Question

6. Consider the weighted voting system [23:8,9,15,8]. Find the Banzhaf power distribution of this weighted voting system. (P1P2,P3) (P1,P2,P4) P1,P3,P4) P2 P3P4) (P1,P2,P3,P4) P1.P2) P1P3) Player Times critical Power index P2.P3) (P2 P4) (P3,P4) P3 7. Cindy, Jamal, Monique, and Ryan are dividing a piece of land using the lone-divider method. The values of the four pieces of land in the eyes of the each player are: Piece 1 35% 20% 25% 15% Piece 2 15% 40% 25% 25% Piece 3 40% 10% 25% 35% Piece 4 1056 30% 25% Monique 25% 8. Jamal, Maggie, and Kendra are dividing an estate consisting of a house, a vacation home, and a small business. Their valuations (in thousands) are shown below. Determine the final allocation Jamal Maggie Kendra $272 $177 $225 House Vacation Home Small Business $306 $194 $252 $166 $278 $300

Answer 1

Voting System is : [23: 8,9,15,8]

The quota is 23 and individual player weights are:

P1	8
P2	9
P3	15
P4	8
Quota	23

Now, we list all combinations and see which combinations are winning ones.

Listing combinations of 2-player combinations, 3-player and 4-player combinations, and then calculating the total weight of the coalition as sum of individual weights of players in that coalition. The winning combinations garnering total weight that meets quota of 23 is shown in bold.

Combinations	Total weight	Quota
{P1,P2}	17	23
{P1,P3}	23	23
{P1,P4}	16	23
{P2,P3}	24	23
{P2,P4}	17	23
{P3,P4}	23	23
{P1,P2,P3}	32	23
{P1,P2,P4}	25	23
{P1,P3,P4}	31	23
{P2,P3,P4}	32	23
{P1,P2,P3,P4}	40	23

The critical player in each winning combination is that player, if who leaves that coalition, the coalition can no longer meet the quota of 23. They are shown in bold. For example, for coalition 1, if any of player P1 or P3 leaves, coalition will no longer be able to meet quota of 23 but for last combination, if any of them leaves, the remaining 3 will still have at least a combined weight of 23.

Winning Coalitions

{P1,P3}

{P2,P3}

{P3,P4}

{P1,P2,P3}

{P1,P2,P4}

{P1,P3,P4}

{P2,P3,P4}

{P1,P2,P3,P4}

Now, we count how many times or in how many Winning combinations each player is critical.

Player	Times Critical	Power Index
P1	2	17%
P2	2	17%
P3	6	50%
P4	2	17%

Total times critical = t = 2+2+6+2 = 12

Power Index for P1 = 2/12 and P3 = 6/12 and likewise.

As per HOMEWORKLIB POLICY, only first full question will be answered. Kindly post others as separate questions.

Please comment if you have any additional questions. I will be happy to clarify. Thanks. :)