Banzhaf Power Index Notes & Practice

Summary

This video explains the concept of the Banzhaf Power Index (BPI) in weighted voting systems, demonstrating how to calculate the power each player holds based on their criticality in coalitions. The process includes identifying winning coalitions, determining critical players, and applying a formula to calculate the BPI. The example uses a four-player system with a quota of 28, showing step-by-step how to analyze coalitions and compute the index. Viewers learn how to calculate each player's share of power by counting the times they are critical to winning coalitions, resulting in a clear and practical understanding of the Banzhaf Power Index.

Takeaways

• 😀 The Banzhaf Power Index (BPI) is used to calculate the power each player holds in a weighted voting system based on their criticality in winning coalitions.
• 😀 A weighted voting system is defined by players with specific votes and a quota that the players must meet in a coalition to win.
• 😀 To calculate the Banzhaf Power Index, you first identify all winning coalitions, then underline the players who are critical to those coalitions.
• 😀 A player is considered critical in a coalition if their removal causes the coalition to lose, meaning they are needed to meet the quota.
• 😀 The winning coalitions are those that add up to or exceed the quota, while losing coalitions fall short.
• 😀 For the given system, no two-player coalitions can meet the quota, meaning all three players are critical in every three-player coalition.
• 😀 The four-player coalition has one critical player (Player 1), while Players 2, 3, and 4 are not critical in that coalition.
• 😀 The Banzhaf Power Index is calculated by counting how many times each player is critical across all winning coalitions and dividing that by the total number of critical instances across all players.
• 😀 Player 1 has the highest BPI of 40% because they are critical in four out of ten winning coalitions.
• 😀 Players 2, 3, and 4 each have a BPI of 20% since each player is critical in two out of ten winning coalitions.
• 😀 To practice and reinforce understanding of the Banzhaf Power Index, additional problems are provided for review, which should be solved before comparing answers.

Q & A

What is the Banzhaf Power Index (BPI) in a weighted voting system?

The Banzhaf Power Index (BPI) measures the power or influence of each player in a voting system by calculating how often each player is critical to the outcome of a winning coalition. A player is considered critical if their removal from a coalition causes it to lose its winning status.

What is the first step in calculating the Banzhaf Power Index for a weighted voting system?

The first step is to list all the winning coalitions in the system. A coalition is considered winning if the total vote count of its members meets or exceeds the quota.

Why are two-player coalitions not considered in this example?

In this example, no two-player coalition can meet the quota of 28. Even the two strongest players, with a combined vote count of 27, fall short of the quota, so the process starts with three-player coalitions.

How do you determine if a player is critical in a winning coalition?

A player is critical in a winning coalition if their removal causes the coalition's total vote count to fall below the quota, turning the coalition from winning to losing.

What role does logical reasoning play in identifying critical players?

Logical reasoning can simplify the identification of critical players. For instance, since no two-player coalition can meet the quota, it’s clear that all players in a three-player coalition are critical, as they are the only ones who can form a winning group.

In the given example, how many winning coalitions are identified?

There are four winning coalitions identified in the example: {Player 1, Player 2, Player 3}, {Player 1, Player 3, Player 4}, {Player 1, Player 2, Player 4}, and {Player 1, Player 2, Player 3, Player 4}.

How do you count the total number of critical occurrences for calculating the Banzhaf Power Index?

To calculate the Banzhaf Power Index, count how many times each player is critical in the winning coalitions. Then, sum these occurrences for all players to find the total number of critical player occurrences, which serves as the denominator in the BPI calculation.

What is the significance of the 10 underlines in the example?

The 10 underlines represent the total number of times players are critical in the winning coalitions. This total is used in the denominator when calculating the Banzhaf Power Index for each player.

How is the Banzhaf Power Index calculated for Player 1 in this example?

Player 1's Banzhaf Power Index is calculated as the number of times Player 1 is critical (4 occurrences) divided by the total number of critical occurrences (10). This gives Player 1 a BPI of 0.4 or 40%.

What can we conclude about the relative power of Player 2, Player 3, and Player 4 in this system?

Players 2, 3, and 4 each have a Banzhaf Power Index of 0.2 or 20%, meaning each of these players has the same level of power in the system, but less than Player 1, who has 40% of the power.