Banzhaf

Banzhaf refers to the Banzhaf power index, a measure of a player’s influence in a voting game. Named after John F. Banzhaf III, who introduced the concept in 1965, the index analyzes how often a player can change a coalition from losing to winning. It is a central tool in cooperative game theory and is widely used in political science and economics to study weighted voting systems.

In a simple voting game, a set of players N has assigned weights and a quota that determines what constitutes a winning coalition. A player i is considered critical in a coalition S if i is in S but removing i from S would cause S to fail to meet the quota. The Banzhaf score β(i) is the number of coalitions in which i is critical. The Banzhaf power index is typically reported as the normalized score B(i) = β(i) / sum_j β(j), so the indices across all players sum to 1.

Example: with three players each holding one vote and a quota of two, there are several coalitions where a player is critical (for example, in {A,B}, A is critical; in {A,C}, A is critical). Each player is critical in two coalitions, yielding β = 2 for each, and a normalized Banzhaf index of 1/3 for all three players.

Applications of the Banzhaf index include analysis of legislative bodies, corporate voting structures, and international organizations with weighted voting. The index contrasts with other power measures, such as the Shapley–Shubik index, and its values depend on the specific weights and quota chosen for the voting game.