Question
In the weighted voting system below, the weights represent voters A, B, C, and so on, in that order. Answer parts (a) through (d). \( [11: 1,2,6,7] \) C has 6 vote(s). D has 7 vote(s). c. Are there any dictators? Yes No 2 d. Which voter(s), if any, have veto power? O. A only B. C and D
Ask by Wheeler Wyatt. in the United States
Feb 13,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
(c) No.
(d) C and D have veto power.
Solution
We are given the weighted voting game [11: 1, 2, 6, 7] for voters A, B, C, and D, respectively, and we wish to answer parts (c) and (d).
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(c) Are there any dictators?
A dictator in a voting game is a voter who can, by themselves, always produce a winning coalition—that is, their vote alone is enough to reach or exceed the quota. In this game the quota is 11, and no single voter’s weight meets or exceeds 11:
• A has 1 vote.
• B has 2 votes.
• C has 6 votes.
• D has 7 votes.
Since no single voter has 11 or more votes, none of them is a dictator.
Answer to (c): No.
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(d) Which voter(s), if any, have veto power?
A voter has veto power if every winning coalition must include that voter. Equivalently, if you remove that voter, no coalition of the remaining voters can reach the quota.
Let’s check each voter:
• Voter A (1 vote): If A is removed, the remaining voters B (2), C (6), and D (7) sum to 15, and winning coalitions can be formed without A (for example, C + D = 6 + 7 = 13 ≥ 11). Thus, A does not have veto power.
• Voter B (2 votes): If B is removed, the remaining voters A (1), C (6), and D (7) total 14, and again there are winning coalitions that do not require B (for example, C + D = 13 ≥ 11). Thus, B does not have veto power.
• Voter C (6 votes): If C is removed, the remaining voters A, B, and D add up to 1 + 2 + 7 = 10, which is less than the quota of 11. So, without C, no coalition can win. Therefore, every winning coalition must include C, giving C veto power.
• Voter D (7 votes): If D is removed, the remaining voters A, B, and C total 1 + 2 + 6 = 9, which is also below the quota of 11. Thus, every winning coalition must include D, giving D veto power.
Therefore, voters C and D have veto power.
Answer to (d): C and D.
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Final Answers:
(c) No.
(d) C and D.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
In this weighted voting system, a dictator is a voter whose votes alone can achieve a decision. In your case, A has 1 vote, which won't dominate the outcome, while B has 2, C has 6, and D has 7. Since no single voter can achieve a winning majority independently, the answer is "No." For veto power, a voter can prevent a winning coalition from forming. Here, C and D, with 6 and 7 votes respectively, can block any coalition that includes A and B since, together, they can’t reach the quota of 11 without either C or D. Hence, the answer is "C and D."
