Computing the Banzhaf Power Index in Network Flow Games

1. Computing the Banzhaf Power Index in Network Flow Games Yoram Bachrach Jeffrey S. Rosenschein

2. 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

3. Weighted Voting Games • 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

4. 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?

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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 :

11. 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

12. #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

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

14. 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

15. 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

16. 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

17. 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

18. 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