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 SciencesDivision of Mathematical Sciences

Nanyang Technological University, Singapore

CoopMAS 2011

Preliminaries◦ Weighted Voting Games◦ The Shapley Value

Manipulation of the Quota in Weighted Voting Games

Our Results Conclusions and Future Work

Outline

A Weighted Voting Game (WVG) on n players is defined as follows:◦ Each player has an integer weight

.◦ A set of players is winning if

Given an ordering of players, player i is pivotal for an ordering if his predecessors are losing, but if he joins, they win.

Weighted Voting Games

614 1 129 7 4

49 q= 50

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 .

The Shapley Value

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.

Manipulation in WVGs

[Faliszewski and 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.

Related 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.

Our Work

Uniformly Distributed Weights

The minimum is at 24…

The peak is at quota 23…

The graph converges to some value when quota is 50%…

The graph is symmetric!

This is the graph for a player whose weight is 23.

Poisson Distributed Weights

Weights are exponents of 2

Weights are a Fibonacci Series

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 .

Theoretical Results

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.

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).

Finding a Minimizing Quota

Uniform Distribution

Uniform Distribution

Normal Distribution

Normal Distribution

Poisson Distribution

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.

Convergence of the graph

Uniformly Distributed Weights

Poisson Distributed Weights

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%.

Conclusions

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%.

Future Work