ASAP Find the shapley shubik power distribution.

Question

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

----------------------------

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}

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)

Answer 1

Quota: 10.5

Player	P1	P2	P3	P4	Total
Votes	5	5	6	3	19

A sequential coalition lists the players in the order in which they joined the coalition. With 4 players there are 4! = 24 sequential coalitions.

A pivotal player is the player in a sequential coalition that changes a coalition from a losing coalition to a winning one (number of votes becomes greater or equal to the quota.) There can only be one pivotal player in any sequential coalition.

Sequential coalition	Pivotal player
<P1, P2, P3, P4>	P3 : 5+5+6 ≥ 10.5
<P1, P2, P4, P3>	P4 : 5+5+3 ≥ 10.5
<P1, P3, P2, P4>	P3 : 5+6 ≥ 10.5
<P1, P3, P4, P2>	P3 : 5+6 ≥ 10.5
<P1, P4, P2, P3>	P2 : 5+3+5 ≥ 10.5
<P1, P4, P3, P2>	P3 : 5+3+6 ≥ 10.5
<P2, P1, P3, P4>	P3 : 5+5+6 ≥ 10.5
<P2, P1, P4, P3>	P4 : 5+5+3 ≥ 10.5
<P2, P3, P1, P4>	P3 : 5+6 ≥ 10.5
<P2, P3, P4, P1>	P3 : 5+6 ≥ 10.5
<P2, P4, P1, P3>	P1 : 5+3+5 ≥ 10.5
<P2, P4, P3, P1>	P3 : 5+3+6 ≥ 10.5
<P3, P1, P2, P4>	P1 : 6+5 ≥ 10.5
<P3, P1, P4, P2>	P1 : 6+5 ≥ 10.5
<P3, P2, P1, P4>	P2 : 6+5 ≥ 10.5
<P3, P2, P4, P1>	P2 : 6+5 ≥ 10.5
<P3, P4, P1, P2>	P1 : 6+3+5 ≥ 10.5
<P3, P4, P2, P1>	P2 : 6+3+5 ≥ 10.5
<P4, P1, P2, P3>	P2 : 3+5+5 ≥ 10.5
<P4, P1, P3, P2>	P3 : 3+5+6 ≥ 10.5
<P4, P2, P1, P3>	P1 : 3+5+5 ≥ 10.5
<P4, P2, P3, P1>	P3 : 3+5+6 ≥ 10.5
<P4, P3, P1, P2>	P1 : 3+6+5 ≥ 10.5
<P4, P3, P2, P1>	P2 : 3+6+5 ≥ 10.5

Player	P1	P2	P3	P4	Total
Times pivotal	6	6	10	2	24

Shapley-Shubik power index (Shapley-Shubik distribution)

P1	P2	P3	P4
6/24 = 1/4	6/24 = 1/4	10/24 = 5/12	2/24 = 1/12