Computing the banzhaf power index in network flow games
Download
1 / 18

Computing the Banzhaf Power Index in Network Flow Games - PowerPoint PPT Presentation


  • 85 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Computing the Banzhaf Power Index in Network Flow Games' - aldis


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
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


ad