Weighted Voting Systems - W.H. Freeman

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CHAPTER 11 Weighted Voting Systems 425(b) Which voters, if any, have veto power?(c) Is any voter a dummy?4. If there are four voters, with voting weights of30, 29, 28, and 13, and if one, and only one, ofthe voters has veto power,(a) which voter has veto power?(b) find the quota.(c) is any voter a dummy?Shapley–Shubik Power Index5. Can a dummy be pivotal in any permutation?Explain why or why not.6. A jury requires a unanimous vote to convict orto acquit. Give a quick way to determine thepivotal voter in any permutation of the jury’smembers.7. For the weighted voting system [51 : 30, 25, 24,21]:(a) List all permutations in which the weight-30voter is pivotal.(b) List all permutations in which the weight-25voter is pivotal.(c) Calculate the Shapley–Shubik index.8. In the voting system [7 : 3, 2, 2, 2, 2, 2]:(a) Describe the set of permutations in which theweight-3 voter is pivotal.(b) How many of these permutations are there?(c) Use the answer that you have given in part (b)to determine the Shapley–Shubik index of thesystem.9. How would the Shapley–Shubik index inExercise 7 change if the quota were increased to(a) 52?(b) 55?(c) 58?10. Refer to the permutation of the 2004presidential election (see Table 11.3). TheRepublican Bush–Cheney ticket carried Nevada(NV), 414,939 to 393,372. Which state would bethe pivot if at the last minute their opponents,Kerry–Edwards, had broadcast an ad thatconvinced 1000 voters to switch from Bush–Cheney to Kerry–Edwards? Assume that no votesare changed outside of Nevada.Banzhaf Power Index11. (a) List the 16 possible combinations of howfour voters, A, B, C, and D, can vote either “yes”(Y) or “no” (N) on an issue.(b) List the 16 subsets of the set {A, B, C, D}.(c) How do the lists in parts (a) and (b)correspond to each other?(d) In how many of the combinations in part (a)is the voteii(i) 4 Y to 0 N?i(ii) 3 Y to 1 N?(iii) 2 Y to 2 N?12. List all winning coalitions in the weightedvoting system, [51 : 30, 25, 24, 21].(a) Identify the critical voters in each coalition.(b) Determine the Banzhaf power index of eachparticipant.(c) List all of the blocking coalitions in which theweight-30 voter is critical, and match each with itsdual winning coalition in which the same voter iscritical.13. The system in Exercise 12 is modified byincreasing the quota to(a) 52.(b) 55.(c) 58.(d) 73.(e) 76.(f) 79.(g) 82.Calculate the Banzhaf index in each case. (Hint:Increasing the quota will reduce the number ofextra votes in each of the original coalitions.When the number of extra votes becomesnegative, the coalition is losing and you can crossit off the list. As the number of extra votesdecreases, a member of a coalition who wasoriginally not a critical voter will become a criticalvoter.)14. Determine the number of extra votes for eachwinning coalition, and calculate the Banzhaf
426 PART III Voting and Social Choiceindex for each of the following weighted votingsystems.(a) [51: 52, 48](b) [3: 2, 2, 1](c) [8: 5, 4, 3](d) [51: 45, 43, 8, 4](e) [51: 45, 43, 6, 6]15. Express the following decimal numbers asbinary:(a) 585(b) 1365(c) 200516. Calculate the following:(a) C 7 3(b) C 5010015(c) C 2(d) C 151317. Calculate the following:(a) C 6 3100(b) C 2100(c) C 98(d) C 5918. The various weighted voting systems used bythe Board of Supervisors of Nassau County, NewYork turned out to be the mathematical quagmiredescribed in Spotlight 11.3. Before the county’sweighted voting was declared unconstitutional by afederal district court in 1993, it was changedseveral times. The weights in use since 1958 wereas follows:WeightsYear Quota H 1 H 2 N B G L1958 16 9 9 7 3 1 11964 58 31 31 21 28 2 21970 63 31 31 21 28 2 21976 71 35 35 23 32 2 31982 65 30 28 15 22 6 7Here H 1 is the presiding supervisor, always fromthe community of Hempstead; H 2 is the secondsupervisor from Hempstead; and N, B, G, and Lare the supervisors from the remaining districts:North Hempstead, Oyster Bay, Glen Cove, andLong Beach.(a) ◆ From 1970 on, more than a simple majoritywas required to pass any measure. Give an argumentin favor of this policy from the viewpoint of asupervisor who would benefit from it, and anargument against the policy from the viewpoint ofa supervisor who would lose some power.(b) In which years were some supervisors dummyvoters?(c) Suppose that the two Hempstead supervisorsalways vote together. In which years are some ofthe supervisors dummy voters?(d) Assume that the two Hempstead supervisorsalways agree, so that the board is in effect a fivevotersystem. Determine the Banzhaf index of thissystem in each year.(e) In 1982, a special supermajority of 72 voteswas needed to pass measures that ordinarilyrequire a two-thirds majority. If the twoHempstead supervisors vote together, what is theBanzhaf index of the resulting five-voter system?(f) ◆ Table 11.6 gives the 1980 census for eachmunicipality, the number of votes assigned to eachsupervisor, and the Banzhaf index for eachsupervisor in 1982. Do you think the votingscheme was fair?Comparing Voting Systems19. Consider a four-person voting system withvoters A, B, C, and D. The winning coalitions are{A, B, C, D}, {A, B, C }, {A, B, D},{A, C, D}, and {A, B}(a) List the minimal winning coalitions.◆ (b) Show that {A} is a minimal blocking coalition.Are there other minimal blocking coalitions?(c) Determine the Banzhaf power index for thisvoting system.(d) Find an equivalent weighted voting system.(e) Calculate the Shapley–Shubik index of thissystem.
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