Understanding the Banzhaf Power Index
The Banzhaf power index measures voting power, not voting weight. A voter may own many votes, yet still have limited influence. Another voter may hold fewer votes, yet often decide whether a coalition wins. This calculator studies that difference through critical swings.
Why It Matters
Weighted voting appears in councils, shareholder boards, game theory, physics teams, and resource models. Each player has a weight. A quota defines the minimum total needed to pass a motion. A coalition is winning when its combined weight reaches that quota. A player is critical when removing that player changes a winning coalition into a losing coalition.
How the Calculator Thinks
The tool checks every possible coalition. It finds the winning groups first. Then it removes each member of those groups, one at a time. When removal breaks the quota, the player receives one swing count. These counts show how often that player can change an outcome.
Reading the Results
The raw Banzhaf score is the number of critical swings. The normalized score divides each player's swings by total swings. This makes all shares add to one. The absolute score divides swings by all possible coalitions that include that player. It shows the chance that a player is decisive in a random coalition.
Advanced Use Cases
You can test majority systems, veto players, equal blocs, and hidden dominance. Try changing the quota while keeping the same weights. A higher quota can make large players stronger. A lower quota can make smaller players more useful. Sometimes two players with different weights receive the same power score.
Practical Interpretation
Use this calculator to compare fairness before final voting rules are accepted. It helps detect players who look weak but are often decisive. It also reveals voters who look strong but rarely change outcomes. Results are useful for classroom work, policy design, board analysis, and cooperative models. Export the results when you need records for reports or further study.
Limitations
The full method checks all coalitions, so large games grow quickly. For exact results, keep the player count reasonable. This page limits input size to protect performance and keep output readable.
Always confirm weights and quota before using results for real group decisions.