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 simple game – 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
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