. Describe the outcomes of the popular and electoral votes.

  4. Example (cont) Solution The difference in popular vote totals between Gore and Bush was 50,999,897 – 50,456,002 ≈ 543,895 Gore won the popular vote over Bush by a margin of more than a half million votes. Neither Gore nor Bush won a majority of the popular vote because a small percentages of the total vote that went to Nader and other candidates.

  5. Example (cont) Because he won a majority of the electoral vote, Bush was elected President of the United States.

  6. Variations on Majority Rule In some cases, a candidate or issue must receive more than a majority of the vote to win─such as 60% of the vote, 75% of the vote, or a unanimous vote. In these cases, a super majority is required. • A 60% super majority is required to end a filibuster in the U.S. Senate. • A 2/3 super majority is required by both the House and Senate, along with at least ¾ of the states, to amend the U.S. Constitution.

  7. Example Evaluate the outcome in each of the following cases. a. Of the 100 senators in the U.S. Senate, 59 favor a new bill on campaign finance reform. The other 41 senators are adamantly opposed and start a filibuster. Will the bill pass? b. A criminal conviction in a particular state requires a vote by 3/4 of the jury members. On a nine-member jury, seven jurors vote to convict. Is the defendant convicted?

  8. Example (cont) c. A proposed amendment to the U.S. Constitution has passed both the House and the Senate with more than the required 2/3 super majority. Each state holds a vote on the amendment. The amendment garners a majority vote in 36 of the 50 states. Is the Constitution amended? d. A bill limiting the powers of the President has the support of 73 out of 100 senators and 270 out of 435 members of the House of Representatives. But the President promises to veto the bill if it is passed. Will it become law?

  9. Example (cont) Solution a. The filibuster can be ended only by a vote of 3/5 of the Senate, or 60 out of the 100 senators. The 59 senators in favor of the bill cannot stop the filibuster. The bill will not become law. b. Seven out of nine jurors represents a super majority of 7/9 = 77.8%. This percentage is more than the required 3/4 (75%), so the defendant is convicted.

  10. Example (cont) Solution c. The amendment must be approved by 3/4, or 75%, of the states. But 36 out of 50 states represents only 36/50 = 72%, so the amendment does not become part of the Constitution. d. The bill has the support of 73/100 = 73% of the Senate and 270/435 = 62% of the House. But overriding a presidential veto requires a super majority vote of 2/3 = 66.7% in both the House and the Senate. The 62% support in the House is not enough to override the veto, so the bill will not become law.

  11. Example The table shows the results for the three major candidates in the 1992 U. S. presidential election. Analyze the outcome. Could Bush have won if Perot had not run?

  12. Example (cont) Solution Bill Clinton won a plurality, but there was no majority winner of the popular vote. If Perot had not run, the 19% of the popular votes would have been divided between Clinton and Bush. (Bush needed an additional 12.3% of the popular vote to reach a 50% total 37.7% + 12.3% = 50%). If 12.3/19 ≈ 65% of the Perot voters preferred Bush to Clinton, Bush could have won the popular vote. Note: winning the popular vote does not guarantee winning the electoral vote.

  13. Preference Schedule Tabulating results from ballots requires a special type of table, called a preference schedule, that tells us how many voters chose each particular ranking order among the candidates.

  14. Voting Methods withThree or More Choices Plurality method: The candidate with the most first-place votes wins. Single (top-two) runoff method: The two candidates with the most first-place votes have a runoff. The winner of the runoff is the winner of the election. Sequential runoff method: A series of runoffs is held, eliminating the candidate with the fewest first-place votes at each stage. Runoffs continue until one candidate has a majority of the first-place votes and is declared the winner.

  15. Voting Methods withThree or More Choices Point system (Borda count): Points are awarded according to the rank of each candidate on each ballot (first, second, third, …). The candidate with the most points wins. Method of pairwise comparisons (Condorcet method):The candidate who wins the most pairwise (one-on-one) contests is the winner of the election.

  16. Example Consider the following three-candidate preference schedule for candidate A, B, and C. Can you find a winner through pairwise comparisons? Explain.

  17. Example (cont) Solution Three pairwise comparisons are possible: A versus B, B versus C, and A versus C. Columns 1 and 2 show A ranked ahead of B, but B is ahead of A in column 3. Therefore, A beats B in a one-on-one contest by 26 to 10. In the B versus C contest, B is ranked ahead of C in columns 1 and 3, so B beats C by 24 to 12. Similarly, A is ranked ahead of C only in column 1, so C beats A by 22 to 14.

  18. Example (cont) Summarizing, we have the following results for the pairwise comparisons: • A beats B • B beats C • C beats A Because A beats B and B beats C, the first two results suggest that A should also beat C. However, the third result shows that A actually loses to C. The results illustrate what is often called the Condorcet paradox, in which pairwise comparisons do not produce a clear winner.

  19. Example Answer the following questions to be sure you understand the preference schedule.

  20. Example (cont) a. How many voters ranked candidates in the order E, B, D, C, A? b. How many voters had candidate E as their first choice? c. How many voters preferred candidate C over candidate A? Solution a. The order E, B, D, C, A appears in the second-to-last column of the table, and the last entry of that column shows that 4 voters preferred this order.

  21. Example (cont) Solution b. Candidate E appears as first choice in the last two columns, which represent a total of 4 + 2 = 6 voters. c. The first column shows candidate A as first choice and candidate C as fourth choice, which means the 18 voters who chose this order prefer candidate A to candidate C. All the remaining columns show candidate C ranked higher than candidate A; for example, the second column shows 12 voters who put C in fourth place and A in fifth place. Therefore, the total number of voters who prefer C over A is 12 + 10 + 9 + 4 + 2 = 37.

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