1. What is the Shapley-Shubik Index?

The Shapley-Shubik Index is a measure of the power of players in a weighted voting system. It calculates the probability that a player will be pivotal in changing the outcome of a vote when all possible orders of voting are considered equally likely.

2. How Does the Calculator Work?

The calculator uses the Shapley-Shubik formula:

Where:

• \( \Sigma \text{ (pivotal positions)} \) — Total count of pivotal positions for a player
• \( N! \) — Factorial of the total number of players

Explanation: The index represents the fraction of all possible voting sequences in which a particular player is pivotal (their vote changes the outcome from losing to winning).

3. Importance of SSI Calculation

Details: The Shapley-Shubik Index is crucial for analyzing power distribution in voting systems, political science applications, corporate governance, and cooperative game theory to determine the relative influence of different players.

4. Using the Calculator

Tips: Enter the count of pivotal positions for a player and the total number of players. Both values must be non-negative integers with total players ≥ 1.

5. Frequently Asked Questions (FAQ)

Q1: What is a pivotal position?
A: A player is pivotal in a voting sequence if their vote changes the outcome from losing to winning when added to the votes that come before them.

Q2: How is N! calculated?
A: N! (N factorial) is the product of all positive integers up to N. For example, 4! = 4 × 3 × 2 × 1 = 24.

Q3: What does the SSI value represent?
A: The SSI value represents the probability that a player will be pivotal when all voting orders are equally likely, ranging from 0 (no power) to 1 (absolute power).

Q4: When is the Shapley-Shubik Index used?
A: It's used in political science, economics, and game theory to analyze power distribution in voting systems, corporate boards, and international organizations.

Q5: How does SSI differ from other power indices?
A: Unlike the Banzhaf index which counts critical coalitions, SSI considers all possible orders of players and identifies pivotal positions in sequential voting.