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

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Computing the banzhaf power index in network flow games

Computing the Banzhaf Power Index in Network Flow Games

Yoram Bachrach

Jeffrey S. Rosenschein


Outline
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
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 games1
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
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
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
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
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
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 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
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
#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


Constructing the inputs
Constructing the Inputs

Copied Graph

Calculate Banzhaf index for this edge


Reduction outline
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 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
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
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
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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