power-bdd 0.0.1

Calculates power indices (Banzhaf/Penrose, Shapley/Shubik) via binary decision diagrams

power_bdd

Creates the ROBDD of a weighted game and calculates power indices according to Banzhaf/Penrose and Shapley/Shubik. This method allows to easily connect bdds with AND or OR and is also suited for voting systems with multiple layers. The method was published by S. Bolus:

• Bolus, S., 2011. Power indices of simple games and vector-weighted majority games by means of binary decision diagrams. European J. Oper. Res. 210 (2), 258–272.
• If you are interested in calculating power indices you should also check out the website, which offers a javascript-version with a lot more features.

Usage

Installation

pip install power_bdd

Import

from power_bdd.bdd import BDD, WeightedGame

one-tier games

This example calculates the power indices for the electoral college, 1996.

w = [54, 33, 32, 25, 23, 22, 21, 18, 15, 14, 13, 13, 12, 12, 11, 11, 11, 11, 10, 10, 9, 9, 8, 8, 8, 8, 8, 8, 7, 7, 7, 6, 6, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3]                
q = 270
game = WeightedGame(q, w)
bdd = BDD(game)
banzhaf = bdd.calc_banzhaf()
shapley = bdd.calc_shapley()
print(banzhaf)
print(shapley)

multi-tier games

This example calculates the power-indices for the U.S. federal system, which can represented by the weighted voting systems G = (G1 and G2 and G3) or (G1 and G4 and G5 and G3) or (G6 and G7), see https://www.fernuni-hagen.de/stochastik/downloads/voting.pdf.

game1 = WeightedGame(218, [0, 0] + [0] * 100 + [1] * 435)
game2 = WeightedGame(51,  [0, 0] + [1] * 100 + [0] * 435)
game3 = WeightedGame(1,   [1, 0] + [0] * 100 + [0] * 435)
game4 = WeightedGame(50,  [0, 0] + [1] * 100 + [0] * 435)
game5 = WeightedGame(1,   [0, 1] + [0] * 100 + [0] * 435)
game6 = WeightedGame(290, [0, 0] + [0] * 100 + [1] * 435)
game7 = WeightedGame(67,  [0, 0] + [1] * 100 + [0] * 435)

federal_system = BDD(game1) & BDD(game2) & BDD(game3) | BDD(game1) & BDD(game4) & BDD(game5) & BDD(game3) | BDD(game6) & BDD(game7)
banzhaf = federal_system.calc_banzhaf()
print(banzhaf)

Complexities

Listed complexities are expected complexities for the computation of all voters. At first glance, the complexities seem partly worse than other methods (e.g. using dynamic programming), but there are several hidden benefitial properties, for instance q is not necessary the q input q, but the smallest integer that is possible to use as quota to represent the game (with any weights).

one-tier games:

power-index     | time          | space       
------------    | ------------- | ------------
Banzhaf/Penrose | O(nq log(q))  | O(nq)
Shapley/Shubik  | O(n^3q)       | O(n^2q)

multi-tier games with m tiers:

power-index     | time                   | space       
------------    | -------------          | ------------
Banzhaf/Penrose | O(n prod_{t=1}^m q^t)  | O(n prod_{t=1}^m q^t)
Shapley/Shubik  | O(n^3 prod_{t=1}^m q^t)| O(n^2 prod_{t=1}^m q^t)

Remarks

• My java-version is much faster (somewhere between 10-100 times) so there should be plenty of room for optimization in python.
• The original version uses an AVL-tree for the create-method. I have replaced that by a SortedList form the sortedcontainers-library.