Enter the number of permutations where each player is pivotal for each coalition, the total number of players, and the rank of the player in a permutation into the calculator to determine the Shapley Shubik Power Index.

Shapley Shubik Power Index Formula

The following formula is used to calculate the Shapley Shubik Power Index. 

SSPI = Σ (P(i, j) * (n - j)!) / n!

Variables:

  • SSPI is the Shapley Shubik Power Index P(i, j) is the number of permutations of all players where player i is pivotal for coalition j
  • n is the total number of players
  • j is the rank of the player in a permutation
  • ! denotes factorial, the product of an integer and all the integers below it

To calculate the Shapley Shubik Power Index, sum the products of the number of permutations where each player is pivotal for each coalition and the factorial of the total number of players minus the rank of the player in a permutation. Divide this sum by the factorial of the total number of players.

What is a Shapley Shubik Power Index?

The Shapley Shubik Power Index is a mathematical method used in game theory and political science to measure the power of a player in a voting game. It considers all possible voting orders and calculates the probability of a player being pivotal, i.e., changing the outcome of a vote. The index assigns each player a power score based on their potential to influence decisions. It was developed by Lloyd Shapley and Martin Shubik to provide a more nuanced understanding of power dynamics in collective decision-making processes.

How to Calculate Shapley Shubik Power Index?

The following steps outline how to calculate the Shapley Shubik Power Index (SSPI).

  1. First, determine the total number of players (n).
  2. Next, determine the number of permutations of all players where each player is pivotal for each coalition (P(i, j)).
  3. Next, calculate the factorial of the total number of players minus the rank of the player in a permutation (n - j)!
  4. Multiply the number of permutations (P(i, j)) by the factorial (n - j)!, for each player and coalition.
  5. Sum up all the products obtained in the previous step.
  6. Finally, divide the sum by the factorial of the total number of players (n!).
  7. The result is the Shapley Shubik Power Index (SSPI).

Example Problem:

Use the following variables as an example problem to test your knowledge.

Total number of players (n) = 5

Number of permutations of all players where player i is pivotal for coalition j (P(i, j)):

P(1, 1) = 2

P(2, 1) = 3

P(3, 1) = 1

P(4, 1) = 4

P(5, 1) = 2

Rank of the player in a permutation (j) = 1

Factorial (n!) = 5!