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The Shapley Value as a Function of the Quota in Weighted Voting Games

The Shapley Value as a Function of the Quota in Weighted Voting Games. Yair Zick , Alexander Skopalik and Edith Elkind School of Physical and Mathematical Sciences Division of Mathematical Sciences Nanyang Technological University, Singapore CoopMAS 2011. Outline. Preliminaries

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The Shapley Value as a Function of the Quota in Weighted Voting Games

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  1. The Shapley Value as a Function of the Quota in Weighted Voting Games YairZick, Alexander Skopalik and Edith Elkind School of Physical and Mathematical Sciences Division of Mathematical Sciences Nanyang Technological University, Singapore CoopMAS 2011

  2. Outline • Preliminaries • Weighted Voting Games • The Shapley Value • Manipulation of the Quota in Weighted Voting Games • Our Results • Conclusions and Future Work

  3. Weighted Voting Games • A Weighted Voting Game (WVG) on n players is defined as follows: • Each player has an integer weight . • A set of players is winningif • Given an ordering of players, player i is pivotal for an ordering if his predecessors are losing, but if he joins, they win. 49 q= 50 14 6 1 9 12 4 7

  4. The Shapley Value • We would like to quantitatively measure the power of a player. • The Shapley value [Shapley, 1953; Shapley and Shubik, 1954], is a very popular measure. • Extensively studied from a theoretical, empirical and computational viewpoint. • The Shapley value of player i is the probability that she is pivotal for a randomly chosen ordering of the players. It is denoted .

  5. Manipulation in WVGs • A central authority wants to maximize/minimize some players’ power. • It can do so by either manipulating the weights or by manipulating the quota. • Our study focuses on quota manipulation and its effect on the power of a player.

  6. Related Work • [Faliszewskiand Hemaspaandra, 2008]: • Deciding which WVG is better for a player is PP-complete. • [Zuckerman, Faliszewski, Bachrach and Elkind, 2008]: • Deciding which quota is better for a player is PP-complete • Finding a quota that makes a player a dummy is in P. • [Leech, 2002], [Leech and Machover, 2003]: • Empirical analysis of real-life WVGs. • [Aziz, Bachrach, Elkind and Paterson, 2011]: • Manipulation by merging and splitting of players’ weights.

  7. Our Work • In order to better understand , we have graphed instances of it in MATLAB for randomly generated weights. • Different distributions lead to different looking graphs. • Empirical results lead to theoretical results, which lead to more empirical experimentation.

  8. Uniformly Distributed Weights This is the graph for a player whose weight is 23. The graph converges to some value when quota is 50%… The peak is at quota 23… The graph is symmetric! The minimum is at 24…

  9. Poisson Distributed Weights

  10. Weights are exponents of 2

  11. Weights are a Fibonacci Series

  12. Theoretical Results • has a global maximum at . • Deciding whether is maximal at q is NP-hard; it is in P for the player with the smallest weight. • Deciding whether is minimal at q is NP-hard; it is in P for the player with the biggest weight. • Players who are below the median always prefer the quota 1 to the quota .

  13. Finding a Minimizing Quota • Appears to be trickier than finding a maximizing quota. • Two quotas are candidates – 1 and • Not always the case • Depends on both the rank of the player (below or above median) and the distribution of weights.

  14. Finding a Minimizing Quota • Even when minimizing quota is not at it is not too far. • We have checked 100 randomly generated weights (distributed according to the uniform, Poisson and normal distributions).

  15. Uniform Distribution

  16. Uniform Distribution

  17. Normal Distribution

  18. Normal Distribution

  19. Poisson Distribution

  20. Convergence of the graph • It is known [Mann and Shapley, 1964] that: • Shapley Value seems to converge to this value as . • If we restrict manipulation to quotas around 50%, manipulators cannot radically increase a player’s power.

  21. Uniformly Distributed Weights

  22. Poisson Distributed Weights

  23. Conclusions • Maximizing quota at . • Deciding if a given quota is minimizing/maximizing is NP-hard. • The quotas 1 and are usually minimizers and when they are not, they are not too far from the minimum. • The Shapley value tends to be for quotas around 50%.

  24. Future Work • A poly-time method of finding a minimizing quota. • The Shapley value of a set of players. • Can one find a maximizing quota in poly-time? • If not, can the maximizing quota be constrained to a range? • Convergence around 50%.

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