`tucoopy.power.shapley_shubik`¶

Shapley–Shubik power index.¶

For simple games, the Shapley–Shubik index coincides with the Shapley value. This module exposes both a game-level wrapper and a dynamic-programming variant for integer weighted voting games.

shapley_shubik_index ¶

shapley_shubik_index(game)

Compute the Shapley–Shubik power index for a simple (0–1) game.

In a simple game, coalitions are either losing or winning: $v(S) \in \{0,1\}$. The Shapley–Shubik index of a player is the probability (under a uniformly random permutation/order of players) that the player is pivotal, i.e. the player whose entry into the coalition turns it from losing to winning.

Equivalently, for simple games the Shapley–Shubik index coincides with the Shapley value of the game:

\[ \text{SSI}(v) = \phi(v). \]

Parameters:

Name	Type	Description	Default
`game`	`Game`	TU game expected to be a simple game (0–1 valued).	required

Returns:

Type	Description
`list[float]`	Shapley–Shubik index for each player (length `n_players`).

Raises:

Type	Description
`InvalidParameterError`	If `game` is not a valid simple game (checked by `tucoopy.properties.simple_games.validate_simple_game`).

Notes

This function is a thin wrapper around tucoopy.solutions.shapley.shapley_value after validating that the input is a simple game.

Examples:

>>> validate_simple_game(g)
>>> ssi = shapley_shubik_index(g)
>>> len(ssi) == g.n_players
True

shapley_shubik_index_weighted_voting ¶

shapley_shubik_index_weighted_voting(weights, quota)

Compute the Shapley–Shubik power index for an integer weighted voting game.

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

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

The Shapley–Shubik index of player $i$ is the probability (under a uniformly random permutation of players) that player $i$ is pivotal, meaning that the total weight of players before $i$ in the permutation is strictly below the quota, but reaches/exceeds the quota when $i$ is added.

This implementation uses dynamic programming to count, for each player $i$, how many coalitions of size $k$ have total weight in the pivotal interval $[q-w_i,\, q-1]$. Those counts are then weighted by the standard Shapley combinatorial coefficients:

\[ \frac{k!\,(n-k-1)!}{n!}. \]