Exam Review Suggestions

•	Be able to identify dictators, dummies, veto power, quotas and weights in a weighted voting system.
•	Be able to find winning coalitions and blocking coalitions, minimal winning coalitions, and minimal blocking coalitions in a weighted voting system. 
•	Be able to determine the winning (blocking) coalitions based on knowing the minimal winning (blocking) coalitions.
•	Be able to determine critical voters in coalitions and the Banzhaf Power Index.

Sample Problems

1) Given the weighted voting system [15: 8, 7, 4, 4, 2] (Assume voters are called A, B, C, D, E with weights in decreasing order)

a) Find all winning coalitions.

b) Find all blocking coalitions.

c) Find all minimal winning coalitions.

d) Find all minimal blocking coalitions.

e) Identify any dictators, dummies, or members with veto power.

f) Compute the Banzhaf Power Index for this system.

2. Give the Banzhaf Power Index for the weighted voting system [10:  8, 1, 1, 1]:

3. Given the weighted voting system [5: 3, 2, 2]

a. Find all minimal winning coalitions.

b. Find all minimal blocking coalitions.

c. Identify any dictators, dummies, or members with veto power.

d. Compute the Banzhaf Power Index for this system.

4. In a weighted voting system with 4 voters, the minimal winning coalitions are: 
{A, B}, {A, C}, {B, C, D}

a. Find all the winning coalitions.

b. Identify any dictators, dummies, or members with veto power.

c. Compute the Banzhaf Power Index for this system.

5. In a weighted voting system with 4 voters, all of the winning coalitions are: 
{A, B}, {A, C}, {A, C, D}, {B, C, D}, {A, B, C}, {A, B, D}, and {A, B, C, D}.  

a. What are the minimal winning coalitions?

b. Are there any dictators?  Explain how you know.

c. Find the Banzhaf Power Index for the system.

6. Give an example of a 3-person voting system with no dummies.  Explain how you know that your example fulfills the given conditions.  If no such example exists, explain why.