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tucoopy.games.apex

Apex (simple voting) games.

An apex game is a weighted voting game with a distinguished apex player \(a\). A coalition \(S\) is winning if it includes the apex and the other members meet the quota:

\[ a \in S \quad \text{and} \quad \sum_{i \in S,\ i \ne a} w_i \ge q. \]
See Also

tucoopy.games.weighted_voting.weighted_voting_game

Examples:

>>> from tucoopy.games.apex import apex_game
>>> g = apex_game(0, [2, 3], quota=3)
>>> g.value(0b11)
1.0

apex_game 

apex_game(
    apex_player,
    weights,
    quota,
    *,
    player_labels=None,
    winning_value=1.0,
    losing_value=0.0,
)

Construct an apex game (simple game).

Let \(a\) be the apex player and let \(w_i\) be the weights. A coalition \(S\) is said to be winning if:

  • \(a \in S\), and
  • \(\sum_{i \in S,\ i \ne a} w_i \ge q\).

The characteristic function is:

\[ v(S) = \begin{cases} w_v, & \text{ if } S \text{ is winning} \\ l_v, & \text{ otherwise} \end{cases} \]

where \(w_v\) is the winning value and \(l_v\) is the losing value.

Parameters:

Name Type Description Default
apex_player int

Index of the apex player.

 required
weights Sequence[float]

Weights for each player (excluding apex).

 required
quota float

Quota required for a coalition to win (excluding apex).

 required
player_labels list of str

Optional labels for players.

 None
winning_value float

Value assigned to winning coalitions (default 1.0).

 1.0
losing_value float

Value assigned to losing coalitions (default 0.0).

 0.0

Returns:

Type Description
Game

Cooperative game instance representing the apex game.

Raises:

Type Description
InvalidParameterError

If fewer than 2 players are provided or apex_player is out of range.

Examples:

>>> from tucoopy.games.apex import apex_game
>>> g = apex_game(0, [2, 3], 3)
>>> g.n_players
2
>>> g.value(1)
0.0
>>> g.value(3)
1.0