Approximating Power Indices

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

2. 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 Hoeffding’s inequality • Adaptations for the Shapley-Shubik power index • Lower bounds • Deterministic approximation algorithms • Randomized approximation algorithms • Related work • Conclusions and future research

3. 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

4. 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?

5. 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

6. 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)

7. The Shapley-Shubik Index • The portion of all permutations where the agent is pivotal • Direct application of the Shapley value for simple coalitional games

8. 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

9. 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, …

10. 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

11. 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

12. 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)

13. Confidence Intervals • Can formulate the problem as building a confidence interval • The interval’s 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 => larger (inaccurate) interval and vice versa

14. 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 - Hoeffding’s inequality

15. Hoeffding’s 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

16. The Number of Samples Required number of samples Confidence interval Simple algorithm for approximating the power index for target accuracy and confidence

17. 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 Hoeffding’s inequality, and get the same equations and algorithm as before

18. 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 ?

19. 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

20. Lower Bounds - Randomized No randomized algorithm can achieve super-polynomial accuracy Use Yao’s 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

21. 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

22. 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