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# Computing the Banzhaf Power Index in Network Flow Games - PowerPoint PPT Presentation

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

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### Computing the Banzhaf Power Index in Network Flow Games

Yoram Bachrach

Jeffrey S. Rosenschein

• 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

• Set of agents

• Each agent has a weight

• A game has a quota

• A coalition wins if

• A simplegame – the value of a coalition is either 1 or 0

• 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?

• 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

• 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

• 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

• 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

• 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 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 :

• 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

• 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

Copied Graph

Calculate Banzhaf index for this edge

• 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

• 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

• 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

• 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

• 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