Question

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: $p_1$: $p_2$: $p_3$:

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: Analyze each permutation for pivotal players

1. Consider permutations:

• Permutation $(P_1,P_2,P_3)$: The weights are $7,4,1$. Cumulative weights: $7$ (not reaching quota $8$), $7 + 4=11$ (reaching quota). So $P_2$ is the pivotal player.
• Permutation $(P_1,P_3,P_2)$: Cumulative weights: $7$ (not reaching quota), $7+1 = 8$ (reaching quota). So $P_3$ is the pivotal player.
• Permutation $(P_2,P_1,P_3)$: Cumulative weights: $4$ (not reaching quota), $4 + 7=11$ (reaching quota). So $P_1$ is the pivotal player.
• Permutation $(P_2,P_3,P_1)$: Cumulative weights: $4$ (not reaching quota), $4+1 = 5$ (not reaching quota), $5 + 7=12$ (reaching quota). So $P_1$ is the pivotal player.
• Permutation $(P_3,P_1,P_2)$: Cumulative weights: $1$ (not reaching quota), $1+7 = 8$ (reaching quota). So $P_1$ is the pivotal player.
• Permutation $(P_3,P_2,P_1)$: Cumulative weights: $1$ (not reaching quota), $1 + 4=5$ (not reaching quota), $5+7 = 12$ (reaching quota). So $P_1$ is the pivotal player.

1. Count the number of times each player is pivotal:

• Player $P_1$ is pivotal $4$ times.
• Player $P_2$ is pivotal $1$ time.
• Player $P_3$ is pivotal $1$ time.

Step3: Calculate Shapley - Shubik power

The Shapley - Shubik power of a player is the number of times the player is pivotal divided by the total number of permutations.

• For $P_1$: $\frac{4}{6}=\frac{2}{3}$.
• For $P_2$: $\frac{1}{6}$.
• For $P_3$: $\frac{1}{6}$.

Answer:

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