The Inverse Shapley Value Problem

For $f$ a weighted voting scheme used by $n$ voters to choose between two candidates, the $n$ \emph{Shapley-Shubik Indices} (or {\em Shapley values}) of $f$ provide a measure of how much control each voter can exert over the overall outcome of the vote. Shapley-Shubik indices were introduced by Lloyd Shapley and Martin Shubik in 1954 \cite{SS54} and are widely studied in social choice theory as a measure of the "influence" of voters. The \emph{Inverse Shapley Value Problem} is the problem of designing a weighted voting scheme which (approximately) achieves a desired input vector of values for the Shapley-Shubik indices. Despite much interest in this problem no provably correct and efficient algorithm was known prior to our work. We give the first efficient algorithm with provable performance guarantees for the Inverse Shapley Value Problem. For any constant \eps > 0 our algorithm runs in fixed poly$(n)$ time (the degree of the polynomial is independent of \eps) and has the following performance guarantee: given as input a vector of desired Shapley values, if any "reasonable" weighted voting scheme (roughly, one in which the threshold is not too skewed) approximately matches the desired vector of values to within some small error, then our algorithm explicitly outputs a weighted voting scheme that achieves this vector of Shapley values to within error \eps. If there is a "reasonable" voting scheme in which all voting weights are integers at most \poly(n) that approximately achieves the desired Shapley values, then our algorithm runs in time \poly(n) and outputs a weighted voting scheme that achieves the target vector of Shapley values to within error $\eps=n^{-1/8}.

An anytime approximation method for the inverse Shapley value problem

Coalition formation is the process of bringing together two or more agents so as to achieve goals that individuals on their own cannot, or to achieve them more efficiently. Typically, in such situations, the agents have conflicting preferences over the set of possible joint goals. Thus, before the agents realize the benefits of cooperation, they must find a way of resolving these conflicts and reaching a consensus. In this context, cooperative game theory offers the voting game as a mechanism for agents to reach a consensus. It also offers the Shapley value as a way of measuring the influence or power a player has in determining the outcome of a voting game. Given this, the designer of a voting game wants to construct a game such that a players Shapley value is equal to some desired value. This is called the inverse Shapley value problem. Solving this problem is necessary, for instance, to ensure fairness in the players voting powers. However, from a computational perspective, finding a players Shapley value for a given game is #p-complete. Consequently, the problem of verifying that a voting game does indeed yield the required powers to the agents is also #P-complete. Therefore, in order to overcome this problem we present a computationally efficient approximation algorithm for solving the inverse problem. This method is based on the technique of successive approximations; it starts with some initial approximate solution and iteratively updates it such that after each iteration, the approximate gets closer to the required solution. This is an anytime algorithm and has time complexity polynomial in the number of players. We also analyze the performance of this method in terms of its approximation error and the rate of convergence of an initial solution to the required one. Specifically, we show that the former decreases after each iteration, and that the latter increases with the number of players and also with the initial approximation error. Copyright © 2008, International Foundation for Autonomous Agents and Multiagent Systems (www.ifaarnas.org). All rights reserved

On the inverse power index problem

Weighted voting games are frequently used in decision making. Each voter has a weight and a proposal is accepted if the weight sum of the supporting voters exceeds a quota. One line of research is the efficient computation of so-called power indices measuring the influence of a voter. We treat the inverse problem: Given an influence vector and a power index, determine a weighted voting game such that the distribution of influence among the voters is as close as possible to the given target value. We present exact algorithms and computational results for the Shapley-Shubik and the (normalized) Banzhaf power index.Comment: 17 pages, 2 figures, 12 table

Ready for the design of voting rules?

The design of fair voting rules has been addressed quite often in the literature. Still, the so-called inverse problem is not entirely resolved. We summarize some achievements in this direction and formulate explicit open questions and conjectures.Comment: 10 page

Matrix approach to consistency of the additive efficient normalization of semivalues

In fact the Shapley value is the unique efficient semivalue. This motivated Ruiz et al. to do additive efficient normalization for semivalues. In this paper, by matrix approach we derive the relationship between the additive efficient normalization of semivalues and the Shapley value. Based on the relationship, we axiomatize the additive efficient normalization of semivalues as the unique solution verifying covariance, symmetry, and reduced game property with respect to the $p-$reduced game

On the performance of the Shapley Shubik and Banzhaf power indices for the allocations of mandates

A classical problem in the power index literature is to design a voting mechanism such as the distribution of power of the different players is equal (or closer) to a pre established target. This tradition is especially popular when considering two tiers voting mechanisms: each player votes in his own jurisdiction to designate a delegate for the upper tier; and the question is to assign a certain number of mandates for each delegate according the population of the jurisdiction he or she represents. Unfortunately, there exist several measures of power, which in turn imply different distributions of the mandates for the same pre established target. The purposes of this paper are twofold: first, we calculate the probability that the two most important power indices, the Banzhaf index and the Shapley-Shubik index, lead to the same voting rule when the target is the same. Secondly, we determine which index on average comes closer to the pre established target.Banzhaf, Shapley-Shubik, power indices

An Accretive Operator Approach to Ergodic Problems for Zero-Sum Games

Mean payoff stochastic games can be studied by means of a nonlinear spectral problem involving the Shapley operator: the ergodic equation. A solution consists in a scalar, called the ergodic constant, and a vector, called bias. The existence of such a pair entails that the mean payoff per time unit is equal to the ergodic constant for any initial state, and the bias gives stationary strategies. By exploiting two fundamental properties of Shapley operators, monotonicity and additive homogeneity, we give a necessary and sufficient condition for the solvability of the ergodic equation for all the Shapley operators obtained by perturbation of the transition payments of a given stochastic game with finite state space. If the latter condition is satisfied, we establish that the bias is unique (up to an additive constant) for a generic perturbation of the transition payments. To show these results, we use the theory of accretive operators, and prove in particular some surjectivity condition.Comment: 4 pages, 1 figure, to appear in Proc. 22nd International Symposium on Mathematical Theory of Networks and Systems (MTNS 2016

Complementary cooperation, minimal winning coalitions, and power indices

We introduce a new simple game, which is referred to as the complementary weighted multiple majority game (C-WMMG for short). C-WMMG models a basic cooperation rule, the complementary cooperation rule, and can be taken as a sister model of the famous weighted majority game (WMG for short). In this paper, we concentrate on the two dimensional C-WMMG. An interesting property of this case is that there are at most $n+1$ minimal winning coalitions (MWC for short), and they can be enumerated in time $O(n\log n)$, where $n$ is the number of players. This property guarantees that the two dimensional C-WMMG is more handleable than WMG. In particular, we prove that the main power indices, i.e. the Shapley-Shubik index, the Penrose-Banzhaf index, the Holler-Packel index, and the Deegan-Packel index, are all polynomially computable. To make a comparison with WMG, we know that it may have exponentially many MWCs, and none of the four power indices is polynomially computable (unless P=NP). Still for the two dimensional case, we show that local monotonicity holds for all of the four power indices. In WMG, this property is possessed by the Shapley-Shubik index and the Penrose-Banzhaf index, but not by the Holler-Packel index or the Deegan-Packel index. Since our model fits very well the cooperation and competition in team sports, we hope that it can be potentially applied in measuring the values of players in team sports, say help people give more objective ranking of NBA players and select MVPs, and consequently bring new insights into contest theory and the more general field of sports economics. It may also provide some interesting enlightenments into the design of non-additive voting mechanisms. Last but not least, the threshold version of C-WMMG is a generalization of WMG, and natural variants of it are closely related with the famous airport game and the stable marriage/roommates problem.Comment: 60 page