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

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?

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

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 :

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

Constructing the Inputs ‘ Copied Graph Calculate Banzhaf index for this edge

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

Connectivity Games and Bounded Layer Graphs 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