power-bdd 0.2.0

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

power_bdd

Let $N={1,2,..,n}$ be a set of voters, each with a voting weight of $w_i$, who vote yes or no over a proposal. A coalition $S\in 2^N$ of yes-voters is considered winning if $\sum_{i\in S} w_i$ is greater or equal to a quota $q$. What is the voting power of each voter?

One very intuitive definition of the voting power of a voter is the probability that she/he influences the outcome by her/his vote given an assumed voting behaviour of all other voters. This leads to the notion of power indices. The most well known two are those according to Banzhaf/Penrose and Shapley/Shubik.

One very cool way to calculated them is by using binary decision diagrams.

This package creates the reduced ordered binary decision diagram ("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

$G$ = ($G_1$ and $G_2$ and $G_3$) or ($G_1$ and $G_4$ and $G_5$ and $G_3$) or ($G_6$ and $G_7$),

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	$O(n \prod\limits_{t=1}^m q^t)$	$O(n \prod\limits_{t=1}^m q^t)$
Shapley/Shubik	$O(n^3 \prod\limits_{t=1}^m q^t)$	$O(n^2 \prod\limits_{t=1}^m q^t)$

Remarks

• My java-version is much faster (somewhere between 10-100 times) so there should be plenty of room for optimization in python. Although beautiful is better than ugly, I guess fast is better than slow... and we need more trees! Not just to save the planet, but also in python.
• The original version uses an AVL-tree for the create-method. I have replaced that by a SortedList form the sortedcontainers-library.