Shapley Shubik Power Index Calculator

The Shapley-Shubik Power Index measures the power of each player in a voting game based on their potential to be pivotal in decision-making. It is calculated by analyzing all possible permutations of players and determining which player can shift the outcome from a loss to a win, contributing to a fair measure of voting power distribution.

Background

The Shapley-Shubik Power Index, developed in 1954 by Lloyd Shapley and Martin Shubik, is a solution concept in cooperative game theory. It extends the Shapley Value concept to weighted voting games, quantifying the influence each player has in coalitions.

Calculation Steps

1. List all permutations of players.
2. For each permutation, identify the pivotal player where the sum of weights first meets or exceeds the quota.
3. Count how many times each player is pivotal across all permutations.
4. The power index for each player is the ratio of their pivotal count to the total number of permutations.

Example

Consider three players with weights [3, 2, 1] and a quota of 4. The Shapley-Shubik Power Index might look like:

• Player 1: 66.67%
• Player 2: 16.67%
• Player 3: 16.67%

Importance

This index is used in political science, economics, and any scenario involving weighted voting systems. It helps assess the fairness of voting power distribution and can influence the design of voting rules.

Common FAQs

1. What is the Shapley Value?

   • The Shapley Value is a broader concept that allocates payouts to players based on their contribution across all possible coalitions.

2. What is a quota?

   • A quota is the minimum weight required for a coalition to be successful in a voting game.

3. How does this index differ from the Banzhaf Index?

   • The Shapley-Shubik Index considers the order of players in permutations, while the Banzhaf Index looks at all possible coalitions without regard to order.