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Question

determine all the winning coalitions and find banzhaf power distribution (Show all work)

[16:5,5,11,6,3]

Answer 1

See http://www.people.vcu.edu/~gasmerom/MAT131/lecture2.html for an explanation.

16 is the quota. Players 1, 2, 3, 4, and 5 have 5, 5, 11, 6, and 3 votes, respectively

Note that the sum of the votes is 5+5+11+6+3 = 30, and to have a majority, 16 votes are required. Thus, the quota makes sense.

I provide the winning coalitions and which players are critical in each one below. A player is critical if, without their vote, the coalition would no longer win

For example, consider {P1,P2,P4,P5}; it s a winning coalition, as their votes are 5+5+6+3=19, and 19 >= 16. Without P1, their vote total is 19 - 5 = 14, and 14 < 16. Thus, P1 is critical. Similarly, P2 and P4 are critical. However, as 19 - 3 = 16, and 16 >= 16, P5 is not critical.

There are 16 winning coalitions.

{P3,P4} P3,

P4

{P3,P4,P5} P3,P4

{P2,P3} P2,P3

{P2,P3,P5} P2,P3

{P2,P3,P4} P3

{P2,P3,P4,P5} P3

{P1,P3} P1,P3

{P1,P3,P5} P1,P3

{P1,P3,P4} P3

{P1,P3,P4,P5} P3

{P1,P2,P4} P1,P2,P4

{P1,P2,P4,P5} P1,P2,P4

{P1,P2,P3} P3

{P1,P2,P3,P5} P3

{P1,P2,P3,P4}

{P1,P2,P3,P4,P5}

P3 is critical in 12 coalitions, and each of P1, P2, and P4 are critical in four coalitions. P5 is never critical

12 + 3*4 = 24

Thus, the power distribution is

P1: 4/24 = 1/6

P2: 4/24 = 1/6

P3: 12/24 = 1/2

P4: 4/24 = 1/6

P5: 0

If you prefer, the power distribution is (1/6, 1/6, 1/2, 1/6, 0)