Banzhaf Index Calculator - Calculator Academy

Choose a solve mode to find a normalized index or an implied swing count, or enter quota and player weights to count critical occurrences for up to 18 players. A zero total swing count leaves normalized power undefined.

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Banzhaf Index Formula

The following formula is used to calculate the normalized Banzhaf Index as a percentage.

BI = (Sᵢ / Sₜₒₜₐₗ) \times 100

Variables:

BI is the (normalized) Banzhaf Index (%)
S_i is the player’s swing count (the number of winning coalitions in which the player is critical)
S_total is the total swing count (the sum of swing counts over all players)

To calculate the normalized Banzhaf Index, divide the player’s swing count (critical occurrences) by the total swing count across all players. Multiply the result by 100 to convert it to a percentage. This measures a player’s voting power in a weighted voting game.

What is a Banzhaf Index?

The Banzhaf Index, also known as the Banzhaf Power Index, is a measure used in game theory and voting theory to quantify the power of a player in a voting game. It is based on the concept of a “swing” (a critical vote): a player is critical in a coalition if the coalition is winning with the player included, but becomes losing if that player is removed. The normalized Banzhaf index is the player’s number of critical occurrences divided by the total number of critical occurrences across all players. The index is named after John F. Banzhaf III. It measures power in the specified mathematical voting model, not the probability of a real-world voting outcome.

How to Calculate Banzhaf Index?

The following steps outline how to calculate the Banzhaf Index.

List all possible coalitions of players (subsets of players).
For each coalition, determine whether it is winning (meets or exceeds the quota/threshold).
For each player, count how many winning coalitions become losing if that player is removed. This is the player’s swing count (S_i).
Add all players’ swing counts to get the total swing count S_total.
Use the formula BI = (S_i / S_total) * 100 to calculate the normalized Banzhaf Index (%), and check your answer with the calculator above.

Example Problem:

Use the following variables as an example problem to test your knowledge.

S_i: 5 (player’s swing count / critical occurrences)

S_total: 20 (total swing count across all players)

BI = (5 / 20) * 100 = 25%