Search

In the popular class of multidimensional poverty measures introduced by Alkire and Foster (2011), a threshold switching function is used to identify who is multidimensionally poor. This paper shows that the weights and cut-off employed in this procedure are generally not unique and that such functions implicitly assume all groups of deprivation indicators of some fixed size are perfect substitutes. To address these limitations, I show how the identification procedure can be extended to incorporate any type of positive switching function, represented by the set of minimal deprivation bundles that define a unit as poor. Furthermore, the Banzhaf power index, uniquely defined from the same set of minimal bundles, constitutes a natural and robust metric of the relative importance of each indicator, from which the adjusted headcount can be estimated. I demonstrate the merit of this approach using data from Mozambique, including a decomposition of the adjusted headcount using a ‘one from each dimension’ non-threshold function.

We aim to identify the most influential members of the Agricultural Committee of the European Parliament (COMAGRI). Unlike previous studies that were based on case studies or interviews with stakeholders, we analyse the voting power of MEPs using a spatial Banzhaf power index. We identify critical members: members whose votes are necessary to form winning coalitions. We found that rapporteurs, EP group coordinators and MEPs from countries with high relative Committee representations, such as Ireland, Poland or Romania are powerful actors. Italy emerges as the most influential member state, while France seems surprisingly weak.

This is a post-peer-review, pre-copyedit version of an article published in Group decision and negotiation. The final authenticated version is available online at: http://dx.doi.org/10.1007/s10726-020-09718-7. The Banzhaf power index for games admits several extensions if the players have more than two ordered voting options. In this paper we prove that the most intu-itive and recognized extension of the index fails to preserve the desirability rela-tion for games with more than three ordered input levels of approval, a failure that undermines the index to be a good measure of power. This leads us to think of an alternative to the Banzhaf index for several input levels of approval. We propose a candidate for which it is proved that: (1) coincides with the Banzhaf index for sim-ple games, (2) it is proportional to its known extension for three levels of approval, and (3) preserves the desirability relation regardless of the number of input levels of approval. This new index is based on measuring the total capacity the player has to alter the outcome. In addition, it can be expressed through a very appropriate mathematical formulation that greatly facilitates its computation. Defining exten-sions of well-established notions in a wider context requires a careful analysis. Dif-ferent extensions can provide complementary nuances and, when this occurs, none of them can be considered to be ‘the’ extension. As shown in this paper, this situa-tion applies when trying to extend the Banzhaf power index from simple games to the broader context of games with several ordered input levels of approval. This research was partially supported by funds from: the Spanish Ministry of Economy and Competitiveness (MINECO) and from the European Union (FEDER funds) under grant MTM2015-66818-P (MINECO/FEDER), and the Spanish Ministry of Science and Innovation grant PID2019-I04987GB-I00.

We introduce the axiom of composition independence for power indices and value maps. In the context of compound (two-tier) voting, the axiom requires the power attributed to a voter to be independent of the second-tier voting games played in all constituencies other than that of the voter. We show that the Banzhaf power index is uniquely characterized by the combination of composition independence, four semivalue axioms (transfer, positivity, symmetry, and dummy), and a mild efficiency-related requirement. A similar characterization is obtained as a corollary for the Banzhaf value on the space of all finite games (with transfer replaced by additivity).

This is a post-peer-review, pre-copyedit version of an article published in Group Decision and Negotiation. The final authenticated version is available online at: https://doi.org/10.1007/s10726-019-09651-4. This article proposes a value which can be considered an extension of the Banzhaf value for cooperative games. The proposed value is defined on the class of j-cooperative games, i.e., games in which players choose among a finite set of ordered actions and the result depends only on these elections. If the output is binary, only two options are available, then j-cooperative games become j-simple games. The restriction of the value to j-simple games leads to a power index that can be considered an extension of the Banzhaf power index for simple games. The paper provides an axiomatic characterization for the value and the index which is closely related to the first axiomatization of the Banzhaf value and Banzhaf power index in the respective contexts of cooperative and simple games.

I introduce a new axiom for power indices on the domain of finite simple games that requires the total power of any given pair i , j of players in any given game v to be equivalent to some individual power, i.e., equal to the power of some single player k in some game w. I show that the Banzhaf power index is uniquely characterized by this new “equivalence to individual power” axiom in conjunction with the standard semivalue axioms: transfer (which is the version of additivity adapted for simple games), symmetry or equal treatment, positivity (which is strengthened to avoid zeroing-out of the index on some games), and dummy.

The Banzhaf index is a function determining the power or influence in the decision of a set of agents. The extension of this index to the family of the cooperative games is named Banzhaf value. The relationships of closeness among the agents should modify their power. Games with a priori unions study situations where the closeness relations among the agents are taken into account. In this model the agents are organized in an a priori partition where each element of the partition represents a group of agents with close interests or ideas. The power is determined in two steps, first as a problem among the unions and later, inside each one, the power of each agent is determined. Proximity relations extend this model considering leveled closeness among the agents. In this paper we analyze a version of the Banzhaf value for games with a proximity relation and we show the interest of this value by applying it to the allocation of the power of the political groups in the European Parliament.

Random Forests are powerful classification and regression tools that are commonly applied in machine learning and image processing. In the majority of random classification forests algorithms, the Gini index and the information gain ratio are commonly used for node splitting. However, these two kinds of node-split methods may pay less attention to the intrinsic structure of the attribute variables and fail to find attributes with strong discriminate ability as a group yet weak as individuals. In this paper, we propose an innovative method for splitting the tree nodes based on the cooperative game theory, from which some attributes with good discriminate ability as a group can be learned. This new random forests algorithm is called Cooperative Profit Random Forests (CPRF). Experimental comparisons with several other existing random classification forests algorithms are carried out on several real-world data sets, including remote sensing images. The results show that CPRF outperforms other existing Random Forests algorithms in most cases. In particular, CPRF achieves promising results in ocean front recognition.

The author analyzes the power distribution among political groups in the European Parliament from 1979 till 2014. Agent’s voting power is treated as the capability to influence the result of collective decision-making process. Often the voting power is not equal to votes share the agent possess. We obtain the values of classical and modified power indices for European political groups and compare their performance. The Banzhaf power index is employed to measure European political groups’ a priori power that does not impose any restrictions on coalition formation, and power indices that take into account groups’ preferences in coalition formation in order to measure actual voting power. Classical power indices are calculated using only votes’ distribution in the committee and a quota to pass a decision. Pairwise preferences allow to have respect to different willingness of groups to form a single coalition. In this paper preferences are modeled using the roll-call data from the votings on important issues of European politics. The less popular is the MP’s position, the smaller is the number of coalitions she can enter and the less is the voting power. We demonstrate that factions can exploit their position and acquire higher relative power than both share of seats and the Banzhaf index measures, and the Liberal group of the European Parliament forming coalitions with both partners from the left and the right is the case. Conversely, hard restrictions on formation of coalitions due to ideological differences can lead to less actual voting power than a priori.

Weighted voting games are important in multiagent systems because of their usage in automated decision making. However, they are not immune from the vulnerability of false-name manipulation by strategic agents that may be present in the games. False-name manipulation involves an agent splitting its weight among several false identities in anticipation of power increase. Previous works have considered false-name manipulation using the well-known Shapley–Shubik and Banzhaf power indices. Bounds on the extent of power that a manipulator may gain exist when it splits into k = 2 false identities for both the Shapley–Shubik and Banzhaf indices. The bounds when an agent splits into k > 2 false identities, until now, have remained open for the two indices. This article answers this open problem by providing four nontrivial bounds when an agent splits into k > 2 false identities for the two indices. Furthermore, we propose a new bound on the extent of power that a manipulator may gain when it splits into several false identities in a class of games referred to as excess unanimity weighted voting games. Finally, we complement our theoretical results with empirical evaluation. Results from our experiments confirm the existence of beneficial splits into several false identities for the two indices, and also establish that splitting into more than two false identities is qualitatively different than the previously known splitting into exactly two false identities.

Critical node discovery plays a vital role in assessing the vulnerability of a computer network to malicious attacks and failures and provides a useful tool with which one can greatly improve network security and reliability. In this paper, we propose a new metric to characterize the criticality of a node in an arbitrary computer network which we refer to as the Combined Banzhaf & Diversity Index (CBDI). The metric utilizes a diversity index which is based on the variability of a node׳s attributes relative to its neighbours and the Banzhaf power index which characterizes the degree of participation of a node in forming shortest paths. The Banzhaf power index is inspired from the theory of voting games in game theory. The proposed metric is evaluated using analysis and simulations. The criticality of nodes in a network is assessed based on the degradation in network performance achieved when these nodes are removed. We use several performance metrics to evaluate network performance including the algebraic connectivity which is a spectral metric characterizing the connectivity robustness of the network. Extensive simulations in a number of network topologies indicate that the proposed CBDI index chooses more critical nodes which, when removed, degrade network performance to a greater extent than if critical nodes based on other criticality metrics were removed.

MacMahon’s Partition Analysis (MPA) is a combinatorial tool used in partition analysis to describe the solutions of a linear diophantine system. We show that MPA is useful in the context of weighte...

© Oxford University Press and Foundation for the European Review of Agricultural Economics 2015.We consider how cost heterogeneity and market power affect voting power in producer referenda for mandatory agricultural marketing organisations with generic promotion programmes in the United States. We measure voting power using the Banzhaf Power Index and propose a new version of this index based on the profit-maximising theory of the firm that provides an improved estimate of voting power. Examining several types of demand shifts and voting rules, we find that both Banzhaf Power and our new measure vary considerably depending on the market structure and level of cost heterogeneity.

Stochastic Asymmetric Blotto Games: Some New Results John Duffy ∗ Abstract. Blotto games. Alexander Matros † We develop some new theoretical results for stochastic asymmetric Keywords: Colonel Blotto game, Contests, Resource Allocation, Lotteries. JEL Classification Nos. C72, C73, D72, D74. 1. Introduction The Colonel Blotto game (Borel 1921), is a two-player non-cooperative game in which players decide how to allocate their given resources across  battlefields. In Borel’s original version of this game, the player who allocates the most resources to any given battlefield wins that battlefield with certainty. The players’ objective function is either to maximize the sum of the value of the battlefields won, or to win a majority value of the  battlefields. In this paper we study stochastic asymmetric versions of the Blotto game under both of these objective functions. In an “asymmetric Blotto” game, the values of the  battlefields may differ from one another though these different values are common to all players. In the “stochastic asymmetric” version of the Blotto game, the deterministic rule for determining which player wins each battlefield is replaced by a lottery contest success function where the chances of winning a given battlefield are increasing with the amount of resources devoted to that battlefield. This stochastic lottery specification makes the payoff function continuous; as a result, if a Nash equilibrium exists, it is unique and in pure strategies, as opposed to the multiplicity of (typically mixed strategy) equilibria that arise in deterministic versions of the Blotto game. There are two main theoretical papers about stochastic asymmetric Blotto games. 1 The first one, Friedman (1958), considers two players who seek to maximize their expected total payoff. We show that Friedman’s result can be extended to any number of players. The second paper, Lake (1979), was the first to study the stochastic asymmetric “majority rule” Blotto game. 2 This version of the game is particularly relevant to understanding electoral competitions in two party systems, e.g., the electoral college system for electing the U.S. president. Lake studied only the case of equal budget constraints. We show that if players’ budgets are the same (as in Lake) or if they are sufficiently similar and the number of items (battlefields) is not too large, then resource allocation under the majority rule version of the stochastic, asymmetric Blotto game is proportional to the Banzhaf power index for each item, while more generally, resource allocation for a particular item will not be proportional to each item’s Banzhaf power index. Our findings thus generalize those of Lake (1979). Department of Economics, University of California, Irvine. Email: duffy@uci.edu. Moore School of Business, University of South Carolina and Lancaster University Management School. Email: alexander.matros@gmail.com See Kovenock and Roberson (2012) for a broader survey of the Blotto game literature. We discovered Lake’s (1979) paper only after we had completed our analysis. His proofs are different from ours, but his main result coincides with our prediction for the case of equal budgets. We thank Steve Brams for providing this reference.

Recent years have witnessed an increased interest, by competition agencies, in assessing the competitive effects of partial acquisitions. We propose a generalization to a partial horizontal acquisition setting of the two most traditional indicators used to screen unilateral anti-competitive effects: the Helfindahl- Hirschman Index and the Gross Upward Price Pressure Index. The proposed generalized indicators can deal with all types of acquisitions that may lessen competition in the industry: acquisitions by owners that are internal to the industry (rival firms) and engage in cross-ownership, as well as acquisitions by owners that are external to the industry and engage in common-ownership. Furthermore, these indicators can deal with direct and indirect acquisitions, which may or may not correspond to control, and nest full mergers as a special case. We provide an empirical application to several acquisitions in the wet shaving industry. The results seem to suggest that (i) a full merger induces higher unilateral anti-competitive effects than a partial controlling acquisition involving the same firms, (ii) a partial controlling acquisition induces higher unilateral anti-competitive effects than a partial non-controlling acquisition involving the same firms and the same financial stakes, and (iii) an acquisition by owners that are internal to the industry induces higher unilateral anti-competitive effects than an acquisition (involving the same firms and the same stakes) by external owners that participate in more than one competitor firm. JEL Classification: L13, L41, L66 Keywords: Antitrust, Partial Horizontal Acquisitions, Oligopoly, Screening Indicators, HHI, GUPPI

Random forests algorithms have been widely used in many classification and regression applications. However, the theory of random forests lags far behind their applications. In this paper, we propose a novel random forests classification algorithm based on cooperative game theory. The Banzhaf power index is employed to evaluate the power of each feature by traversing possible feature coalitions. Hence, we call the proposed algorithm Banzhaf random forests (BRFs). Unlike the previously used information gain ratio, which only measures the power of each feature for classification and pays less attention to the intrinsic structure of the feature variables, the Banzhaf power index can measure the importance of each feature by computing the dependency among the group of features. More importantly, we have proved the consistency of BRFs, which narrows the gap between the theory and applications of random forests. Extensive experiments on several UCI benchmark data sets and three real world applications show that BRFs perform significantly better than existing consistent random forests on classification accuracy, and better than or at least comparable with Breiman's random forests, support vector machines (SVMs) and k-nearest neighbors (KNNs) classifiers.

The portmanteau Brexit describes the withdrawal of the United Kingdom (UK) from the European Union (EU) which will cause a shift of power in the European institutions. The departure of one of the largest EU members will affect the voting power of member countries in the European Council significantly. This Council is the central hub of political decision making in the EU, defining the overall political direction and priorities and setting the policy agenda for the entirety of the EU. Using the Banzhaf power index, we have measured the voting power before and after the Brexit and analyzed the increasing power of the members of the Visegrád Group compared to other European states. We have found that there is growth in the voting power of all Visegrád states, with Poland experiencing the biggest increase. However, the extent by which the Visegrád Group will profit from this statistically growing power depends on the coordination of their voting behavior in the future.

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.

Nowadays, enlargement is one of the most important issue for European Union. The situation that makes this issue important is the influence of membership of the candidate states on the voting power distribution. With the enlargements, European Union decision-making processes were regulated several times. The last regulation to the European Union voting system was brought by the Treaty of Lisbon. In this study, voting power distribution in Council of the European Union is evaluated for candidate states and member states considering the acts adopted by the Treaty of Lisbon and determined as candidate states determined as Turkey, Croatia, Iceland, Former Yugoslav Republic of Macedonia. In the analysis, member states are grouped the according to the enlargement of the EU and the effect of the states that take part in these groups on the decision and changes of voting power are calculated using Banzhaf power index for power measurement.

We consider a simple model of cooperation among agents called Coalitional Skill Games (CSGs). This is a restricted form of coalitional games, where each agent has a set of skills that are required to complete various tasks. Each task requires a set of skills in order to be completed, and a coalition can accomplish the task only if the coalition@?s agents cover the set of required skills for the task. The gain for a coalition depends only on the subset of tasks it can complete. We consider the computational complexity of several problems in CSGs, such as testing if an agent is a dummy or veto agent, computing the core and core-related solution concepts, and computing power indices such as the Shapley value and Banzhaf power index.

This paper is a twofold contribution. First, it contributes to the problem of enumerating some classes of simple games and in particular provides the number of weighted games with minimum and the number of weighted games for the dual class as well. Second, we focus on the special case of bipartite complete games with minimum, and we compare and rank these games according to the behavior of some efficient power indices of players of type 1 (or of type 2). The main result of this second part establishes all allowable rankings of these games when the Shapley-Shubik power index is used on players of type 1., 26 pages, 2 figures, to appear in Annals of Operations Research

In a recent article, Riggs et al. (2009) aim to measure the ‘Electoral College winner's advantage’—in particular, the extent to which the winner’s electoral vote margin of victory is magnified as a result of (i) the ‘two electoral vote add-on’ given to each state and (ii) the ‘winner-take-all’ mode of casting state electoral votes. Their results are based on two sets of one million simulated two-candidate elections. This note has two purposes. The first is to demonstrate that RHR’s simulation estimates can be calculated precisely using the theory of voting power measurement. The second is to correct several flaws in RHR’s analysis, the most substantial of which pertains to the effect of the two electoral vote add-on, which actually has a negative effect on the winner’s advantage.

Many multiagent domains where cooperation among agents is crucial to achieving a common goal can be modeled as coalitional games. However, in many of these domains, agents are unequal in their power to affect the outcome of the game. Prior research on weighted voting games has explored power indices, which reflect how much "real power" a voter has. Although primarily used for voting games, these indices can be applied to any simple coalitional game. Computing these indices is known to be computationally hard in various domains, so one must sometimes resort to approximate methods for calculating them. We suggest and analyze randomized methods to approximate power indices such as the Banzhaf power index and the Shapley---Shubik power index. Our approximation algorithms do not depend on a specific representation of the game, so they can be used in any simple coalitional game. Our methods are based on testing the game's value for several sample coalitions. We show that no approximation algorithm can do much better for general coalitional games, by providing lower bounds for both deterministic and randomized algorithms for calculating power indices. We also provide empirical results regarding our method, and show that it typically achieves much better accuracy and confidence than those required.

A voting power analysis was made to estimate the power of IMF members within the Executive Board and the Fund in general through the existing constituency system. For this purpose we introduced two absolute power indices extending the classical Penrose (non-normalized Banzhaf) and Coleman (''the power of the body to act'') indices by taking into account the members' preferences to coalesce, based on their membership in the currently functioning political-economic blocs with varying degrees of economic integration and members' regional proximity as well. We considered an indirect voting system where countries are contending for power at two levels: within their constituencies and within the Executive Board. Accordingly, each of the proposed power indices determines a member's power at the appropriate level. The total power of a country is then defined by a compound power index, which is the product of the above-mentioned indices. Experiments displayed the robustness of the indices to small variations in the weight coefficients assigned to the preferences considered. Using our approach, we analysed countries' voting power within the Executive Board under the three cases of majority voting adopted by the IMF, namely, a simple majority and qualified majorities of 70% and 85% of the total votes. The power indices for all members (except those with voting rights suspended) and their constituencies were calculated and compared with the Penrose and Banzhaf indices. The results show that members' absolute voting power as measured by our indices produces a power distribution different from that of Penrose and Banzhaf power in both directions. The constituencies' relative voting power is rather close to their IMF voting weights for the case of the simple majority voting and tends to redistribute between constituencies as the threshold number of votes for a decision to be taken increases; thus, for the 85% supermajority, the maximum power share is that of the Netherlands's constituency and not that of the US's, as the Banzhaf power index suggests.

The generating functions representative of the Banzhaf power index of the parties, or political groups of electoral bodies, is applied to the case where particular parties simultaneously (in pairs) cannot be part of any coalition. For such types of restrictions on coalition formation, the two-stage procedure for evaluating the Banzhaf index using generating functions is presented. Throughout the steps of the procedure the adjusted Banzhaf index generating function polynomial is transformed to the standard representation used for the calculation of this index. The presented approach is discussed and the procedure is illustrated by several examples.

The International Monetary Fund is one of the largest international organizations using a weighted voting system. The weights of its 188 members are determined by a fixed amount of basic votes plus some extra votes for so-called Special Drawing Rights (SDR). On January 26, 2016, the conditions for the SDRs were increased at the 14th General Quota Review, which drastically changed the corresponding voting weights. However, since the share of voting weights in general is not equal to the influence, of a committee member on the committees overall decision, so-called power indices were introduced. So far the power distribution of the IMF was only computed by either approximation procedures or smaller games than then entire Board of Governors consisting of 188 members. We improve existing algorithms, based on dynamic programming, for the computation of power indices and provide the exact results for the IMF Board of Governors before and after the increase of voting weights. Tuned low-level details of the algorithms allow the repeated routine with sparse computational resources and can of course be applied to other large voting bodies. It turned out that the Banzhaf power shares are rather sensitive to changes of the quota., Comment: 19 pages, 2 figures, 13 tables

In this paper, we present methods to compute Banzhaf–Coleman and Shapley–Shubik power indices for weighted majority games when some players are incompatible. We use the so-called generating functions as a tool. Ministerio de Ciencia e Innovación | Ref. MTM2011-27731-C03

In this paper, we suppose that players join in coalitions and form a coalition configuration, and we provide a generalization of the normalized Banzhaf-Coleman (1965, 1971) index to this framework.

This paper develops a new axiomatic characterization of the Banzhaf index of power using four axioms from four different contributions to the area. A nice feature of the characterization is independence of the axioms showing importance of each of them in the exercise.

The decisiveness index introduced in this paper is designed to provide a normalized measure of the agility of all simple games, primarily viewed as collective decision-making mechanisms. We study the mathematical properties of the index and derive different axiomatic characterizations for it. Moreover, a close relationship is shown to the Banzhaf index of power––for which twice the decisiveness index plays the role of potential function––that gives rise to an effective computational procedure. Some real-world examples illustrate the usefulness of the decisiveness index, together with the Banzhaf power index, in applications to political science.

This paper studies the effects of a voting scheme change on the stock market prices of Electrolux and SKF AB using standard event study methodology and a clinical approach. The economic effect of the voting scheme change is assessed using the market model. We investigate the loss of control due to the change in the voting scheme. The degree of change in power is calculated using the Shapley-Shubik power index and the Banzhaf power index. There is a wealth transfer from the high vote shareholders to the low vote shareholders in the process since in both cases the high vote shareholders required no compensation. We expect share price to respond positively to an announcement of a forthcoming voting scheme change, due to the reduced power discount and corporate governance improvement. The magnitude of the response on the event day depends also on the information structure of the period leading up to the announcement. A bigger effect on the value of the firm is to be expected if the voting powers of the major owner(s) shift away from absolute control to moderate control, indicating a significant change in governance pattern.

In this paper we first evaluate thirteen seat distributions inthe Second Chamber of the Dutch parliament by means of severalindices of voting power. Subsequently, we search for theoccurrence of the paradox of redistribution, the paradox ofnew members, and the paradox of large size for each powerindex. The indices used are the Shapley-Shubik index, thenormalized Banzhaf index, the Penrose-Banzhaf index, theHoller index, and the Deegan-Packel index.

The Shapley–Shubik power index in a voting situation depends on the number of orderings in which each player is pivotal. The Banzhaf power index depends on the number of ways in which each voter can effect a swing. If there are n players in a voting situation, then the function which measures the worst case running time for computing these indices is in O( n 2 n ). We present a combinatorial method based in generating functions to compute these power indices efficiently in weighted double or triple majority games and we study the time complexity of the algorithms. Moreover, we calculate these power indices for the countries in the Council of Ministers of the European Union under the new decision rules prescribed by the Treaty of Nice.

In this paper we introduce a new coalitional value in the context of TU games with an a priori system of unions, which it is called the symmetric coalitional Banzhaf value. This value satisfies the property of symmetry in the quotient game, the quotient game property, and it is a coalitional value of Banzhaf. Several characterizations are provided and two political examples illustrate the differences with respect to the Owen value and the Banzhaf–Owen value.

There are many situations in which different groups make collective decisions by committee voting, with each group represented by a single person. A natural question is what voting system such a committee should use. Concepts based on voting power provide guidelines for this choice. The two most prominent concepts require the Banzhaf power index to be proportional to the square root of group size or the Shapley-Shubik power index to be proportional to group size. Instead of studying the choice of voting systems based on such theoretical concepts, in this paper, I ask which systems individuals actually prefer. To answer this question, I design a laboratory experiment in which participants choose voting systems. I find that people behind the veil of ignorance prefer voting systems following the rule of proportional Shapley-Shubik power; in front of the veil subjects pr efer voting systems benefiting their own group. Participants' choices can only partially be explained by utility maximization or other outcome based concepts.

In this paper, we prove that both problems for calculating the Banzhaf power index and the Shapley–Shubik power index for weighted majority games are NP-complete.