Question

. A law firm is run by four partners (A; B; C; D) . Each partner has one vote and decisions are made by majority rule, but in the case of a 2-2 tie, the coalition with A (the senior partner) wins.

Answer 1

Answer:

1) (ABCD) (ABC) (ABD) (ACD) (BCD) (AB) (AC) (AD)

2) A= 0.5000; B = 0.1667; C = 0.1667; D = 0.1667

Step-by-step explanation:

The questions are missing and should be

1) List all the winning coalitions in this voting system and the critical players in each.

2) Find the Banzhaf power index of this law firm.

1) Any combination of three partners, four partners, or a combination with two partners containing A is a winning coalition:

(ABCD) (ABC) (ABD) (ACD) (BCD) (AB) (AC) (AD)

2) The Banzhaf power index for each partner is given by the number of outcomes that said partner has a swing vote, divided by the number of possible swing vote situations.

Partners B, C, D will only have swing votes (change the outcome of the voting) when they are all in the same coalition, or when they are alone in a coalition with partner A:

$B_B=2\\B_C=2\\B_D=2$

As for partner A, they will only be powerless to change the outcome when all of the other partners are in the same coalition (2 out of 8 possible combinations):

$B_A=6$

The Banzhaf power index for each partner is:

$B=\frac{2}{2+2+2+6}=0.1667\\C=\frac{2}{2+2+2+6}=0.1667 \\D=\frac{2}{2+2+2+6}=0.1667 \\A=\frac{6}{2+2+2+6}=0.5000$