Shubik Power Index Research Articles

Cross invariance, the Shapley value, and the Shapley–Shubik power index

In this paper we propose a simple axiom which, along with the axioms of additivity (transfer) and dummy player, characterizes the Shapley value (the Shapley–Shubik power index) on the domain of TU (simple) games. The new axiom, cross invariance, demands payoff invariance on symmetric players across “quasi-symmetric games,” that is, games where excluding null players, all players are symmetric. Additionally, we demonstrate that the axiom of additivity can be replaced by a new axiom called strong monotonicity, or it can be completely dropped if a stronger version of cross invariance is employed. We also show that the weighted Shapley values can be characterized using a weighted variant of cross invariance. Efficiency is derived rather than assumed in our characterizations. This fresh perspective contributes to a deeper understanding of the Shapley value and its applicability.

Aggregated Power Indices for Measuring Indirect Control in Complex Corporate Networks with Float Shareholders.

The purpose of this paper is to introduce new methods to measure the indirect control power of firms in complex corporate shareholding structures using the concept of power indices from cooperative game theory. The proposed measures vary in desirable properties satisfied, as well as in the bargaining models of power indices used to construct them. Hence, they can be used to produce different pictures of the coalitional strength of firms in control of other firms in mutual shareholding networks with the presence of cycles. Precisely, in the framework of Karos and Peters from 2015, ten power indices substitute the original Shapley and Shubik power index in a modular fashion. In this way, we obtain a set of new measures called aggregated indices. The float shareholders typically hold less than 5 percent of the outstanding shares, which is an uncertain element of indirect control in complex shareholding structures. The fuzzy number seems appropriate to model these shareholders' behavior. The novelty is that we model the behavior of float using Z-fuzzy numbers. The new methods are tested in an example.

Monte Carlo Methods for the Shapley–Shubik Power Index

This paper deals with the problem of calculating the Shapley–Shubik power index in weighted majority games. We propose an efficient Monte Carlo algorithm based on an implicit hierarchical structure of permutations of players. Our algorithm outputs a vector of power indices preserving the monotonicity, with respect to the voting weights. We show that our algorithm reduces the required number of samples, compared with the naive algorithm.

Brexit and Power in the Council of the European Union

The exit of the United Kingdom from the European Union has had profound economic and political effects. Here, we look at a particular aspect, the power distribution in the Council of the European Union. Using the Shapley–Shubik power index, we calculate the member states’ powers with and without the United Kingdom and update earlier power forecasts using the Eurostat’s latest population projections. There is a remarkably sharp relation between population size and the change in power: Brexit increases the largest members’ powers while decreasing the smallest ones’ powers.

Axiomatizations for the Shapley–Shubik power index for games with several levels of approval in the input and output

The Shapley-Shubik index is a specialization of the Shapley value and is widely applied to evaluate the power distribution in committees drawing binary decisions. It was generalized to decisions with more than two levels of approval both in the input and the output. The corresponding games are called $(j,k)$ simple games. Here we present a new axiomatization for the Shapley-Shubik index for $(j,k)$ simple games as well as for a continuous variant, which may be considered as the limit case.

An Axiomatization for Two Power Indices for (3,2)-Simple Games

The aim of this work is to give a characterization of the Shapley–Shubik and the Banzhaf power indices for (3,2)-simple games. We generalize to the set of (3,2)-simple games the classical axioms for power indices on simple games: transfer, anonymity, null player property and efficiency. However, these four axioms are not enough to uniquely characterize the Shapley–Shubik index for (3,2)-simple games. Thus, we introduce a new axiom to prove the uniqueness of the extension of the Shapley–Shubik power index in this context. Moreover, we provide an analogous characterization for the Banzhaf index for (3,2)-simple games, generalizing the four axioms for simple games and adding another property.

The Shapley–Shubik power index for dichotomous multi-type games

This work focuses on multi-type games in which there are a number of non-ordered types in the input, while the output consists of a single real value. When considering the dichotomous case, we extend the Shapley–Shubik power index and provide a full characterization of this extension. Our results generalize the literature on classical cooperative games.

Spatial power indices with applications on real voting data from the Chamber of Deputies of the Czech Parliament

Participants in parliamentary voting—usually political parties—are evaluated with a value that assigns them predicted power with respect to their strength by so called power indices. However, in real world, political parties’ representatives act not strictly as predicted in theory. One possibility how the representatives differ from theory is the way they form coalitions; the coalitions can be announced (for example official governmental coalition parties in multiparty parliaments) or hidden. To incorporate the coalition forming influence, Bilal et al. (http://aei.pitt.edu/2052/1/001591_1.pdf, 2001) proposed to consider additional weights to possible coalitions into power indices. This article applies the concept of additional weights to calculate an ex post power distribution using Shapley–Shubik power index together with Banzhaf power index on real voting data, namely the data from the Chamber of Deputies of the Czech Parliament with the emphasis on the State Budget voting issues during 2006–2010 parliamentary period.

Power theories for multi-choice organizations and political rules: Rank-order equivalence

Voting power theories measure the ability of voters to influence the outcome of an election under a given voting rule. In general, each theory gives a different evaluation of power, raising the question of their appropriateness, and calling for the need to identify classes of rules for which different theories agree. We study the ordinal equivalence of the generalizations of the classical power concepts–the influence relation, the Banzhaf power index, and the Shapley–Shubik power index–to multi-choice organizations and political rules. Under such rules, each voter chooses a level of support for a social goal from a finite list of options, and these individual choices are aggregated to determine the collective level of support for this goal. We show that the power theories analyzed do not always yield the same power relationships among voters. Thanks to necessary and/or sufficient conditions, we identify a large class of rules for which ordinal equivalence obtains. Furthermore, we prove that ordinal equivalence obtains for all linear rules allowing a fixed number of individual approval levels if and only if that number does not exceed three. Our findings generalize all the previous results on the ordinal equivalence of the classical power theories, and show that the condition of linearity found to be necessary and sufficient for ordinal equivalence to obtain when voters have at most three options to choose from is no longer sufficient when they can choose from a list of four or more options.

Confidence Intervals for the Shapley–Shubik Power Index in Markovian Games

We consider simple Markovian games, in which several states succeed each other over time, following an exogenous discrete-time Markov chain. In each state, a different simple static game is played by the same set of players. We investigate the approximation of the Shapley–Shubik power index in simple Markovian games (SSM). We prove that an exponential number of queries on coalition values is necessary for any deterministic algorithm even to approximate SSM with polynomial accuracy. Motivated by this, we propose and study three randomized approaches to compute a confidence interval for SSM. They rest upon two different assumptions, static and dynamic, about the process through which the estimator agent learns the coalition values. Such approaches can also be utilized to compute confidence intervals for the Shapley value in any Markovian game. The proposed methods require a number of queries, which is polynomial in the number of players in order to achieve a polynomial accuracy.

Power measures derived from the sequential query process

We study a basic sequential model for the formation of winning coalitions in a simple game, well known from its use in defining the Shapley–Shubik power index. We derive in a uniform way a family of measures of collective and individual decisiveness in simple games, and show that, as for the Shapley–Shubik index, they extend naturally to measures for TU-games. These individual measures, which we call weighted semivalues, form a class whose intersection with that of the class of weak semivalues yields the class of all semivalues.We single out the simplest measure in this family for more investigation, as it is new to the literature as far as we know. Although it is very different from the Shapley value, it is closely related in several ways, and is the natural analogue of the Shapley value under a nonstandard, but natural, definition of simple game. We illustrate this new measure by calculating its values on some standard examples.

Characterization of the Shapley–Shubik power index without the efficiency axiom

We show that the Shapley–Shubik power index on the domain of simple (voting) games can be uniquely characterized without the efficiency axiom. In our axiomatization, the efficiency is replaced by the following weaker requirement that we term the gain-loss axiom: any gain in power by a player implies a loss for someone else (the axiom does not specify the extent of the loss). The rest of our axioms are standard: transfer (which is the version of additivity adapted for simple games), symmetry or equal treatment, and dummy.

The complexity of power-index comparison

We study the complexity of the following problem: Given two weighted voting games G ′ and G ″ that each contain a player p , in which of these games is p ’s power index value higher? We study this problem with respect to both the Shapley–Shubik power index and the Banzhaf power index. Our main result is that for both of these power indices the problem is complete for probabilistic polynomial time (i.e., is PP -complete). We apply our results to partially resolve some recently proposed problems regarding the complexity of weighted voting games. We also study the complexity of the raw Shapley–Shubik power index. Deng and Papadimitriou showed that the raw Shapley–Shubik power index is #P -metric-complete. We strengthen this by showing that the raw Shapley–Shubik power index is many–one complete for #P . And our strengthening cannot possibly be further improved to parsimonious completeness, since we observe that, in contrast with the raw Banzhaf power index, the raw Shapley–Shubik power index is not #P -parsimonious-complete.

Voters' power in voting games with abstention: Influence relation and ordinal equivalence of power theories

The influence relation was introduced by Isbell [Isbell, J.R., 1958. A class of simple games. Duke Math. J. 25, 423–439] to qualitatively compare the a priori influence of voters in a simple game, which by construction allows only “yes” and “no” votes. We extend this relation to voting games with abstention (VGAs), in which abstention is permitted as an intermediate option between a “yes” and a “no” vote. Unlike in simple games, this relation is not a preorder in VGAs in general. It is not complete either, but we characterize VGAs for which it is complete, and show that it is a preorder whenever it is complete. We also compare the influence relation with recent generalizations to VGAs of the Shapley–Shubik and Banzhaf–Coleman power indices [Felsenthal, D.S., Machover, M., 1997. Ternary voting games. Int. J. Game Theory 26, 335–351; Freixas, J., 2005a. The Shapley–Shubik power index for games with several levels of approval in the input and output. Dec. Support Systems 39, 185–195; Freixas, J., 2005b. The Banzhaf index for games with several levels of approval in the input and output. Ann. Operations Res. 137, 45–66]. For weakly equitable VGAs, the influence relation is a subset of the preorderings defined by these two power theories. We characterize VGAs for which the three relations are equivalent.

12. L. S. Shapley and Martin Shubik. “A Method for Evaluating the Distribution of Power in a Committee System,” American Political Science Review 48 (1954): 787–92 Cited 360 times.

Next to having one's book prominently displayed on the racks of every airport bookstore in the country, receiving recognition for an article written more than half a century ago is a close second. Indeed, for some of us it is the very highest form of intellectual acknowledgment. The Shapley–Shubik power index, invented more than 50 years ago, is a familiar concept in the analytical lexicon of political science. In brief, it is a measure of the ex ante likelihood that an individual will be pivotal in transforming a nonwinning coalition into a winning one. This depends on the voting or decision rule, on the one hand, and on the number of votes or decision weight possessed by an individual, on the other. The “power” of an individual, so determined, may be aggregated to measure the power of a faction, party, proto-coalition, or voting chamber (the latter in a multicameral setting)

An Asymmetric Shapley–Shubik Power Index

This paper extends the traditional “pivoting” and “swing” schemes in the Shapley–Shubik (S-S) power index and the Banzhaf index to the case of “blocking”. Voters are divided into two groups: those who vote for the bill and those against the bill. The uncertainty of the division is described by a probability distribution. We derive the S-S power index, based on a priori ignorance about the random bipartition.

Faster algorithms for computing power indices in weighted voting games

We consider weighted voting games with n players. We show how to compute the Banzhaf power index for every player within a running time of O( n 2 1.415 n ), and how to compute the Shapley–Shubik power index within a running time of O( n 1.415 n ). Our result improves on the straightforward running times of O( n 2 2 n ) and O( n 2 n ), respectively, that are implicit in the definitions of these power indices.

A power analysis of linear games with consensus

Linear simple games with consensus are considered. These games are obtained by intersecting a linear simple game (a game where the desirability order for players is total) and a symmetric ( k-out-of- n) game. We investigate the behavior of the Shapley–Shubik power index when passing from a linear game with consensus level q 1 to the same game with consensus level q 2> q 1. We also introduce a “range notion”, which intuitively represents the egalitarianism of the Shapley–Shubik index, obtain an upper bound on this measure, and characterize when it is achieved. Finally, the theory developed here is illustrated with several real-world voting systems.

The Shapley–Shubik power index for games with several levels of approval in the input and output

Voting systems with several levels of approval in the input and output are considered in this paper. That means games with n≥2 players, j≥2 ordered qualitative alternatives in the input level and k≥2 possible ordered quantitative alternatives in the output. We introduce the Shapley–Shubik power index notion when passing from ordinary simple games or ternary voting games with abstention to this wider class of voting systems. The pivotal role of players is analysed by means of several examples and an axiomatization in the spirit of Shapley and Dubey is given for the proposed power index.

On authority distributions in organizations: equilibrium

One assumption in the Shapley–Shubik power index is that there is no interaction nor influence among the voting members. This paper will apply the command structure of Shapley (1994) to model members' interaction relations by simple games. An equilibrium authority distribution is then formulated by the power-in/power-out mechanism. It turns out to have much similarity to the invariant measure of a Markov chain and therefore some similar interpretations are followed for the new setting. In some sense, one's authority distribution quantifies his general administrative power in the organization and his long-run influence on all members. We provide a few applications in conflict resolution, college and journal ranking, and organizational choice.