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Computing the Banzhaf Power Index in Network Flow Games

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Computing the Banzhaf Power Index in Network Flow Games Powered By Docstoc
					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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