tucoopy.power.deegan_packel¶
Deegan–Packel power index.¶
The Deegan–Packel index is based on minimal winning coalitions and distributes a unit contribution across members inversely proportional to coalition size.
deegan_packel_index ¶
deegan_packel_index(game)
Compute the Deegan–Packel power index for a complete simple game.
This index is based on the minimal winning coalitions of the game. A coalition \(S\) is minimal winning if:
- \(v(S) = 1\), and
- for every \(i \in S\), \(v(S \setminus \{i\}) = 0\).
Each minimal winning coalition distributes one unit of power equally among its members. That is, for each minimal winning coalition \(S\), every player \(i \in S\) receives:
\[ \frac{1}{|S|}. \]
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] | Deegan–Packel index for each player (length |
Notes
- Like the Holler (Public Good) index, this index considers only minimal winning coalitions.
- The key difference is that the Holler index counts how many minimal winning coalitions contain a player, while the Deegan–Packel index weights each coalition by the inverse of its size.
- This reflects the idea that being essential in a small coalition should count more than being essential in a large one.
Examples:
>>> dpi = deegan_packel_index(g)
>>> sum(dpi)
1.0