1. § 2.4 - 2.5 The Shapley-Shubik Power Index

2. The Shapley-Shubik Power Index
When discussing power of a coalition in terms of the Banzhaf Index we did not care about the order in which player’s cast their votes.
In other words, yesterday we considered {P1 ,P2} and {P2 ,P1} to be the same coalition.

3. The Shapley-Shubik Power Index
The Shapley-Shubik Power Index concerns itself with sequential coalitions--coalitions in which the order that players join matters.

4. Example: Let us consider the coalition {P1 ,P2,P3}
Example: Let us consider the coalition {P1 ,P2,P3}. How many sequential coalitions contain these players?
We have the following sequential coalitions:
P1 , P2, P3   P1 , P3, P2   P2 , P1, P3   P2 , P1, P3   P3 , P1, P2   P3 , P2, P1 
We can see that there are a total of 6. (In the first sequential coalition what we are saying is that P1 starteed the coaliton, then P2 joined who in turn was followed byP3.)

5. The Shapley-Shubik Power Index
The Shapley-Shubik Power Index concerns itself with sequential coalitions--coalitions in which the order that players join matters.
In general, the number of sequential coalitions with N players is: N ! = (N)(N - 1). . .(2)(1)

6. The Shapley-Shubik Power Index
In each (winning) sequential coalition there is a pivotal player--a player whose joining causes the coalition to change from a losing coalition to a winning coalition.
We will use the concept of the pivotal player to define the Shapley-Shubik Power Index.

7. Example: Let us return to our example of the countries with the funny names. (ie - The [10: 6, 5, 4] example from yesterday.)
We have already seen that the 6 possible sequential coalitions are:  P1 , P2, P3   P1 , P3, P2   P2 , P1, P3   P2 , P1, P3   P3 , P1, P2   P3 , P2, P1 

8. Example: Let us return to our example of the countries with the funny names. (ie - The [10: 6, 5, 4] example from yesterday.)
We have already seen that the 6 possible sequential coalitions are: Sequential Coalition Pivotal Player  P1 , P2, P3  P2  P1 , P3, P2  P3  P2 , P1, P3  P1  P2 , P1, P3  P1  P3 , P1, P2  P1  P3 , P2, P1  P1

9. Example: Let us return to our example of the countries with the funny names. (ie - The [10: 6, 5, 4] example from yesterday.)
We have already seen that the 6 possible coalitions are: Sequential Coalition Pivotal Player  P1 , P2, P3  P2  P1 , P3, P2  P3  P2 , P1, P3  P1  P2 , P1, P3  P1  P3 , P1, P2  P1  P3 , P2, P1  P1

10. Example: Let us return to our example of the countries with the funny names. (ie - The [10: 6, 5, 4] example from yesterday.)
P1 is pivotal four times.
P2 is pivotal one time.
P3 is pivotal one time.

11. Example: Let us return to our example of the countries with the funny names. (ie - The [10: 6, 5, 4] example from yesterday.)
P1 is pivotal four times.
P2 is pivotal one time.
P3 is pivotal one time. Since there are a total of 6 sequential coalitions, under the Shapley-Shubik Power Index we have: P1 : 4/6 P2 : 1/6 P3 : 1/6

12. The Shapley-Shubik Power Index
Finding the Shapley-Shubik Power Index of Player P : Step 1. Make a list of all sequential coalitions containing all N players. Step 2. In each sequential coalition determine the pivotal player. Step 3. Count the number of times P is pivotal--call this number S. The Shapley-Shubik Power Index for the player P is the fraction S/(N !).

13. The Shapley-Shubik Power Index
The list of all of the Shapley-Shubik Power Indices for a given election is the Shapley-Shubik power distribution of the weighted voting system.

14. Example: (Example 2.15) Let us consider a city with a 5 member council that operates under the “strong-mayor” system. (This is a system in which a motion will pass if the mayor and two other council members vote for it or if all four ‘normal’ members vote for it.) It is obvious that the four ‘normal’ council members all have the same level of power--so if we want to characterize the distribution for this town all we need to do is examine the mayor.

15. Example: The European Union (revisited). There are a total of 15
Example: The European Union (revisited). There are a total of 15! = 1,307,674,368,000 possible sequential coalitions (and = 32,767 ‘normal’ coalitions) to consider.
Country
Votes
Banzhaf Power
Shapley-Shubik Power
France, Germany, Italy, UK 10
1849/16,565 ≈ 11.16%
11.67%
Spain 8
1531/16,565 ≈ 9.24%
9.55%
Belgium, Greece, Netherlands, Portugal 5
973/16,565 ≈ 5.87%
5.52%
Austria, Sweden 4
793/16,565 ≈ 4.79%
4.54%
Denmark, Finland, Ireland 3
595/16,565 ≈ 3.59%
3.53%
Luxembourg 2
375/16,565 ≈ 2.26%
2.07%