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Computing the Banzhaf Power Index in Network Flow GamesPowerPoint Presentation

Computing the Banzhaf Power Index in Network Flow Games

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Presentation Transcript

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 simplegame – 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

- Shapley-Shubik 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

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

- 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

- Matsui and Matsui

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

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