Calculate the Banzhaf power index for a weighted voting system: enter the quota and player weights to see each player’s swings and share of power.

Count every coalition for 1โ€“18 players. A coalition wins when its weight meets or exceeds the quota.

Positive decimal, up to 6 decimal places; maximum 1 billion.

Separate 1โ€“18 nonnegative weights with commas or semicolons. Use a decimal point, no thousands separators; up to 6 decimal places.

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Banzhaf Index Formula

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

BI = (Sแตข / Sโ‚œโ‚’โ‚œโ‚โ‚—) ร— 100

Variables:

  • BI is the (normalized) Banzhaf Index (%)
  • Si is the playerโ€™s swing count (the number of winning coalitions in which the player is critical)
  • Stotal is the total swing count (the sum of swing counts over all players)

To calculate the normalized Banzhaf Index, divide the playerโ€™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 playerโ€™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 playerโ€™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.

  1. List all possible coalitions of players (subsets of players).
  2. For each coalition, determine whether it is winning (meets or exceeds the quota/threshold).
  3. For each player, count how many winning coalitions become losing if that player is removed. This is the playerโ€™s swing count (Si).
  4. Add all playersโ€™ swing counts to get the total swing count Stotal.
  5. Use the formula BI = (Si / Stotal) * 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.

Si: 5 (playerโ€™s swing count / critical occurrences)

Stotal: 20 (total swing count across all players)

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