A) consider the weighted voting system (q: 8,4,2,1). find the banzhaf power distribution of this weighted voting systwm when q=12.

(q: 8,4,2,1)

q = 12

List all coalitions ( 2 pair)

Those whose total weight is equal to q or more than q will go further in the list of winning coalitions

So, will go further in winning coalition

Coalitions (  3 pair or more)

Those whose total weight is equal to q or more than q will go further in the list of winning coalitions

winning coalitions:

In case of

If Player 1 leaves

So, total weight will be 4

So, Player 1 is critical

If Player 2 leaves

So, total weight will be 8

So, Player 2 is critical

In case of

If Player 1 leaves

So, total weight will be 4+2=6

So, Player 1 is critical

If Player 2 leaves

So, total weight will be 8+2=10

So, Player 2 is critical

If Player 3 leaves

So, total weight will be 8+4=12

So, Player 3 is not critical since total weight is equal to q

In case of

If Player 1 leaves

So, total weight will be 4+1=5

So, Player 1 is critical

If Player 2 leaves

So, total weight will be 8+1=9

So, Player 2 is critical

If Player 4 leaves

So, total weight will be 8+4=12

So, Player 4 is not critical since total weight is equal to q

If Player 1 leaves

So, total weight will be 4+2+1=7

So, Player 1 is critical

If Player 2 leaves

So, total weight will be 8+2+1=11

So, Player 2 is critical

If Player 3 leaves

So, total weight will be 8+4+1=13

So, Player 3 is not critical since total weight is greater than q

If Player 4 leaves

So, total weight will be 8+4+2=14

So, Player 4 is not critical since total weight is greater than q

Player          Times critical           Banzhaf power index

 1                        4                  

 2                       4                

  3                      0                                        0

  4                       0                                       0

                     Sum = 8

.25

step-by-step explanation:

answer: cos [   arcsin (3/5) - arccos (1/2) ] =

step-by-step explanation: × = 3/5 = 0.6--> x = 36.87 and x = 180 - 36.87 = 143.13 deg

cos y = 1/2 --> y = +- 60 deg a. cos (x - y) = cos (143.13 - 60) = cos 83,13  

=

0.12

b. cos (36.87 - 60) = cos (-23.23) = 0.92