Computing the Banzhaf Power Index in Network Flow Games

Yoram Bachrach

Jeffrey S. Rosenschein

Outline Power indices The Banzhaf power index Network flow games - NFGs

Motivation The Banzhaf power index in NFGs #P-Completeness

Restricted case Connectivity games Bounded layer graphs Polynomial algorithm for a restricted case

Related work Conclusions and future directions

Weighted Voting Games

Set of agents Each agent has a weight A game has a quota A coalition wins if A simple game – the value of a coalition is

either 1 or 0

N

ia A iw R

C Ni

ia C

w q

q

Weighted Voting Games

Consider No single agent wins, every coalition of 2 agents wins, and

the grand coalition wins No agent has more power than any other

Voting power is not proportional to voting weight Your ability to change the outcome of the game with your

vote How do we measure voting power?

1 2 351, 50, 26, 26q w w w

Power Indices

The probability of having a significant role in determining the outcome Different assumptions on coalition formation Different definitions of having a significant role

Two prominent indices Shapley-Shubik Power Index

Similar to the Shapley value, for a simple game Banzhaf Power Index

The Banzhaf Power Index

Critical (swinger) agent in a winning coalition is an agent that causes the coalition to lose when removed from it

The Banzhaf Power Index of an agent is the portion of all coalitions where the agent is critical

Network Flow Game

A network flow graph G=<V,E> Capacities Source vertex s, target vertex t Agent i controls A coalition C controls the edges

The value of a coalition C is the maximal flow it can send between s and t

:c E R

{ | }c iE e i C ie E

Simple Network Flow Game

A network flow game, with a target required flow k

A coalition of edges wins if it can send a flow of at least k from s to t

Motivation Bandwidth of at least k is required from s to t in a communication

network Edges require maintenance

Chances of a failure increase when less resources are spent Limited amount of total resources

“Powerful” edges are more critical Edge failure is more likely to cause a failure in maintaining the

required bandwidth More maintenance resources

The Banzhaf Power in Simple Network Flow Games

The Banzhaf index of an edge The portion of edge coalitions which allow the

required flow, but fail to do so without that edge Let The Banzhaf index of :

{ | }ie iC C E e C

ie

NETWORK-FLOW-BANZHAF Given an NFG, calculate the Banzhaf power index of the edge e

Graph G=<V,E> Capacity function c Source s and target t Target flow k Edge e

Easy to check if an edge coalition allows the target flow, but fails to do it without e Run a polynomial algorithm to calculate maximal flow Check if its above k Remove e Check if the maximal flow is still above k

But calculating the Banzhaf power index required finding out how many such edge coalitions exist

#P-Completeness of NETWORK-FLOW-BANZHAF Proof by reduction from #MATCHING #MATCHING

Given a biparite G=<U,V,E>, |U|=|V|=k Count the number of perfect matchings in G A prominent #P-complete problem

The reduction builds two identical inputs to NETWORK-FLOW-BANZHAF With different target flows:

#MATCHING result is the difference between the results,k k

Constructing the Inputs

Copied GraphCalculate Banzhaf index for this edge

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Reduction Outline We make sure Any subset of edges missing even one edge on the first layer or last two

layers does not allow a flow of k We identify an edge subset in G’ with an edge subset (matching

candidate) in G Any perfect matching allows a flow of k

But any matching that misses a vertex does not allow such a flow of k (but only less)

Matching a vertex more than once would allow a flow of more than k The Banzhaf index counts the number of coalitions which allow a k flow

This is the number of perfect matchings and overmatchings But giving a target flow of more than k counts just the overmatchings

1k

Connectivity Games and Bounded Layer Graphs

Connectivity games Restricted form of NFGs Purpose of the game is to make sure there is a path from s to t All edges have the same capacity (say 1) Target flow is that capacity

Layer graphs Vertices are divided to layers L0={s},…,Ln={t} Edges only go between consecutive layers

C-Bounded layer graphs (BLG) Layer graphs where there are at most c vertices in each layer No bound on the number of edges

Polynomial Algorithm for CONNECTIVITY-BLG-BANZHAF

Dynamic programming algorithm for calculating the Banzhaf power index in bounded layer graphs Iterate through the layer, and update the number of coalitions

which contain a path to vertices in the next layer Polynomial due to the bound on the number of vertices in a layer

Related Work The Banzhaf and Shapley-Shubik power indices

Deng and Papadimitriou – calculating Shapley values in weighted votings games is #P-complete

Network Flow Games Kalai and Zemel – certain families of NFGs have non empty cores Deng et al. – polynomial algorithm for finding the nucleolus of restricted NFGs

Power indices complexity Matsui and Matsui

Calculating the Banzhaf and Shapley-Shubik power indices in weighted voting games is NP-complete

Survey of algorithms for approximating power indices in weighted voting games

Conclusion & Future Directions

Shown calculating the Banzhaf power index in NFGs is #P-complete

Gave a polynomial algorithm for a restricted case Possible future work

Other power indices Approximation for NFGs Power indices in other domains