Banzhaf Power Index — Definition, Formula & Examples

The Banzhaf Power Index is a number that measures how much real influence a voter has in a weighted voting system. It counts how often that voter can turn a winning group into a losing one by leaving.

Consider a weighted voting game $[q: w_1, w_2, \dots, w_n]$ . A coalition $S$ is winning if $\sum_{j \in S} w_j \ge q$ . Player $i$ is critical (a swing voter) in a winning coalition $S$ if $S \setminus \{i\}$ is losing. Let $B_i$ be the number of winning coalitions in which player $i$ is critical. The normalized Banzhaf power index of player $i$ is $\beta_i = B_i / \sum_{j=1}^{n} B_j$ . Some texts instead use the non-normalized form $B_i / 2^{n-1}$ .

Key Formula

\beta_i = \frac{B_i}{B_1 + B_2 + \cdots + B_n}

Where:

$\beta_i$ = Normalized Banzhaf power index of player i, a value from 0 to 1
$B_i$ = Number of winning coalitions in which player i is critical
$n$ = Number of players in the weighted voting game

Worked Example

Problem: Three shareholders A, B, and C hold 4, 3, and 2 votes. A motion passes with at least 6 votes, so the game is

[6: 4, 3, 2]

. Find each shareholder's Banzhaf power index.

Step 1: Check every coalition with two or more players. A single player can never win, because the largest weight is 4, which is less than 6. {A, B}, {A, C}, and {A, B, C} reach the quota, so they are winning. {B, C} falls short, so it is losing.

\{A,B\}: 4+3=7, \quad \{A,C\}: 4+2=6, \quad \{B,C\}: 3+2=5<6, \quad \{A,B,C\}: 4+3+2=9

Step 2: In each winning coalition, mark a player as critical if removing that player drops the total below 6. In {A, B} and {A, C}, both members are critical. In {A, B, C}, only A is critical: without A the total is 5, while removing B or C still leaves at least 6.

B_A = 3, \quad B_B = 1, \quad B_C = 1

Step 3: Add the critical counts to get the total, then divide each player's count by that total.

3+1+1=5, \qquad \beta_A = \frac{3}{5}, \quad \beta_B = \frac{1}{5}, \quad \beta_C = \frac{1}{5}

Answer: A has power index

\frac{3}{5}

(60%). B and C each have power index

\frac{1}{5}

(20%). B and C have equal power even though B holds more votes.

Visualization

Why It Matters

Courts and political scientists have used the Banzhaf index to judge whether weighted voting is fair. Examples include county boards in New York, where it was cited in legal challenges, and analyses of the U.S. Electoral College. It is a standard topic in liberal-arts math courses such as "Excursions in Modern Mathematics" and in introductory social choice theory.

Common Mistakes

Mistake: Assuming power is proportional to vote weight. With 4, 3, and 2 votes out of 9, students predict shares of about 44%, 33%, and 22%.

Correction: Power depends on how often a player is critical, not on raw weight. In

[6: 4, 3, 2]

the shares are 60%, 20%, and 20%. Always list the winning coalitions and count the critical players.

Related Terms

Game Theory — Broader field that studies cooperative voting games
Borda Count — Another social choice tool, used to rank candidates
Fair Game — Related idea of fairness in mathematical games
Prisoner's Dilemma — Classic noncooperative game, in contrast to coalition voting
Optimization — Used to design weights that achieve target power