`tucoopy.power.banzhaf`¶

Banzhaf power index.¶

This module provides the Banzhaf index for simple (0–1) games, as well as an efficient dynamic-programming variant for integer weighted voting games.

banzhaf_index ¶

banzhaf_index(game, *, normalized=True)

Compute the Banzhaf power index for a simple game.

In a simple game (\(v(S) \in \{0,1\}\)), a player \(i\) is critical (or a swing player) in a coalition \(S\) if:

\[ v(S) = 1 \quad \text{and} \quad v(S \setminus \{i\}) = 0. \]

The (raw) Banzhaf value of player \(i\) is the number of coalitions in which \(i\) is critical. The normalized Banzhaf index divides these counts by the total across all players so that the index sums to 1.

Parameters:

Name	Type	Description	Default
`game`	`GameProtocol`	A simple game.	required
`normalized`	`bool`	If True, return the normalized Banzhaf index. If False, return the raw Banzhaf value (number of swings scaled by \(2^{-(n-1)}\)).	`True`

Returns:

Type	Description
`list[float]`	Banzhaf index for each player.

Raises:

Type	Description
`InvalidParameterError`	If the game is not a valid simple game.

Notes

The Banzhaf index measures criticality across all coalitions, without weighting by coalition size or permutations.
It differs from the Shapley–Shubik index, which is based on pivotality in permutations.

Examples:

>>> bi = banzhaf_index(g)
>>> len(bi) == g.n_players
True

banzhaf_index_weighted_voting ¶

banzhaf_index_weighted_voting(
    weights, quota, *, normalized=True
)

Compute the Banzhaf index for an integer weighted voting game using dynamic programming (without enumerating all \(2^n\) coalitions).

A weighted voting game is defined by weights \(w_1,\ldots,w_n\) and a quota \(q\). A coalition \(S\) is winning if:

\[ \sum_{i \in S} w_i \ge q. \]

A player \(i\) is critical in a coalition if removing \(i\) changes the coalition from winning to losing. This implementation counts how many subsets of the other players have total weight in the pivotal interval \([q-w_i,\, q-1]\).