Approximating Power Indices

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Approximating Power Indices

• Yoram Bachrach(Hebew University)
• Evangelos Markakis(CWI)
• Ariel D. Procaccia (Hebrew University)
• Jeffrey S. Rosenschein (Hebrew University)
• Amin Saberi (Stanford University)

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Outline

• Power indices
• Weighted Voting Games
• The Banzhaf and Shapley-Shubik power indices
• Applications of power indices
• Computational hardness results
• Approximating Power Indices by Sampling
• Estimating the power index
• Confidence interval through Hoeffdings
  inequality
• Adaptations for the Shapley-Shubik power index
• Lower bounds
• Deterministic approximation algorithms
• Randomized approximation algorithms
• Related work
• Conclusions and future research

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Weighted Voting Games

• Set of agents
• Each agent has a weight
• A game has a quota
• A coalition wins if
• A simple game the value of a coalition is
  either 1 or 0

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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?

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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

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The Banzhaf Power Index

• Pivotal (critical) 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
  pivotal (critical)

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The Shapley-Shubik Index

• The portion of all permutations where the agent
  is pivotal
• Direct application of the Shapley value for
  simple coalitional games

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Applications of Power Indices

• Measuring political power in decision making
  bodies
• US electoral college
• EU Council of Ministers
• International Monetary Fund
• Cost sharing schemes
• Cost allocation
• Network reliability

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Computational Considerations

• Many applications, so computing them is of high
  importance
• Naïve algorithms are exponential
• Banzhaf - Go over all possible coalitions
• Shapley-Shubik Go over all possible
  permutations of the agents
• Can power indices be computed tractably in
  interesting domains?
• Voting games, netowrk domains, cost sharing,

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Computational Hardness of Computing Power Indices

• Weighted voting games
• Banzhaf is NP-hard to compute
• Shapley-Shubik is even worse P-complete
• Polynomial algorithms for very restricted
  domains
• Network reliability
• Network flow domains P-complete
• Polynomial for very restricted networks
• Connectivity games P-complete
• Polynomial algorithms for trees
• Hardness results for many other cooperative
  domains

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Approximating By Sampling

• Use randomized algorithms to approximate the
  power index
• Probably approximately correct (PAC) algorithm
• Return an approximately correct power index with
  high probability
• For a given the probability of
  returning a value which misses the correct index
  by more than depends on the number of samples
• Basic operation - coalition value queries
• Randomly sample coalitions, and check if target
  agent is pivotal for that coalition

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Estimating the Power Index

• Estimate the Banzhaf power index as the
  proportion of all samples coalitions where target
  agent is pivotal
• Determine the required number of samples
  according to the required
• Confidence level (probability of a big error)
• Approximation accuracy (maximal allowed
  distance from the correct value)

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Confidence Intervals

• Can formulate the problem as building a
  confidence interval
• The intervals width depends on the maximal
  allowed inaccuracy
• Build the interval so that the probability of
  having the correct index outside the interval is
  at most
• Given the same number of samples, we can build
  different confidence intervals
• Higher confidence gt larger (inaccurate) interval
  and vice versa

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Accuracy, Confidence and Samples

• Higher accuracy and confidence require more
  sampled coalitions
• But, how many?
• Tying the variables together
• Unconservative - Normal approximation for the
  Binomial distribution
• Conservative - Hoeffdings inequality

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Hoeffdings Inequality

• Each coalition sampled is a random variable 1 if
  target agent is critical, 0 if not
• The expectancy is the power index
• Conservative confidence interval

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The Number of Samples

• Required number of samples
• Confidence interval
• Simple algorithm for approximating the power
  index for target accuracy and confidence

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Adaptations for the Shapley-Shubik Power Index

• Apply the same method for Shapley-Shubik
• Randomly sample permutations
• Rather than coalitions
• The Shapley-Shubik index is the proportion of all
  permutations where an agent is pivotal
• Each permutation sampled is a random variable 1
  if target agent is critical, 0 if not
• The expectancy is the power index
• Use Hoeffdings inequality, and get the same
  equations and algorithm as before

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Lower Bounds

• Obtained a PAC method for approximating the power
  index
• Polynomial accuracy
• Number of samples
• The number of samples is polynomial even if is
  exponentially small
• Can we achieve this with a deterministic
  algorithm with polynomial number of queries?
• Can a randomized algorithm achieve
  super-polynomial accuracy, i.e. where
  or even ?

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Lower Bounds - Deterministic

• With deterministic algorithms, we need an
  exponential number of queries to achieve
  polynomial accuracy
• There is a constant c such that any deterministic
  algorithm that approximates the Banzhaf index
  with accuracy better than requires
  samples
• Consider a deterministic algorithm that uses less
  then the above stated queries
• Consider an input I where the power index of an
  agent is 0
• Show a family of inputs F with high power index
• The algorithm is deterministic, so it is always
  possible to construct an input for which the
  queries regarding the coalition are all answered
  by 0, so the algorithm cannot differentiate among
  I and F

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Lower Bounds - Randomized

• No randomized algorithm can achieve
  super-polynomial accuracy
• Use Yaos Minimax principse to show a lower
  bound for a randomized algorithm it is enough to
  construct a distribution on a family of inputs
  and show a lower bound for a deterministic
  algorithm on this distribution

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Related Work

• The Banzhaf and Shapley-Shubik power indices
• Power indices hardness results
• Matsui Matsui Banzhaf and Shapley in WVGs is
  NPC
• Deng Papadimitriou Shapley in WVG is P-C
• Bachrach Rosenschein Banzhaf in network flow
  games is P-C
• Power index calculation and approximation methods
• Mann and Shapley Monte-Carlo simulations and
  exact computation improvements via generating
  functions
• Owen multilinear extension methods
• Fatima, Wooldridge and Jennings approximate
  method for voting games with empirical analysis

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Conclusions

• Suggested an randomized approximation method for
  the power index
• PAC analysis build a confidence interval
• Express the relation between the required number
  of queries, accuracy and confidence
• Running time is polynomial in accuracy and
  confidence
• Lower bounds
• No deterministic algorithm can achieve comparable
  accuracy with polynomial number of queries
• No randomized algorithm can achieve
  super-polynomial accuracy
• Future research
• Computing power indices exactly in restricted
  domains
• Better approximation for restricted domains
• Empirical analysis of confidence / accuracy