Weighted Voting Systems - W.H. Freeman

424 PART III Voting and Social Choice11. A critical voter in a coalition is a voter(a) who has the most votes.(b) who has fewer votes than the number of extravotes of the coalition.(c) whose defection changes the coalition from awinning coalition to a losing coalition.12. The weight of a voter is(a) the number of votes assigned to the voter.(b) the number of times the voter is pivotal.(c) the number of times the voter is part of awinning coalition.13. How large is the number 8! (eight factorial)?(a) More than a million(b) Less than a million but more than 10,000(c) Less than 10,00014. In how many different ways can six voters beordered from first to last?(a) 30(b) 64(c) More than 50015. In how many ways can six voters respond to a“yes–no” question?(a) 12(b) 36(c) 6416. What number is missing from this row ofPascal’s triangle?(a) 20(b) 25(c) 3017. A pivotal voter1 6 15 ___ 15 6 1(a) casts the deciding vote in a permutation ifeach voter votes in turn.(b) is a dictator.(c) can defeat any motion by voting “no.”18. A voter with veto power(a) always wins.(b) has the greatest voting weight.(c) is pivotal in every permutation where he orshe is the last voter.19. C 9 12 (a) C 312(b) 220(c) Both of the above20. In the weighted voting system [7 : 3, 2, 2, 2]the number of voting combinations with exactlythe quota to win is(a) 2(b) 3(c) 4EXERCISES ■ Challenge ◆ DiscussionHow Weighted Voting Works◆ 1. In the United States Senate, each of the 100senators has one vote, and the vice president ofthe United States can vote also if it is necessary tobreak a tie.(a) A simple majority is needed to pass a bill.What constitutes a winning coalition, and whatconstitutes a blocking coalition?(b) One seat in the Senate is vacant, so that only99 senators can vote. What constitutes a winningcoalition, and what constitutes a blocking coalitionfor passing a bill?(c) To ratify a treaty, a two-thirds majority isrequired. What constitutes a winning coalition,and what constitutes a blocking coalition?◆ 2. Is it possible to have a weighted votingsystem in which more votes are required to blocka measure than to pass a measure?3. In the weighted voting system [9 : 5, 4, 3]:(a) Is there a dictator?