1. What is the Banzhaf Power Index?

The Banzhaf Power Index measures the power of a player in a voting game. It calculates the probability that a player's vote is critical to the outcome, i.e., the number of times a player can change a losing coalition into a winning one.

2. How Does the Calculator Work?

The calculator uses the Banzhaf formula:

Where:

• Critical Votes — Number of times a player's vote is critical
• N — Total number of players in the voting game

Explanation: The index represents the proportion of all possible voting combinations in which a particular player's vote is decisive.

3. Importance of Banzhaf Power Index

Details: The Banzhaf Power Index is crucial for analyzing voting systems, determining the actual power distribution among voters, and ensuring fair representation in weighted voting systems.

4. Using the Calculator

Tips: Enter the number of critical votes and the total number of players. Both values must be valid integers (critical votes ≥ 0, total players > 0).

5. Frequently Asked Questions (FAQ)

Q1: What is a critical vote?
A: A critical vote occurs when a player's vote can change the outcome from losing to winning in a voting coalition.

Q2: How is this different from the Shapley-Shubik index?
A: While both measure voting power, Banzhaf considers all possible coalitions equally, while Shapley-Shubik considers the order of voting.

Q3: What are typical Banzhaf index values?
A: Values range from 0 to 1, where 0 means no power and 1 means complete control over outcomes.

Q4: When is the Banzhaf index most useful?
A: It's particularly useful for analyzing weighted voting systems like corporate shareholder voting or political electoral systems.

Q5: Are there limitations to this index?
A: The index assumes all voting combinations are equally likely and may not account for strategic voting or coalition formation.