Respuesta :
Giving the weighted voting system
[tex]\lbrack11\colon11,6,2,1\rbrack[/tex]We are asked to find the Shapley Shubik power distribution of
To solve the question, we would find the number of sequential coalitions.
Since the number of weights is 4, therefore;
[tex]\begin{gathered} Sequential\text{ coalitions = 4!} \\ =4\times3\times2\times1 \\ =24 \end{gathered}[/tex]This means there would be 24 coalitions, this would be difficult to list out so we would have to make a few assumptions.
From the question, we have four voters. Let each voter be the following respectively.
[tex]\sigma_1,\sigma_2,\sigma_3,\sigma_4[/tex]The Shapley shubik power index is given by the formula below
[tex]\begin{gathered} \sigma_i=\frac{SS_i}{\text{Total number of sequential coalitions}} \\ \text{where }SS_i\text{ is the number of sequential coalitions where each player is pivotal} \end{gathered}[/tex]Now, let's make the assumption.
1) The sum of the last three voters weight will not reach the quota 11
2) The quota is only reached when any of the last three voters' weights is added to the first voter's weight.
Therefore, the first voter's weight is pivotal in each sequential coalition.
From this assumption, we can have
[tex]\begin{gathered} SS_1=24 \\ SS_2=0 \\ SS_3=0 \\ SS_4=0 \end{gathered}[/tex]Therefore the power index for each voter would be;
[tex]\begin{gathered} \sigma_1=\frac{24}{24}=1 \\ \sigma_2=\frac{0}{24}=0 \\ \sigma_3=\frac{0}{24}=0 \\ \sigma_4=\frac{0}{24}=0 \end{gathered}[/tex]Answer A
[tex]\sigma_1=1,\sigma_2=0,\sigma_3=0,\sigma_4=0_{}[/tex]Otras preguntas
