`tucoopy.power.johnston`¶

Johnston power index.¶

The Johnston index refines Banzhaf criticality by splitting each winning coalition's contribution equally among its critical players.

johnston_index(game)

Compute the Johnston power index for a complete simple game.

In a simple game (\(v(S) \in \{0,1\}\)), a player \(i\) is critical in a winning coalition \(S\) if removing \(i\) makes the coalition losing:

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

The Johnston index assigns to each critical player an equal share of the coalition's criticality. That is, each winning coalition \(S\) contributes:

\[ \frac{1}{|C(S)|} \]

to each \(i \in C(S)\).

The resulting vector is then normalized to sum to 1.

Parameters:

Name	Type	Description	Default
`game`	`Game`	A complete simple game.	required

Returns:

Type	Description
`list[float]`	Johnston index for each player (length `n_players`), normalized to sum to 1.

Raises:

Type	Description
`InvalidGameError`	If the game is not a complete simple game (checked by :func:`require_complete_simple_game`).
`NotSupportedError`	If `n_players` exceeds the supported limit for completeness checks.

Notes

The Johnston index refines the idea of criticality used in the (unnormalized) Banzhaf index.
In the Banzhaf index, each critical occurrence contributes equally (1). In the Johnston index, the contribution of a coalition is divided equally among its critical players.
This index captures how often a player is decisive and how many other players share that decisiveness.

Examples:

>>> ji = johnston_index(g)
>>> sum(ji)
1.0