would an electoral college be fair explain based on solutions below
Find the shapley shubik power distribution.
Determine all the sequential coalitions and find the shapley shubik power distribution:
First you need to understand the notation [10.5:5,5,6,3]
Quota = the number you need to have to reach your goal or to win
Then you need to get the number of permutations of A,B,C and D and then for each permutation, you need to add the votes from left to right to get the pivotal vote.
example:
In permutation: ABCD
A = 5 (A is not the pivotal vote)
A+B =10 (B is not the pivotal vote)
A+B+C = (C is the pivotal vote because it reaches or passes the goal or quota of 10.5)
In permutation: CBDA
C= 6
C+B = 11 (B is the pivotal vote)
In permutation: BCAD
B = 5
B+C = 11(C is the pivotal vote)
and so on.......
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The number of permutations of A, B,C and D is 4!= 4*3*2*1 = 24
So if you get all the pivotal votes for each permutation, you'll get the following:
A as pivotal -> 5
B as pivotal -> 7
C as pivotal -> 10
D as pivotal -> 2
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Total pivotals->24
The Shapley -Shubik Power Index or Distribution (SSPI) for a voter is the number of times the voter was pivotal divided by the total number of permutation
SSPI(A) = 5/24
SSPI(B) = 7/24
SSPI(C) = 10/24 = 5/12
SSPI(D) = 2/24 = 1/12
16 is the quota. Players 1, 2, 3, 4, and 5 have 5, 5, 11, 6, and 3 votes, respectively
Note that the sum of the votes is 5+5+11+6+3 = 30, and to have a majority, 16 votes are required. Thus, the quota makes sense.
I provide the winning coalitions and which players are critical in each one below. A player is critical if, without their vote, the coalition would no longer win
For example, consider {P1,P2,P4,P5}; it s a winning coalition, as their votes are 5+5+6+3=19, and 19 >= 16. Without P1, their vote total is 19 - 5 = 14, and 14 < 16. Thus, P1 is critical. Similarly, P2 and P4 are critical. However, as 19 - 3 = 16, and 16 >= 16, P5 is not critical.
There are 16 winning coalitions.
{P3,P4} P3,
P4
{P3,P4,P5} P3,P4
{P2,P3} P2,P3
{P2,P3,P5} P2,P3
{P2,P3,P4} P3
{P2,P3,P4,P5} P3
{P1,P3} P1,P3
{P1,P3,P5} P1,P3
{P1,P3,P4} P3
{P1,P3,P4,P5} P3
{P1,P2,P4} P1,P2,P4
{P1,P2,P4,P5} P1,P2,P4
{P1,P2,P3} P3
{P1,P2,P3,P5} P3
{P1,P2,P3,P4}
{P1,P2,P3,P4,P5}
would an electoral college be fair in this senario based on the info below. Explain detailed!
P3 is critical in 12 coalitions, and each of P1, P2, and P4 are critical in four coalitions. P5 is never critical
12 + 3*4 = 24
Thus, the power distribution is
P1: 4/24 = 1/6
P2: 4/24 = 1/6
P3: 12/24 = 1/2
P4: 4/24 = 1/6
P5: 0
If you prefer, the power distribution is (1/6, 1/6, 1/2, 1/6, 0)
