consider the weighted voting system 8: 7, 4, 1
find the shapley - shubik power distribution of this weighted voting system.
list the power for each player as a fraction:
$p_1$:
$p_2$:
$p_3$:

Question

Accepted Answer

# Explanation:
## Step1: Calculate total number of permutations
The number of players is $n = 3$. The total number of permutations of $n$ players is $n!=3!=3\times2\times 1=6$.
## Step2: Define the players and quota
Let $P_1$ have weight $w_1 = 7$, $P_2$ have weight $w_2 = 4$, and $P_3$ have weight $w_3 = 1$. The quota $q = 8$.
## Step3: Analyze each permutation for pivotal - player
1. Permutation $(P_1,P_2,P_3)$: The cumulative weights are $7$ (after $P_1$), $7 + 4=11$ (after $P_2$). $P_2$ is the pivotal - player.
2. Permutation $(P_1,P_3,P_2)$: The cumulative weights are $7$ (after $P_1$), $7+1 = 8$ (after $P_3$). $P_3$ is the pivotal - player.
3. Permutation $(P_2,P_1,P_3)$: The cumulative weights are $4$ (after $P_2$), $4 + 7=11$ (after $P_1$). $P_1$ is the pivotal - player.
4. Permutation $(P_2,P_3,P_1)$: The cumulative weights are $4$ (after $P_2$), $4 + 1=5$ (after $P_3$), $5+7 = 12$ (after $P_1$). $P_1$ is the pivotal - player.
5. Permutation $(P_3,P_1,P_2)$: The cumulative weights are $1$ (after $P_3$), $1 + 7=8$ (after $P_1$). $P_1$ is the pivotal - player.
6. Permutation $(P_3,P_2,P_1)$: The cumulative weights are $1$ (after $P_3$), $1 + 4=5$ (after $P_2$), $5+7 = 12$ (after $P_1$). $P_1$ is the pivotal - player.
## Step4: Calculate Shapley - Shubik power
The Shapley - Shubik power of a player is the fraction of permutations in which the player is the pivotal - player.
The number of times $P_1$ is the pivotal - player is $4$. So, the Shapley - Shubik power of $P_1$ is $\frac{4}{6}=\frac{2}{3}$.
The number of times $P_2$ is the pivotal - player is $1$. So, the Shapley - Shubik power of $P_2$ is $\frac{1}{6}$.
The number of times $P_3$ is the pivotal - player is $1$. So, the Shapley - Shubik power of $P_3$ is $\frac{1}{6}$.

# Answer:
$P_1:\frac{2}{3}$
$P_2:\frac{1}{6}$
$P_3:\frac{1}{6}$

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QUESTION IMAGE

Question

Explanation:

Step1: Calculate total number of permutations

Step2: Define the players and quota

Step3: Analyze each permutation for pivotal - player

Step4: Calculate Shapley - Shubik power

Answer: