Shapley Shubik Power Distribution Calculator

Shapley-Shubik Index Formula:

\[ SSI = \frac{\text{Number of Winning Coalitions}}{\text{Total Coalitions}} \]

Shapley-Shubik Indices:

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1. What is the Shapley-Shubik Power Index?

Definition: The Shapley-Shubik Power Index measures the power of each voter in a weighted voting system by calculating their likelihood of being pivotal.

Purpose: It helps analyze voting systems where voters may have different weights (like shareholders in a company or countries in international organizations).

2. How Does the Calculator Work?

The calculator uses the formula:

\[ SSI = \frac{\text{Number of Winning Coalitions}}{\text{Total Coalitions}} \]

Where:

\( SSI \) — Shapley-Shubik Index (dimensionless)
Number of Winning Coalitions — Coalitions where voter is pivotal (dimensionless)
Total Coalitions — Total possible coalitions (dimensionless)

Explanation: For each possible ordering of voters, we identify the pivotal voter (the one whose addition reaches the quota). The SSI is the fraction of orderings where each voter is pivotal.

3. Importance of Power Distribution

Details: Understanding power distribution helps design fair voting systems and analyze influence in collective decision-making processes.

4. Using the Calculator

Tips: Enter the number of voters, their weights (comma separated), and the quota (minimum votes needed to win). The calculator will show each voter's power index.

5. Frequently Asked Questions (FAQ)

Q1: What does it mean to be a pivotal voter?
A: A voter is pivotal in an ordering if the votes of all preceding voters don't meet the quota, but adding this voter's weight does.

Q2: How is this different from Banzhaf power index?
A: Shapley-Shubik considers orderings of voters, while Banzhaf considers all possible coalitions without regard to order.

Q3: What's a typical application of this index?
A: It's used in political science, corporate governance, and game theory to analyze voting power distribution.

Q4: Does a higher weight always mean more power?
A: Not necessarily. The relationship between weight and power depends on the quota and other voters' weights.

Q5: What's the computational complexity?
A: The calculation grows factorially with the number of voters, so it's practical only for small numbers of voters.