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.

{A,B}, {A, B, C}, {A, B, D}, {A, B, E}, {A, C, D}, {B, C, D}, {B, C, D, E}, {A, C, D, E}, {A, B, D, E}, {A, B, C, E},  {A, B, C, D}, {A, B, C, D, E}

b) Find all blocking coalitions.

{A, B}, {A, C}, {A, D}, {B, C}, {B, D}, {A, B, C}, {A, B, D}, {A, B, E}, {A, C, D}, {A, C, E}, 
{A, D, E}, {B, C, D}, {B, C, E}, {B, D, E}, {B, C, D, E},{A, C, D, E}, {A, B, D, E},{A, B, C, E},  
{A, B, C, D}, {A, B, C, D, E}

c) Find all minimal winning coalitions.

{A,B}, {B, C, D}, {A, C, D}

d) Find all minimal blocking coalitions.

{A, B}, {A, C}, {A, D}, {B, C}, {B, D}

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

E is a dummy; there are no dictators or members with veto power.

f) Compute the Banzhaf Power Index for this system.

(16, 16, 8, 8, 0)

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

Let’s start with a weight-1 voter, M. He will be a critical voter in a winning coalition if and only if the votes of the other members in the coalition add up to exactly 9. There is only one way to achieve this: the weight-8 voter, and one of the other two weight-1 voters, could join with M. There are such coalitions. Thus, M is a critical voter in 2 winning coalitions; doubling this, we find that his Banzhaf power index is 4.

Now we need to determine the Banzhaf power index of the weight-8 voter, N. She is a critical voter in any winning coalition because, without her, other voters do not have enough combined weight to pass a motion. Also, any winning coalition must have either 10 or 11 weights.
A 10-weight-winning coalition must contain N and two out of the three weight-1 voters. There are:  such coalitions.
The only 11-weight-winning coalition is the one that contains all the voters.
Thus, N is a critical voter in 3 winning coalitions; doubling this, we find that her Banzhaf power index is 6. The Banzhaf power index of this committee is (6, 4, 4, 4).

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

a. Find all minimal winning coalitions.

{A, B}, {A, C}

b. Find all minimal blocking coalitions.

{A}, {B,C}

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

There are no dictators or dummies. A has veto power.
d. Compute the Banzhaf Power Index for this system.
(6, 2, 2)

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.

{A, B}, {A, C}, {B, C, D}, {A, B, C}, {A, B, D},{A, C, D}, {A, B, C, D}

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

No dictators, so dummies, no members with veto power.

c. Compute the Banzhaf Power Index for this system.

(10, 6, 6, 2)

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?

Let’s start with the winning coalition {A, B}. A is a critical voter in this winning coalition since, without A, B can not win alone. We know this because {B} is not one of the winning coalitions.

Similarly, B is also a critical voter since {A} is not a winning coalition. Thus, each voter in this winning coalition is a critical voter or, by definition, this coalition is one of the minimal winning coalitions.

Similarly, {A, C} is also a minimal winning coalition.

However, {A, C, D} is not a minimal winning coalition since D is not a critical voter in this coalition. We know this because, without D, {A, C} can still win. Similarly, {A, B, C} is not minimal since C is not critical; {A, B, D} is not minimal since D is not critical; {A, B, C, D} is not minimal since C and D are not critical; 

Even though {B, C, D} has 3 voters, it is still minimal since each voter is critical.

Conclusion: The minimal winning coalitions are {A, B}, {A, C}, and {B, C, D}.

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

There is no dictator because none of the voters appears in all the minimal winning coalitions. This means for any voter, there is at least one winning coalition in which he is not critical.

c. Find the Banzhaf Power Index for the system.

Let’s start with A. He is critical in {A, B}, {A, C}, {A, C, D}, {A, B, C}, and {A, B, D}.
Thus, his power index is 
B is critical in {A, B}, {B, C, D}, and {A, B, D}. His power index is 
C is critical in {A, C}, {A, C, D}, and {B, C, D}. His power index is 	
D is critical in {B, C, D}. His power index is 

Banzhaf Power Index: (10, 6, 6, 2)

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.  

 is an example of a 3-person voting system with no dummies. Since the only winning coalition contains all the voters, none of them can be a dummy.