consider the weighted voting system q : 13,7,3,1. find the banzhaf power distribution of this weighted voting system for the following cases.
(a) when the quota is 16, the banzhaf index for each player is (please enter your answer as a fraction):
p1: p2: p3: p4:
(b) when the quota is 20, the banzhaf index for each player is (please enter your answer as a fraction):
p1: p2: p3: p4:
(c) when the quota is 23, the banzhaf index for each player is (please enter your answer as a fraction):
p1: p2: p3: p4:

Question

Accepted Answer

# Explanation:
## Step1: Understand the Banzahf power index
The Banzahf power index of a player is the number of times the player is a critical voter divided by the total number of times any player is a critical voter. A player is a critical voter in a coalition if removing the player from the coalition causes the coalition to no longer meet the quota.

## Step2: Analyze case (a) with quota $q = 16$
- List all winning coalitions:
    - Coalition $\{P1,P2\}$: $13 + 7=20\geq16$. Critical voters: $P1$ and $P2$ (if $P1$ is removed $7<16$, if $P2$ is removed $13<16$)
    - Coalition$\{P1,P3\}$: $13 + 3=16\geq16$. Critical voters: $P1$ and $P3$ (if $P1$ is removed $3<16$, if $P3$ is removed $13<16$)
    - Coalition$\{P1,P4\}$: $13+1 = 14<16$ (not winning)
    - Coalition$\{P2,P3,P4\}$: $7 + 3+1=11<16$ (not winning)
    - Coalition$\{P1,P2,P3\}$: $13 + 7+3=23\geq16$. Critical voters: $P1$ (if $P1$ is removed $7 + 3=10<16$), $P2$ (if $P2$ is removed $13+3 = 16$, not critical), $P3$ (if $P3$ is removed $13 + 7=20\geq16$, not critical)
    - Coalition$\{P1,P2,P4\}$: $13+7 + 1=21\geq16$. Critical voters: $P1$ (if $P1$ is removed $7+1=8<16$), $P2$ (if $P2$ is removed $13 + 1=14<16$)
    - Coalition$\{P1,P3,P4\}$: $13+3 + 1=17\geq16$. Critical voters: $P1$ (if $P1$ is removed $3+1=4<16$), $P3$ (if $P3$ is removed $13+1=14<16$)
    - Coalition$\{P2,P3,P4\}$: $7 + 3+1=11<16$ (not winning)
    - Coalition$\{P1,P2,P3,P4\}$: $13+7 + 3+1=24\geq16$. Critical voters: $P1$ (if $P1$ is removed $7+3 + 1=11<16$)

- Count the critical - voter appearances:
        - $P1$: appears as critical voter in $\{P1,P2\}$,$\{P1,P3\}$,$\{P1,P2,P3\}$,$\{P1,P2,P4\}$,$\{P1,P3,P4\}$,$\{P1,P2,P3,P4\}$ (6 times)
        - $P2$: appears as critical voter in $\{P1,P2\}$,$\{P1,P2,P4\}$ (2 times)
        - $P3$: appears as critical voter in $\{P1,P3\}$,$\{P1,P3,P4\}$ (2 times)
        - $P4$: 0 times
    - Total number of critical - voter appearances $=6 + 2+2+0=10$
    - Banzahf indices: $P1:\frac{6}{10}=\frac{3}{5}$, $P2:\frac{2}{10}=\frac{1}{5}$, $P3:\frac{2}{10}=\frac{1}{5}$, $P4:0$

## Step3: Analyze case (b) with quota $q = 20$
- List all winning coalitions:
    - Coalition$\{P1,P2\}$: $13 + 7=20\geq20$. Critical voters: $P1$ and $P2$ (if $P1$ is removed $7<20$, if $P2$ is removed $13<20$)
    - Coalition$\{P1,P2,P3\}$: $13 + 7+3=23\geq20$. Critical voters: $P1$ (if $P1$ is removed $7 + 3=10<20$), $P2$ (if $P2$ is removed $13+3 = 16<20$), $P3$ (if $P3$ is removed $13 + 7=20\geq20$, not critical)
    - Coalition$\{P1,P2,P4\}$: $13+7 + 1=21\geq20$. Critical voters: $P1$ (if $P1$ is removed $7+1=8<20$), $P2$ (if $P2$ is removed $13 + 1=14<20$)
    - Coalition$\{P1,P3,P4\}$: $13+3 + 1=17<20$ (not winning)
    - Coalition$\{P2,P3,P4\}$: $7 + 3+1=11<20$ (not winning)
    - Coalition$\{P1,P2,P3,P4\}$: $13+7 + 3+1=24\geq20$. Critical voters: $P1$ (if $P1$ is removed $7+3 + 1=11<20$), $P2$ (if $P2$ is removed $13+3 + 1=17<20$)
    - Coalition$\{P1,P2\}$, $\{P1,P2,P3\}$, $\{P1,P2,P4\}$, $\{P1,P2,P3,P4\}$
    - Count the critical - voter appearances:
        - $P1$: appears as critical voter in $\{P1,P2\}$,$\{P1,P2,P3\}$,$\{P1,P2,P4\}$,$\{P1,P2,P3,P4\}$ (4 times)
        - $P2$: appears as critical voter in $\{P1,P2\}$,$\{P1,P2,P3\}$,$\{P1,P2,P4\}$,$\{P1,P2,P3,P4\}$ (4 times)
        - $P3$: 0 times (except in non - winning coalitions)
        - $P4$: 0 times
    - Total number of critical - voter appearances $=4 + 4+0+0=8$
    - Banzahf indices: $P1:\frac{4}{8}=\frac{1}{2}$, $P2:\frac{4}{8}=\frac{1}{2}$, $P3:0$, $P4:0$

## Step4: Analyze case (c) with quota $q = 23$
- List all winning coalitions:
    - Coalition$\{P1,P2,P3\}$: $13 + 7+3=23\geq23$. Critical voters: $P1$ (if $P1$ is removed $7 + 3=10<23$), $P2$ (if $P2$ is removed $13+3 = 16<23$), $P3$ (if $P3$ is removed $13 + 7=20<23$)
    - Coalition$\{P1,P2,P3,P4\}$: $13+7 + 3+1=24\geq23$. Critical voters: $P1$ (if $P1$ is removed $7+3 + 1=11<23$), $P2$ (if $P2$ is removed $13+3 + 1=17<23$), $P3$ (if $P3$ is removed $13 + 7+1=21<23$)
    - Other coalitions ($\{P1,P2\}$: $13 + 7=20<23$, $\{P1,P3\}$: $13+3 = 16<23$, etc.) are not winning
    - Count the critical - voter appearances:
        - $P1$: appears as critical voter in $\{P1,P2,P3\}$ and $\{P1,P2,P3,P4\}$ (2 times)
        - $P2$: appears as critical voter in $\{P1,P2,P3\}$ and $\{P1,P2,P3,P4\}$ (2 times)
        - $P3$: appears as critical voter in $\{P1,P2,P3\}$ and $\{P1,P2,P3,P4\}$ (2 times)
        - $P4$: 0 times
    - Total number of critical - voter appearances $=2 + 2+2+0=6$
    - Banzahf indices: $P1:\frac{2}{6}=\frac{1}{3}$, $P2:\frac{2}{6}=\frac{1}{3}$, $P3:\frac{2}{6}=\frac{1}{3}$, $P4:0$

# Answer:
(a) $P1:\frac{3}{5}$, $P2:\frac{1}{5}$, $P3:\frac{1}{5}$, $P4:0$

(b) $P1:\frac{1}{2}$, $P2:\frac{1}{2}$, $P3:0$, $P4:0$

(c) $P1:\frac{1}{3}$, $P2:\frac{1}{3}$, $P3:\frac{1}{3}$, $P4:0$

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QUESTION IMAGE

Question

Explanation:

Step1: Understand the Banzahf power index

Step2: Analyze case (a) with quota \(q = 16\)

Step3: Analyze case (b) with quota \(q = 20\)

Answer: