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 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?

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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

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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

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Network Flow Game

• A network flow graph GltV,Egt
• 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

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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

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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

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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

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NETWORK-FLOW-BANZHAF

• Given an NFG, calculate the Banzhaf power index
  of the edge e
• Graph GltV,Egt
• 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

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P-Completeness of NETWORK-FLOW-BANZHAF

• Proof by reduction from MATCHING
• MATCHING
• Given a biparite GltU,V,Egt, UVk
• 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

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Constructing the Inputs

Copied Graph
Calculate Banzhaf index for this edge
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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

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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 L0s,,Lnt
• 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

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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

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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

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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