`tucoopy.solutions.banzhaf`¶

Banzhaf values for TU games.¶

This module provides the (raw) Banzhaf value and common variants as semivalues.

banzhaf_value(game)

Compute the (raw) Banzhaf value for a TU game.

The Banzhaf value measures how often a player is critical across all coalitions, treating all coalitions of the other players as equally likely.

For player \(i\):

\[ \beta_i = \frac{1}{2^{n-1}} \sum_{S \subseteq N \setminus \{i\}} \big( v(S \cup \{i\}) - v(S) \big). \]

Parameters:

Name	Type	Description	Default
`game`	`GameProtocol`	TU game.	required

Returns:

Type	Description
`list[float]`	Banzhaf value vector of length `n_players`.

Notes

Examples:

>>> beta = banzhaf_value(g)
>>> len(beta) == g.n_players
True

normalized_banzhaf_value(game)

Compute the normalized Banzhaf value for a TU game.

The raw Banzhaf values are rescaled so that the resulting allocation sums to the value of the grand coalition:

\[ \hat{\beta}_i = \frac{\beta_i}{\sum_j \beta_j} \, v(N). \]

Parameters:

Name	Type	Description	Default
`game`	`GameProtocol`	TU game.	required

Returns:

Type	Description
`list[float]`	Normalized Banzhaf allocation.

Notes

If all raw Banzhaf values are zero, the function returns a zero vector.
This normalization makes the Banzhaf value comparable to other allocation rules such as the Shapley value.

Examples:

>>> nb = normalized_banzhaf_value(g)
>>> sum(nb) == g.value(g.grand_coalition)
True

weighted_banzhaf_value(game, *, p=0.5)

Weighted Banzhaf value (p-binomial semivalue).

Interprets the marginal contribution as an expectation where each other player joins independently with probability p.

For p=0.5 this equals the (raw) Banzhaf value.

Parameters:

Name	Type	Description	Default
`game`	`GameProtocol`	TU game.	required
`p`	`float`	Inclusion probability in [0, 1].	`0.5`

Returns:

Type	Description
`list[float]`	Allocation vector of length n_players.

Raises:

Type	Description
`InvalidParameterError`	If p is outside [0, 1].