A weighted voting system is represented by [7: 2, 5, 5] The winning coalitions a

Question

A weighted voting system is represented by [7: 2, 5, 5] The winning coalitions are shown in the following table. Beside each winning coalition list the members of the coalition who are crntical members. Separate multiple member, then type none" critic al members by commas. If no voter in a coalition is a critical Winning Coalitions Critic al Members AB,C AB A,C B,C What is the Banzhaf power index for each voter in the system? Express your answer as an integer or a fraction reduced to simplest form Banzhaf power Banzhaf power index for B Banzhaf power index for C index for A 2

Explanation / Answer

The voting system is {7 : 2, 5, 5}

Where quota q = 7

Part A:

i) Coalation ABC: 2+ 5 +5 = 12

Without A; 12 – 2 = 10 >q (winning)

Without B; 12 – 5 = 7>q (winning)

Without C: 12 – 5 = 7 >q (winning)

Hence, NO Critical Members

ii) Coalation AB: 2+ 5 = 7

Without A; 7 – 2 = 5 <q (loosing)

Without B; 7 – 5 = 2 <q(loosing)

A, B are critical

iii) Coalation AC: 2+ 5 = 7

Without A; 7 – 2 = 5 <q (loosing)

Without C; 7 – 5 = 2 <q (loosing)

A, C are critical

iv) Coalation BC: 5+ 5 = 10

Without B; 10 – 5 = 5 <q (loosing)

Without C; 10 – 5 = 5 <q (loosing)

B, C are critical

Part B: Banzhaf Power index

We have the winning coalitions and the critical players for each case in part A above

Player A has been critical 2 times

Player B has been critical 2 times

Player C has been critical 2 times

Total critical times = 6

Banzhaf power Index for each voter is given as

BPI (A) = 2/6 = 1/3

BPI (B) = 2/6 = 1/3

BPI (C) = 2/6 = 1/3

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