Consider the weighted voting system $$ [15: 1,3,6,9] $$ a) How many voters are there? b) What is the quota? c) List all winning coalitions. a) There are 4 voters. (Type a whole number.) b) The quota is $$ \square $$ 15 (Type a whole number.) c) Let A, B, C, ... represent the voters in the given order. Identify all the winning coalitions. Select all that apply. A. $$ \{B, D\} $$ B. $$ \{A, B, C, D\} $$ $$ C_{4}\{C\} $$ D. $$ \{B, C\} $$ E. $$ \{A, D\} $$ F. $$ \{A\} $$ G. $$ \{C\} $$ H. $$ \{B\} $$ l. $$ \{A C\} $$ J. $$ \{C, D\} $$ K. $$ \{A, B\} $$ L. $$ \{B, C, D\} $$ M. $$ \{A, C, D\} $$ N. $$ \{A, B, C\} $$ 0. $$ \{A, B, D\} $$

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UpStudy AI · Accepted Answer

We are given the weighted voting game [15: 1, 3, 6, 9]. Label the voters as follows:

• A has weight 1  
• B has weight 3  
• C has weight 6  
• D has weight 9

Step (a): The number of voters is simply the number of weights given.

A) There are 4 voters.

Step (b): The quota is the number given before the colon.

B) The quota is 15.

Step (c): A coalition wins if the sum of the weights of its members is at least 15.

Let’s list all the possible coalitions and see which ones have total weight ≥15.

1. Single-voter coalitions:
 • {A}: 1  
 • {B}: 3  
 • {C}: 6  
 • {D}: 9  
None of these reach 15.

2. Two-voter coalitions:
 • {A, B}: 1+3 = 4  
 • {A, C}: 1+6 = 7  
 • {A, D}: 1+9 = 10  
 • {B, C}: 3+6 = 9  
 • {B, D}: 3+9 = 12  
 • {C, D}: 6+9 = 15  → Winning

3. Three-voter coalitions:
 • {A, B, C}: 1+3+6 = 10  
 • {A, B, D}: 1+3+9 = 13  
 • {A, C, D}: 1+6+9 = 16  → Winning  
 • {B, C, D}: 3+6+9 = 18  → Winning

4. The four-voter coalition:
 • {A, B, C, D}: 1+3+6+9 = 19  → Winning

Now, match these with the options given. The options (with the letters) are:

A. {B, D}    (Weight 3+9 = 12 → Losing)  
B. {A, B, C, D}  (Weight 19 → Winning)  
C. {C}      (Weight 6 → Losing)  
D. {B, C}    (Weight 3+6 = 9 → Losing)  
E. {A, D}    (Weight 1+9 = 10 → Losing)  
F. {A}      (Losing)  
G. {C}      (Losing)  
H. {B}      (Losing)  
I. {A, C}    (Weight 1+6 = 7 → Losing)  
J. {C, D}    (Weight 6+9 = 15 → Winning)  
K. {A, B}    (Weight 1+3 = 4 → Losing)  
L. {B, C, D}  (Weight 3+6+9 = 18 → Winning)  
M. {A, C, D}  (Weight 1+6+9 = 16 → Winning)  
N. {A, B, C}  (Weight 1+3+6 = 10 → Losing)  
O. {A, B, D}  (Weight 1+3+9 = 13 → Losing)

Thus, the winning coalitions are the ones whose total weight is at least 15:  
• {A, B, C, D} (Option B)  
• {C, D} (Option J)  
• {B, C, D} (Option L)  
• {A, C, D} (Option M)

Final Answers:
a) 4  
b) 15  
c) The winning coalitions are B, J, L, and M.
a) There are 4 voters. b) The quota is 15. c) The winning coalitions are: - {A, B, C, D} - {C, D} - {B, C, D} - {A, C, D}

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