Please show all of your work for full credit.
MATH 105: Exam 2
Name:
1. (10 points) For the weighted voting system [7 : 5, 2, 1] which player(s) have veto power?
Why?
2. (10 points) Construct a weighted voting system which can only pass measures by unanimity.
3. (10 points) Construct a weighted voting system where there is a dictator.
4. (25 points) For the weighted voting system [8 : 5, 3, 2, 1, 1]
a) List all winning coalitions.
b) Identify all critical players in each of the coalitions you listed.
c) Find the total critical count of the system.
d) Determine the Banzhaf power indices for each player.
5. (10 points) How many sequential coalitions would there be for the weighted voting system [5 : 4, 3, 2, 1]?
6. (10 points) How many total (not necessarily winning) coalitions would there be for the
weighted voting system [5 : 4, 3, 2, 1]?
7. (25 points) For the weighted voting system [6 : 4, 2, 1]
a) List all sequential coalitions.
b) Identify the pivotal player for each sequential coalition.
c) Find the pivotal counts for each player.
d) Find the Shapley-Shubik power indices for each player.
8. (5 points (bonus)) What comes next in this sequence?
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