# Computing the Banzhaf Power Index in Network Flow Games

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```					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
 Polynomial algorithm for a restricted case
   Related work
   Conclusions and future directions
Weighted Voting Games
   Set of agents N
   Each agent ai  A has a weight wi  R
   A game has a quota q
   A coalition C  N wins if  wi  q
a C
i

   A simple game – the value of a coalition is
either 1 or 0
Weighted Voting Games
   Consider q  51, w1  50, w2  26, w3  26
   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
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 c : E  R
   Source vertex s, target vertex t
   Agent i controls ei  E
   A coalition C controls the edges   Ec  {ei | i  C}
   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 Cei  {C  E | ei  C}

 The Banzhaf index of ei :
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: k , k  
   #MATCHING result is the difference between the results
Constructing the Inputs

‘

Calculate Banzhaf
Copied Graph    index for this edge
Reduction Outline
   We make sure k  1
   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
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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 views: 40 posted: 9/5/2011 language: English pages: 18