Computing the Banzhaf
Power Index in Network
Flow Games
Yoram Bachrach
Jeffrey S. Rosenschein
Outline
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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
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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
A simple game – the value of a coalition is
either 1 or 0
i
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Weighted Voting Games
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Consider q  51, w1  50, w2  26, w3  26
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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
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Your ability to change the outcome of the game with your
vote
How do we measure voting power?
Power Indices
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The probability of having a significant role in
determining the outcome
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Different assumptions on coalition formation
Different definitions of having a significant role
Two prominent indices
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Shapley-Shubik Power Index
 Similar to the Shapley value, for a simple game
Banzhaf Power Index
The Banzhaf Power Index
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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
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A network flow graph G=<V,E>
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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
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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
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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
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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 :
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NETWORK-FLOW-BANZHAF
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Given an NFG, calculate the Banzhaf power index of the edge e
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Easy to check if an edge coalition allows the target flow, but fails to do it
without e
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Graph G=<V,E>
Capacity function c
Source s and target t
Target flow k
Edge 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 NETWORKFLOW-BANZHAF
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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-FLOWBANZHAF
 With different target flows: k , k  
#MATCHING result is the difference between the results
Constructing the Inputs
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Copied Graph
Calculate Banzhaf
index for this edge
Reduction Outline
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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
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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
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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
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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
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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
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The Banzhaf and Shapley-Shubik power indices
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Network Flow Games
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Deng and Papadimitriou – calculating Shapley values in weighted votings games
is #P-complete
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
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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
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Shown calculating the Banzhaf power index in NFGs
is #P-complete
Gave a polynomial algorithm for a restricted case
Possible future work
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Other power indices
Approximation for NFGs
Power indices in other domains