11 Weighted Voting Systems

Chapter 11 Exercises

11.1 How Weighted Voting Works

Question 11.31

1. How would you explain to the weight-12 voter in the weighted voting system that he is a dummy?

Any two of the other three voters have enough votes to carry a motion, but the weight-12 voter cannot combine his vote with any one voter to make a winning coalition. Therefore, his vote can never affect a committee decision.

Question 11.32

2. Consider a weighted voting system in which there are five voters who have weights 5, 4, 3, 2, and 1.

What is the least possible quota, and what is the greatest?
If the weight-5 voter has veto power, and no other voter does, what is the quota?
If the weight-1 voter is a dummy, what is the quota?

Question 11.33

3. Which voters, if any, have veto power in the weighted voting system ? Is any voter a dummy?

The voters with weights 5 and 4 each have veto power, and the weight-3 voter is a dummy.

Question 11.34

4. Given a voting system , such that exactly one of the voters has veto power, answer the following questions:

The weight- _____________ voter is the one with veto power.
Find .
Is any voter a dummy?
The voter with veto power wields more power than the others. Is there any difference in the power between the other three voters?

Question 11.35

5. The various weighted voting systems used by the Board of Supervisors of Nassau County, New York, turned out to be the mathematical quagmire described in Spotlight 11.4 (page 480). Before the county’s weighted voting was declared unconstitutional by a federal district court in 1993, it was changed several times. The weights in use since 1958 were as follows:

		*Weights*
Year	Quota
1958	16	9	9	7	3	1	1
1964	58	31	31	21	28	2	2
1970	63	31	31	21	28	2	2
1976	71	35	35	23	32	2	3
1982	65	30	28	15	22	6	7

Here, is the presiding supervisor, always from the community of Hempstead; is the second supervisor from Hempstead; and , , , and are the supervisors from the remaining districts: North Hempstead, Oyster Bay, Glen Cove, and Long Beach.

List the dummy voters in each year.
Suppose that the two Hempstead supervisors always vote together. Now list the dummy voters in each year.

(a) In 1958, , , and were dummies. In 1964, , , and were dummies. There were no dummies in 1970 and after.

(b) In 1958 and 1964, the Hempstead supervisors would be dictators; , , , and would be dummies. Following 1970 there would have been no dummies until 1982, when G was a dummy.

Question 11.36

6. In the weighted voting system , for which whole numbers , not less than 21 and not greater than 41,

does the weight-10 voter not have veto power?
is the weight-10 voter the only one with veto power?
is the weight-3 voter a dummy?

11.2 The Shapley–Shubik Model

Question 11.37

7. Voters use the weighted voting system .

List all permutations in which is pivotal.
List all permutations in which is pivotal.
Calculate the Shapley–Shubik power index of the system.

(a) is pivotal in these permutations: , , , , , , , , , , , and .

(b) is pivotal in these permutations: , , , and .

(c) The Shapley-Shubik index of , which is pivot in 12 permutations, is . Each of the other voters has four pivots and a Shapley-Shubik power index of .

Question 11.38

8. How would the Shapley–Shubik power index in Exercise 7 change if the quota were increased to

Question 11.39

9. Voters , and use the weighted voting system .

Describe the set of permutations in which Voter is pivotal.
What is the number of permutations that you found in part (a)?
Use the answer that you have given in part (b) to determine the Shapley–Shubik power index of the system.

(a) Voter is pivotal in the third or fourth position of any permutation.

(b)

(c) Voter has a Shapley-Shubik power index of . Each of the five weight-2 voters has a Shapley-Shubik power index equal to of the remaining of the power; that is, .

Question 11.40

10. Refer to the permutation of the 2012 presidential election (see Table 11.1 on page 468). The Democratic ticket received 2,990,274 votes in Pennsylvania, to the Republican ticket’s 2,680,434. Which state would be the pivotal voter if, at the last minute, a television ad convinced 150,000 voters to switch from the Democratic ticket to the Republican ticket? Assume that no votes are changed outside of Pennsylvania.

497

Question 11.41

11. was a member of a committee until he resigned, after discovering that with the voting system in use, he was a dummy. Did the Shapley-Shubik power indices of the other committee members change as a result of ’s departure?

11.

No. Before left, he was pivot in no permutations. The Shapley-Shubik power indices of the other participants were their shares of the permutations in which they were pivot. With removed, the shares would remain the same.

Question 11.42

12. If a state uses the District System (described in Spotlight 11.1 on page 465) to choose its electors in a two-candidate presidential election, as Maine and Nebraska do, then some electoral permutations are impossible because the electors corresponding to the senators must not appear in a permutation before all of the electors representing the congressional districts, or after all of them. How would this affect the calculation of the Shapley-Shubik power index?

11.3 The Banzhaf Model

Question 11.43

13. A committee has four members, , , , and . It makes decisions by majority rule.

What is the quota if each member has a voting weight of 1?
List all the winning coalitions in which is critical.

13.

(a) 3

(b) , , and

Question 11.44

14. , , , , , use the weighted voting system . Decide which of the following coalitions are winning and which are losing. Identify the critical voters in the winning coalitions.

Question 11.45

15. For each of the following weighted voting systems, make a list of all winning coalitions. Identify the critical voters in each coalition, and calculate the Banzhaf power index for each voter. Give the voters the names .

15.

In listing winning coalitions, the critical voters are marked with an asterisk.

(a) is dictator; the winning coalitions are and and is the only critical voter in each. Hence has a Banzhaf power index of 2, and has a Banzhaf power index of 0.

(b) This is a majority-rule system: The winning coalitions are all with two voters, and the grand coalition. The grand coalition has no critical voters, and each member of a two-voter coalition is critical. For example, is critical in and , so the Banzhaf power index of is 2. and also have Banzhaf power indices of 2.

(c) The winning coalitions are , , and . In the first two, both voters are critical, but in the grand coalition, only is critical. Hence the Banzhaf power index of is 3, while the Banzhaf power indices of and are both equal to 1.

(d) Any two of , , and can form a winning coalition. cannot form a winning coalition by joining with a single other voter, and so is a dummy. The winning coalitions are: , , , , , , and . Thus, , , and have Banzhaf power indices equal to 4, while the Banzhaf power index of is 0.

(e) can form a winning coalition with any one of the other three voters. Without it takes all three of the other voters to form a winning coalition. Thus, the winning coalitions are , , , , , , , and . The Banzhaf power indices are: , 6; , 2:, , 2; and , 2.

Question 11.46

16. Four voters, , , , and , use the weighted voting system . Make a table showing all winning coalitions in the left column. In the second column, put the number of extra votes. This will enable you to identify the critical voters of the winning coalitions. List the critical voters in a third column, and use this table to determine the Banzhaf power index of each participant.

Question 11.47

17.This exercise is intended for a group of 2 to 4 students. If the quota for the voting system in Exercise 16 increases, you can quickly modify the table you made by reducing the extra votes—ignore a coalition when its extra votes become negative. As the number of extra votes decreases, more of a coalition’s voters will be critical—until the coalition switches from winning to losing.

For example, is a winning coalition with 0 extra votes when the quota . If , it is a losing coalition, so and are no longer credited with critical votes for that coalition. On the other hand, and gain critical votes in and , respectively.

Determine the Banzhaf index for this system with the following quotas:

17.

For each quota, the winning coalitions, with the number of extra votes, are listed. Asterisks identify the critical voters. The notation for Banzhaf power indices is such that (5,3,3,1) designates 5 for , 3 for and , and 1 for .

The abbreviation “e.v.” means “ extra votes.”

(a)

(b)

(c)

(d)

(e)

(f)

(g)

Question 11.48

18. Calculate the following terms²:

498

Question 11.49

19. Calculate the following terms.

19.

(a) 20 (b) 4950 (c) 4950 (d) 126

Question 11.50

20. A committee has 10 members and decides measures by weighted voting. The chairperson, Franklin, has voting weight 4; each of the 9 other members has weight 1, and the quota is 7. Determine the Shapley-Shubik and Banzhaf power indices of each member.

Question 11.51

21. Agnes, Boris, and Carla, weight-1 voters in the committee described in Exercise 20, cede their votes to a fourth weight-1 voter, Essie. Use the following steps to recalculate the power indices.

Use the notation to describe the weighted voting system as it stands after the pact.
Gerry, a weight-1 member of the committee, will be pivot in permutations where she is in the middle, with Essie before her and Franklin after, or the other way around. How many of these permutations are there?
Determine Gerry’s Shapley-Shubik power index, and then the Shapley-Shubik power index of each of the other members.
Describe the winning coalitions that would have Gerry as a critical voter.
Describe the winning coalitions in which Franklin is a critical voter.
Determine the Banzhaf power index for each member of the committee after the pact.

21.

(a)

(b) Three voters must precede Gerry in the permutation. One must be Essie or Franklin, and two must be chosen from the four weight-1 voters other than Gerry, who already has been placed in the middle of the permutation. The number of ways to select these voters is . Once selected, there are ways to order them. The remaining three voters, who will be after Gerry in the permutation, can also be ordered in 6 ways, so the number of permutations in which Gerry is pivotal is .

(c) Gerry’s Shapley-Shubik index is , and each of the other four weight-1 voters has the same index. The sum of the indices of the weight-1 voters is . The remaining of the power is shared equally by Essie and Franklin, so the Shapley-Shubik power index of each is .

(d) Such a coalition would include either Essie or Franklin and two weight-1 voters in addition to Gerry. There are such coalitions. That is Gerry’s Banzhaf power index.

(e) To be a winning coalition, at least 3 votes are needed in addition to Franklin’s. For Franklin to be a critical voter, the coalition must not include more than 6 votes in addition to Franklin’s. There are ways to assemble three weight-1 voters to get 3 more votes to join Franklin. To get 4 more votes, Franklin could be joined by Essie alone or by four weight-1 voters; there are of these coalitions. To get 5 more votes, one needs either all of the weight-1 voters or Essie and four weight-1 voters. There are 6 ways to assemble a coalition of this type. To get 6 more votes, we would need Essie and two weight-1 voters; there are such coalitions.

(f) Franklin’s Banzhaf power index, which is the same as Essie’s, is , and we have noted that Gerry, and each other weight-1 voter, has an index of 12.

Question 11.52

22. Refer to Exercise 5 (page 496) for a brief history of weighted voting in the Nassau County Board of Supervisors. Assume that the two Hempstead supervisors always agree, so that the board is effectively a five-voter system. Determine the Banzhaf power index of this system in each year. You should be able to do this by hand. If you would like to find the index for the full system each year, you may use the Power Index Calculator (see page 502).

Question 11.53

23. If each member of a 12-person jury in which a unanimous decision is required tosses a coin to determine his or her decision, the probability that a given juror will cast a critical vote is . In some states, civil cases are tried before a six-person jury, and the quota for a decision is 5 votes. With such a jury, what is the probability that a given juror will cast a critical vote, if each juror uses a coin toss to determine his or her vote?

23.

A juror is critical in a coalition with four other like- minded jurors. There are four-member coalitions of the five other jurors, so the Banzhaf power index of this juror is 5. The probability is this index, divided by .

Question 11.54

24. A voting system with voters follows majority rule. That is, each voter has weight 1, and the quota is . How many minimal winning coalitions are there? What does this number have to do with the Banzhaf power index?

Question 11.55

25. This exercise is about the District System for allocating votes in the Electoral College (see Spotlight 11.1 on page 465). You will need to use the Power Index Calculator for this (see page 502). Find the total percentage of power that Nebraska has in the Electoral College by adding the percentages of power of the two electors representing the state and the three individual electors who represent congressional districts, according to the Banzhaf power index. Next, consider what would happen if Nebraska changed its law so that all of its electors would be committed to vote for the ticket that won the statewide contest: There would be 53 participants in the Electoral College, and Nebraska would have a voting weight of 5. Would this change result in an increase in Nebraska’s percentage of Banzhaf power, or a decrease?

25.

There would be no significant difference in Nebraska’s share of the voting power in the Electoral College.

11.4 Voting Systems—Without Weights

Question 11.56

26. Consider a four-person voting system with voters , , , and . The winning coalitions are , , , , and .

List the minimal winning coalitions.
A minimal blocking coalition is a set of voters that is voting against a measure, has sufficient voting weight to prevent the measure from passing, and in which the votes of each participant is essential to do so. Show that has veto power and therefore that is a minimal blocking coalition.
Find another minimal blocking coalition.
Determine the Banzhaf power index of each voter.
Find an equivalent weighted voting system. (Hint: If two voters have the same Banzhaf index, give them the same weight.)
Calculate the Shapley-Shubik index of each voter.

Question 11.57

27. In Exercise 26, the term minimal blocking coalition was defined. Must minimal blocking coalitions overlap, as minimal winning coalitions do?

27.

No. For example, consider a jury whose vote must be unanimous to pass a motion. A single juror would be a minimal blocking coalition, and two such minimal blocking coalitions would not intersect.

Question 11.58

28. A five-member committee has a voting system in which the chairperson can pass or block any motion that she supports or opposes, provided that at least one other member is on her side. Show that this voting system is equivalent to the weighted voting system.

499

Question 11.59

29. Find weighted voting systems that are equivalent to the following:

A committee of three faculty members and the dean. To pass a measure, at least two faculty members and the dean must vote “yes.”
A committee of five faculty members, the dean, and the provost. To pass a measure, three faculty, the dean, and the provost must vote “yes.”

29.

(a)

(b)

Question 11.60

30. A four-member faculty committee and a three- member administration committee vote separately on each issue. The measure passes if it receives the support of a majority of each committee. Show that this system is not equivalent to a weighted voting system.

Question 11.61

31. Calculate the Banzhaf index of the voting system in Exercise 30. Who is more powerful according to the Banzhaf model, a faculty member or an administrator?

31.

A faculty member will be a critical voter in a coalition that includes, in addition to , exactly two of the other three faculty members and at least two administrators. There are ways to assemble the other faculty members and ways to assemble the administrators. Thus, each faculty member has a Banzhaf power index of .

An administrator will be critical in a winning coalition that includes at least three faculty members and exactly one of the other two administrators. There are ways to assemble the faculty members and ways to get the other administrator. An administrator, with a Banzhaf power index of , is a bit less powerful on this committee than a faculty member, according to the Banzhaf model.

Question 11.62

32. Determine the Shapley-Shubik index of the system in Exercise 30. Who is more powerful according to the Shapley-Shubik model, a faculty member or an administrator?

Question 11.63

33. Explain why a voting system in which no voter has veto power must have at least three minimal winning coalitions.

33.

If there is only one minimal winning coalition, then every voter who belongs to that coalition has veto power, and every voter who doesn’t belong is a dummy. If there are just two minimal winning coalitions, then they must intersect. Those voters in the intersection will have veto power.

Question 11.64

34. How many distinct (nonequivalent) voting systems with four voters can you find? Systems that have dummies don’t count. The challenge is to find all nine distinct systems and to find—if possible—weighted voting systems equivalent to each.

Question 11.65

35. A corporation has four shareholders and a total of 100 shares. The quota for passing a measure is the votes of shareholders owning 51 or more shares. The number of shares owned by each shareholder is as follows:

Shareholder	Shares Owned
	40
	26
	24
	10

There is also an investor, , who is interested in buying shares but does not own any shares at present. Sales of fractional shares are not permitted.

List the winning coalitions and compute the number of extra votes for each. Make a separate list of the losing coalitions, and compute the number of votes that would be needed to make the coalition winning.
How many shares can sell to without causing any of the winning coalitions listed in part (a) to lose or any of the losing coalitions in part (a) to win?
How many shares can sell to without changing the sets of winning or losing coalitions?
How many shares can sell to without changing the winning coalitions? Because is now a dummy, he must remain a dummy after the trade.

35.

(a) In the following table, all coalitions that include are omitted.

Winning Coalition	Extra Votes	Losing Coalition	Vote Deficit
	15		51
	13		11
	39		25
	25		27
	49		41
	23		1
	9		1
			15
			17

(b) can sell zero shares, because if gains just one share, the losing coalition will become a winning coalition.

(c) can sell 13 shares. All winning coalitions that include have enough votes to support that, and the losing coalitions that include would still be losing after the sale. ( is one share short of meeting the quota, but any transfer of shares between and would not affect its total.)

(d) can sell zero shares. If had just one share, the losing coalition would become a winning coalition.

Question 11.66

36. Which of the following voting systems is equivalent to the voting system in use by the corporation in Exercise 35?

Question 11.67

37. A nine-member committee has a chairperson and eight ordinary members. A motion can pass if and only if it has the support of the chairperson and at least two other members, or if it has the support of all eight ordinary members.

Find an equivalent weighted voting system.
Determine the Banzhaf power index.
Determine the Shapley-Shubik power index.
Compare the results of parts (b) and (c). Do the power indices agree on how power is shared in this committee?

37.

(a)

(b) The Banzhaf power index of the chairperson is 246, and each ordinary member’s index is 8. There are critical votes in all. According to the Banzhaf model, the chair has of the power, and each of the other members has 2.90% of the power.

(c) The chair is pivot in a permutation when located in positions 3 through 8 of a permutation; that is, in 6 of the 9 locations of a permutation. Hence the Shapley-Shubik power index of the chair is . The other members share the remaining of the power, so the Shapley-Shubik power index of each is .

(d) The two models do not agree closely in this case.

Question 11.68

38. The New York City Board of Estimate consists of the mayor, the comptroller, the city council president, and the presidents of each of the five boroughs. It used to employ a voting system in which the city officials each had 2 votes and the borough presidents each had 1; the quota to pass a measure was 6. This voting system was declared unconstitutional by the U.S. Supreme Court in 1989. Although the boroughs had unequal populations, they had equal representation on the board, in violation of the equal protection clause of the 14th amendment to the U.S. Constitution (New York City Board of Estimate v. Morris).

Describe the minimal winning coalitions.
Determine the Banzhaf power index.

Question 11.69

39. A proposed weighted voting system for the New York City Board of Estimate (see Exercise 38) that is based on the populations of the boroughs is . Find a simpler system of weights that yields an equivalent voting system. Do you think this system would satisfy the Supreme Court’s objections?

39.

The borough presidents, taken together, have a total voting weight of 36, and the voting weight of each borough president is more than 1. Thus, the minimal winning coalitions consist of one of the following:

All three city officials (34 extra votes)
Two city officials and one borough president (at least 0.8 and not more than 10.3 extra votes)
One city official and all five borough presidents (0 extra votes)

Therefore, the weighted voting system is equivalent to the system with minimal winning coalitions (a) through (c).

The Supreme Court would reject this system since— like the system that it was intended to replace—all of the borough presidents wield the same amount of voting power. It is interesting to note that the system where the city officials had 2 votes and the quota was 6, the Banzhaf and Shapley-Shubik models both gave each city official about 19% of the power. With the proposed change, the city officials each had about 29% of the power with the Banzhaf model (27% with the Shapley-Shubik model).

500

Question 11.70

40. The voting system in use by the U.N. Security Council is described in Example 21 (page 490).

Show that this voting system is equivalent to the weighted voting system in which each of the 5 permanent members has 7 votes, the 10 nonpermanent members each have 1 vote, and the quota is 39.
Compute the Banzhaf power index for a permanent member and for a nonpermanent member.
The Security Council originally had 5 permanent members and 6 members who served two-year terms. Each permanent member had veto power, and 6 votes were required to resolve an issue. Devise an equivalent weighted voting system and compute its Banzhaf index.
Do you think that the addition of four more nonpermanent members caused the permanent members to lose significant power?

Question 11.71

41. A committee has senior members , , and , and junior members , , and . Each senior member has voting weight 3, and junior members each have voting weight 1. The quota for passing a measure is 7. Find the minimal winning coalitions of the committee and determine the Banzhaf power index for members of each class.

41.

The minimal winning coalitions would be formed by either all three senior members or two senior members and one junior member. The winning coalitions that include as a critical voter would be ; , with being either or and being one of , , (there are of these); , with as before and representing two of , , (there are 6 of these coalitions, too); or (two coalitions). Hence the Banzhaf power index of each senior member is .

A junior member can only be critical in a coalition with two senior members and no other junior members. There are 3 such coalitions—that is the Banzhaf power index of a junior member.

Question 11.72

42. A new junior member joins the system described in Exercise 41. Again, describe the minimal winning coalitions and determine the Banzhaf power index for members of each class. According to the Banzhaf model, does the presence of this new junior member increase or decrease the share of power of each junior member?

Question 11.73

43. Use the Shapley-Shubik model to compare the share of power of a junior member in the systems described in Exercises 41 and 42.

43.

Use , to denote a senior member (either or ) and to represent junior members. In the committee with three junior members, is pivotal in the following permutations:

(12 permutations)
( permutations)
(also 72 permutations)
( permutations)

Thus, the Shapley-Shubik power index of is . By the Shapley-Shubik model, the three senior members hold between them of the power. The remaining 15% is divided between the junior members, so each has a Shapley-Shubik power index of 5%.

With four junior members, would be pivotal in these permutations:

(48 permutations)
( permutations)
( permutations)
(384 permutations)
( permutations)

This comes to a total of 1296 permutations. Dividing by , we obtain the Shapley-Shubik power index of , 25.714%. The three senior members have 77.142% of the power in the Shapley-Shubik model. The remaining 22.858% belongs to the junior members, so each has 5.714% of the voting power in this model. Thus, by including another junior member, the power of each junior member has slightly increased.

Question 11.74

44. Is the committee in Self Check 23 on page 491 equivalent to a weighted voting system? If so, find suitable weights.

Question 11.75

45. An alumni committee consists of 3 rich alumni and 12 recent graduates. To pass a measure, a majority, including at least 2 of the rich alumni, must approve.

Describe the minimal winning coalitions.
Suppose this is a weighted voting system. Give the recent graduates each a weight of 1, and let be the weight of each of the rich alumni. Find the total weight of each minimal winning coalition. For example, the coalition with all three rich alumni and five of the recent graduates would have weight 3r + 5.
Compare the total weight of the losing coalition consisting of all recent graduates and one rich alumna with the total weight of one of the minimal winning coalitions to show that .
Compare the total weight of the largest losing coalition that includes all three rich alumni with the same winning coalition you used in part (c). Does any weight that works here also satisfy the inequality in part (c)?
Is this voting system equivalent to a weighted voting system?

45.

(a) The winning coalitions must have a total of at least 8 members, including at least 2 rich alumni. The minimal winning coalitions will have exactly 8 members: either 3 rich alumni and 5 recent graduates or 2 rich alumni and 6 recent graduates.

(b) The total weight of a coalition including 3 rich alumni and 5 recent graduates would be , and the total weight of a coalition with 2 rich alumni and 6 recent graduates would be .

(c) This losing coalition has total weight of . Its total weight is less than the total weight of the winning coalition with 2 rich almuni and 6 recent graduates. Therefore, . Subtract from both sides to get .

(d) The largest losing coalition with 3 rich alumni would include 4 recent graduates, and its total weight would be ; thus . Subtract from both sides to get ; in other words, is less than 2. But the inequality from part (c) says is greater than 6. No number can be both more than 6 and less than 2, so the two inequalities are inconsistent.

(e) No

Question 11.76

46. Find the Banzhaf power index of each participant in the voting system of Exercise 45.

Question 11.77

47. List the minimal winning coalitions in each of the following three-voter systems. Match each system with an equivalent system listed in Table 11.8 on page 490.

47.

We will name the voters , , .

(a) Minimal winning coalitions: , . This system is known as the “chair veto.”

(b) Minimal winning coalitions: , , . This is the “majority"system.

(d) Minimal winning coalition: . This is the “clique” system.

(e) Minimal winning coalition: . This is the “consensus” system.

Question 11.78

48. In Skills Checks 14–16, we considered a five-voter system in which the minimal winning coalitions are , , and .

Show that this system is not equivalent to any weighted voting system.
Explain why and have equal power and why and have equal power.
Make a list of all of the winning coalitions and use it to determine the Banzhaf power index of each voter.

Chapter Review

Question 11.79

49. A 5-member committee will use weighted voting. The members are assigned voting weights of 12, 7, 4, 3, and 1. They would like to set the quota so that no member has veto power and no member is a dummy voter. Help them by giving them a list of the quotas that meet their specifications. For each quota that you list, find all of the minimal winning coalitions.

49.

Call the voters , , , , and , in decreasing order by weight. The total weight of this system is 27, so the quota must be at least 14. If is not to have veto power, then the remaining voters, whose total weight is 15, must be able to pass a motion. Thus, the only possible quotas that prevent any voter from having veto power are 14 and 15. If the quota is 14, will be a dummy voter. The winning coalitions that include are , which has total weight 15, and combined with some nonempty subset of , which would have total weight at least equal to 16, which is the weight of . Voter would not be critical in any of these coalitions, since they all have at least 1 extra vote.

If the quota is 15, then has 0 extra votes, which makes it a minimal winning coalition. The weight-1 voter is not a dummy. Therefore, 15 is the only quota that meets the specifications. The other minimal winning coalitions are , , and .

Question 11.80

50. A committee has two co-chairs and five other members. The voting weight of each co-chair is 4, the other members each have voting weight 1, and the quota is 7. Find the Banzhaf and Shapley-Shubik power indices for each member.

Question 11.81

51. For a bill to become a federal law, it must be passed by a majority of the 435-member House of Representatives, as well as of the 101-member Senate (we count the vice president as a voting member of the Senate); then it must be approved by the president. If the president vetoes the bill, it can still become law if a two-thirds majority of each house of Congress votes to override the veto. (The vice president does not participate in votes to override a veto, so for veto overrides, the Senate has 100 members.) Describe the minimal winning coalitions in this voting system.

51.

The minimal winning coalitions consist of the following:

The president, the vice president, 50 senators, and 218 members of the House of Representatives
The president, 51 senators, and 218 members of the House of Representatives
67 senators and 290 members of the House of Representatives

501

Question 11.82

52. In the system the weight-3 voter votes “yes” and each of the weight-1 voters uses a coin toss to determine his or her vote. What is the probability that the weight-3 voter has cast a critical vote?

Question 11.83

53. Find the Shapley-Shubik power index for each voter in the system .

53.

Focus on the weight-3 voter, who is singular. This voter is pivot if and only if placed in the third, fourth, or fifth position in a permutation. The permutation has length 7, so the Shapley-Shubik power index for this voter is . The six weight-1 voters share the remaining of the power; thus, each has Shapley-Shubik power index equal to .