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A method for evaluating the distribution of power in a committee system

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  • Yale University, Santa Fe Institute
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... However, for all chosen characteristic functions, the obtained values respect the key axioms of Shapley values [Shapley 1953]. In the case of game theory, concretely when measuring a priori voting power, Shapley values are instantiated starting from a given characteristic function, which was first proposed in 1954 by Shapley & Shubik [Shapley and Shubik 1954]. The same characteristic function has been used, explicitly or implicitly, in most other proposals of power indices, i.e. measures of relative voting power. ...
... The novel characteristic function is inspired by those commonly used in game theory [Shapley and Shubik 1954]. The motivation then and now is to assign importance to the elements (features or voters) which are critical for changing the value of a decision of interest. ...
... Also, the following result is immediate. 12 We adopt the concept of critical elements from game theory, that can be traced at least to the work of Shapley&Shubik on voting power [Shapley and Shubik 1954]. Clearly, Crit( , S) holds iff Δ (S) holds (see (6)). ...
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Recent work demonstrated the existence of critical flaws in the current use of Shapley values in explainable AI (XAI), i.e. the so-called SHAP scores. These flaws are significant in that the scores provided to a human decision-maker can be misleading. Although these negative results might appear to indicate that Shapley values ought not be used in XAI, this paper argues otherwise. Concretely, this paper proposes a novel definition of SHAP scores that overcomes existing flaws. Furthermore, the paper outlines a practically efficient solution for the rigorous estimation of the novel SHAP scores. Preliminary experimental results confirm our claims, and further underscore the flaws of the current SHAP scores.
... In cooperative game theory, various kinds of power indexes are used to measure the influence that a given player has on the outcome of the game or to define a way of sharing the benefits of the game among the players. The best known power indexes are due to Shapley [21,22] and Banzhaf [2,8]. ...
... where C denotes a random coalition. Notice that formula (22) can also be obtained from (20) by using the random indicator vector X = (X 1 , . . . , X n ). ...
... Proof. Partitioning T ⊆ N into K ⊆ N ∖ S and L ⊆ S, we can rewrite the sum in (22) as ...
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The Banzhaf power index was introduced in cooperative game theory to measure the real power of players in a game. The Banzhaf interaction index was then proposed to measure the interaction degree inside coalitions of players. It was shown that the power and interaction indexes can be obtained as solutions of a standard least squares approximation problem for pseudo-Boolean functions. Considering certain weighted versions of this approximation problem, we define a class of weighted interaction indexes that generalize the Banzhaf interaction index. We show that these indexes define a subclass of the family of probabilistic interaction indexes and study their most important properties. Finally, we give an interpretation of the Banzhaf and Shapley interaction indexes as centers of mass of this subclass of interaction indexes.
... The measure of voting power in assemblies of voters has attracted the interest of researchers since at least the work of L. Penrose in the 1940s (Penrose 1946), with important contributions in the following decades (Shapley and Shubik 1954;Banzhaf III 1965). More recently, measures of importance have been studied in other domains, that include inconsistent knowledge bases Konieczny 2006, 2010;Raddaoui, Straßer, and Jabbour 2023), intensity of attacks in argumentation (Amgoud, Ben-Naim, and Vesic 2017), set covering (Gusev 2020(Gusev , 2023, database management ), but also explainable artificial intelligence (XAI) (Lundberg and Lee 2017; Biradar et al. 2024;. ...
... Since the 1940s (Penrose 1946), there has been interest in assigning relative importance to voters of weighted voting games; these measures are referred to as power indices (Felsenthal and Machover 1998). In this paper, we focus on a few well-known power indices, namely those of Shapley-Shubik (Shapley and Shubik 1954), Banzhaf (Banzhaf III 1965) and Deegan-Packel (Deegan and Packel 1978). ...
... Existing measures of relative importance of a voter i ∈ N (e.g., (Shapley and Shubik 1954;Banzhaf III 1965;Deegan and Packel 1978)) analyze all possible coalitions S ⊆ N . For each coalition S ⊆ N , one accounts for the contribution of i for the coalition, i.e., ∆ i (S), weighted by a factor ς(S), that depends on the power index being considered. ...
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Measures of voting power have been the subject of extensive research since the mid 1940s. More recently, similar measures of relative importance have been studied in other domains that include inconsistent knowledge bases, intensity of attacks in argumentation, different problems in the analysis of database management, and explainability. This paper demonstrates that all these examples are instantiations of computing measures of importance for a rather more general problem domain. The paper then shows that the best-known measures of importance can be computed for any reference set whenever one is given a monotonically increasing predicate that partitions the subsets of that reference set. As a consequence, the paper also proves that measures of importance can be devised in several domains, for some of which such measures have not yet been studied nor proposed. Furthermore, the paper highlights several research directions related with computing measures of importance.
... The power or influence of a player in a weighted voting game is not proportional to its weight. Many power indices have been proposed in the literature, as ways to measure a player's voting power in a voting game [2,5,12,25]. Among these power indices, Shapley-Shubik and Banzhaf-Coleman indices are the most reputed. ...
... This value represents the fraction of power for a voter i according to bargaining model of that value [25]. The Shapley-Shubik index is normalized since ...
... The constraints (54) replace the lines (21) to (24) and (28) to (30). The model keeps the constraints (12) to (17), (20), (25) to (27), (32) to (34) of the ILP model of Figure 1. Only the expression of the objective function of the model of Figure 5 changes for the Chebyshev norm. ...
... In power indices, voter importance measures how likely each voter is to be critical for a set of voters to represent a winning coalition (e.g. see (Shapley and Shubik, 1954)). In contrast, the existing definitions of SHAP scores overlook the criticality of features, and instead measure feature contributions starting from the expected values of the ML model. ...
... Shapley-Shubik (Shapley and Shubik, 1954), Banzhaf (Banzhaf III, 1965), Deegan-Packel (Deegan and Packel, 1978), Johnston (Johnston, 1978), Holler-Packel (Holler and Packel, 1983), Andjiga (Andjiga, Chantreuil, and Lepelley, 2003), and the Responsibility index (Biradar et al., 2024) (which can be related with earlier work (Chockler, Halpern, and Kupferman, 2008)). ...
... Moreover, early work on voting power was clear about how to measure the importance of each voter in voting games. Indeed, and quoting from (Shapley and Shubik, 1954): "Our definition of the power of an individual member depends on the chance he has of being critical to the success of a winning coalition". All of the power indices briefly summarized above explicitly measure the importance of an individual member (in our case a feature) at being critical. ...
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A central goal of eXplainable Artificial Intelligence (XAI) is to assign relative importance to the features of a Machine Learning (ML) model given some prediction. The importance of this task of explainability by feature attribution is illustrated by the ubiquitous recent use of tools such as SHAP and LIME. Unfortunately, the exact computation of feature attributions, using the game-theoretical foundation underlying SHAP and LIME, can yield manifestly unsatisfactory results, that tantamount to reporting misleading relative feature importance. Recent work targeted rigorous feature attribution, by studying axiomatic aggregations of features based on logic-based definitions of explanations by feature selection. This paper shows that there is an essential relationship between feature attribution and a priori voting power, and that those recently proposed axiomatic aggregations represent a few instantiations of the range of power indices studied in the past. Furthermore, it remains unclear how some of the most widely used power indices might be exploited as feature importance scores (FISs), i.e. the use of power indices in XAI, and which of these indices would be the best suited for the purposes of XAI by feature attribution, namely in terms of not producing results that could be deemed as unsatisfactory. This paper proposes novel desirable properties that FISs should exhibit. In addition, the paper also proposes novel FISs exhibiting the proposed properties. Finally, the paper conducts a rigorous analysis of the best-known power indices in terms of the proposed properties.
... To this end so-called power indices where introduced. For the binary case, the Shapley-Shubik index, introduced in [17], is one of the most commonly used power indices. Besides an axiomatic foundation of the Shapley-Shubik index [2], there is also a picturesque description: Assume that the voters express their support for a proposal one after the other. ...
... A classical question in this context asks for the influence of a committee member (or voter) on the aggregated decision. For a simple gameṽ : 2 N → {0, 1} the so-called Shapley-Shubik index, see [17], of player 1 ≤ i ≤ n inṽ is given bỹ ...
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In 1996 Dan Felsenthal and Mosh\'e Machover considered the following model. An assembly consisting of n voters exercises roll-call. All n! possible orders in which the voters may be called are assumed to be equiprobable. The votes of each voter are independent with expectation 0<p<10<p<1 for an individual vote {\lq\lq}yea{\rq\rq}. For a given decision rule v the \emph{pivotal} voter in a roll-call is the one whose vote finally decides the aggregated outcome. It turned out that the probability to be pivotal is equivalent to the Shapley-Shubik index. Here we give an easy combinatorial proof of this coincidence, further weaken the assumptions of the underlying model, and study generalizations to the case of more than two alternatives.
... Finally, the issues of bargaining and (inter)governmental coalitions were addressed in the work of Shapley and Shubik (1954), which also constitutes an application of the theory of coalitions mentioned above and which focuses on decision-making power in organisations within the framework of game theory. Here the coalition is formed on the basis of the preferences of the actors and the question is how power is shared between them with regard to their marginal contribution to the coalition (Péreau 2009;Shapley 1953). ...
... Once the prerequisites are established, the construction of each simple index becomes possible and is inspired by the methodology for calculating voting power proposed by Shapley and Shubik (1954), also used by several authors such as Abadoma et Eze (2023), Dia and Kamwa (2020), Abidi et al. (2020), Interactions at the WHO are thus defined as the game defined as follows: let there be a game consisting of n players i (countries), each with a weight Π i (i = 1, 2… n) with a quota q the minimum required to take a given action. The n countries belong to a large set N (WHO) and any subset S of N is a coalition (the different regions of the organisation). ...
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This article analyses the role of the economic coalition in improving the bargaining power of countries in the African region at the World Health Organisation (WHO), using data from the World Health Assembly (WHA) from 2010 to 2019. The study is based on the observation that this region’s bargaining power is the weakest in the organisation. This bargaining power is obtained by linearly combining two simple indices (calculated using the Shapley–Shubik method). He therefore analyses two ways of improving this indicator: a coalition of countries by regional grouping within the meaning of the WHO, then a reconfiguration of this coalition on the basis of economic unions. The results show that by forming intra-regional alliances, Africa increases its bargaining power and gains an average of two places on the podium. This position is even better when coalitions are formed on the basis of economic unions. This leads to an increase in this region’s bargaining power of twice that of the initial coalition, and also improves its position on the podium.
... Consequently, concerning the cooperative willingness of the agents and capturing their strategic interactions are the key instrument to settle water geopolitical conflict. The Shapley-Shubik power index method [36] can quantitatively simulate agent's cooperation willingness, providing a good mathematical framework for identifying stable solutions for shared resources allocations. Since Loehman et al. [37] used this index method to select stable solutions for wastewater treatment cost allocation, it has been widely applied and gained a good reputation in the water resources management [1,17,23,[38][39][40]. ...
... The Shapley-Shubik power index [36], defined as the ratio of an agent's loss due to its departure from the grand coalition to the sum of all other agents' losses after they leave the coalition [37], proved to be an appropriate method for simulating agent's cooperation willingness and identifying the most stable solution in the cooperative game-based water management literature [1,23,[38][39][40]. However, this method still fails to find fair and self-enforceable continuous solutions for transboundary water allocation due to agent's asymmetric negotiation power and their disagreement utility points are not simultaneously concerned and included. ...
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Cooperative and self-enforceable water allocation is a key instrument to manage geopolitical conflict induced by water scarcity, which necessitates concerning the cooperative willingness of the agents and considering their heterogeneity in geography, climate, hydrology, environment and social economy. Based on a multi-indicator system that contains asymmetric information on water volume contribution, current water consumption, water economic efficiency and efforts for eco-environmental protection, this study proposed a water allocation framework by combining the asymmetric power index approach with bankruptcy theory for solving the transboundary water allocation problem under scarcity. The proposed method was applied to the Yellow River Basin in northern China, which is mainly shared by nine provincial districts and frequently suffers from severe water shortage, and its results were compared with six alternative methods. The results highlight the necessity of quantifying agent’s willingness to cooperate under the background of asymmetric negotiation power when making decisions on transboundary water allocations. The proposed method allows to perform transboundary water allocations through simultaneous consideration of the agent’s cooperation willingness, asymmetric negotiation power as well as disagreement allocation points, which ensure the stability, fairness and self-enforceability of allocation results. Therefore, it can offer practical and valuable decision-making insights for transboundary water management under water scarcity.
... Consequently, the cooperative willingness of the agents and capturing their strategic interactions are the key instruments to settle water geopolitical conflict. The Shapley-Shubik power index method [37] can quantitatively simulate agents' willingness to cooperate, providing a good mathematical framework for identifying stable solutions for the allocation of shared resources. Since Loehman et al. [38] used this index method to select stable solutions for wastewater treatment cost allocation, it has been widely applied and gained a good reputation in water resources management [1,16,24,[39][40][41]. ...
... where u i x * i is the utility function value of the agent i; x * i is the water allocation of the agent i (decision variable); and wc i is the water claim of the agent i. The Shapley-Shubik power index [37], defined as the ratio of an agent's loss due to its departure from the grand coalition to the sum of all other agents' losses after they leave the coalition [38], proved to be an appropriate method for simulating agents' willingness to cooperate and identifying the most stable solution in the cooperative game-based water management literature [1,24,[39][40][41]. However, this method still fails to find fair and self-enforceable continuous solutions for transboundary water allocation due to agents' asymmetric negotiation power and their disagreement utility points not being simultaneously considered and included. ...
Article
Full-text available
Cooperative and self-enforceable water allocation is a key instrument to manage geopolitical conflict induced by water scarcity, which necessitates the cooperative willingness of the agents and considers their heterogeneity in geography, climate, hydrology, environment and social economy. Based on a multi-indicator system that contains asymmetric information on water volume contribution, current water consumption, water economic efficiency and efforts for eco-environmental protection, this study proposed a water allocation framework by combining the asymmetric power index approach with bankruptcy theory for solving the transboundary water allocation problem under scarcity. The proposed method was applied to the Yellow River Basin in northern China, which is mainly shared by nine provincial districts and frequently suffers from severe water shortages, and its results were compared with six alternative methods. The results highlight the necessity of quantifying agents’ willingness to cooperate under the condition of asymmetric negotiation power when making decisions on transboundary water allocations. The proposed method allows for transboundary water allocations through simultaneous consideration of the agent’s willingness to cooperate and asymmetric negotiation power, as well as disagreement allocation points, which ensure the stability, fairness and self-enforceability of allocation results. Therefore, it can offer practical and valuable decision-making insights for transboundary water management under water scarcity.
... Especially important is the analysis of how significant players are in WVGs, i.e., what they contribute to forming winning coalitions. Their influence can be measured by so-called power indices among which some well-known examples are: the Shapley-Shubik index due to Shapley and Shubik [27], the probabilistic Penrose-Banzhaf index due to Dubey and Shapley [9], and also the normalized Penrose-Banzhaf index due to Penrose [21] and Banzhaf [2]. We are concerned with the former two. ...
... One of them is the probabilistic Penrose-Banzhaf power index, which was introduced by Dubey and Shapley [9] as an alternative to the original normalized Penrose-Banzhaf index [21,2]. The other index we will study is the Shapley-Shubik power index, introduced by Shapley and Shubik [27] as follows: We assume familiarity with the basic concepts of computational complexity theory, such as the well-known complexity classes P (deterministic polynomial time), NP (nondeterministic polynomial time), and PP (probabilistic polynomial time [14]). NP PP is the class of problems that can be solved by an NP oracle Turing machine accessing a PP oracle. ...
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Weighted voting games are a well-known and useful class of succinctly representable simple games that have many real-world applications, e.g., to model collective decision-making in legislative bodies or shareholder voting. Among the structural control types being analyzing, one is control by adding players to weighted voting games, so as to either change or to maintain a player's power in the sense of the (probabilistic) Penrose-Banzhaf power index or the Shapley-Shubik power index. For the problems related to this control, the best known lower bound is PP-hardness, where PP is "probabilistic polynomial time," and the best known upper bound is the class NP^PP, i.e., the class NP with a PP oracle. We optimally raise this lower bound by showing NP^PP-hardness of all these problems for the Penrose-Banzhaf and the Shapley-Shubik indices, thus establishing completeness for them in that class. Our proof technique may turn out to be useful for solving other open problems related to weighted voting games with such a complexity gap as well.
... A negative vote or a "veto" by a permanent member prevents adoption of a proposal, even if it has received the required votes. As an alternative of the usual simple game used to model this voting system (see Shapley and Shubik, 1954, for instance), we can define the game with diversity constraint (N, v, B, d) where ...
... Any value f restricted to SGD is called an index and assigns to each game (N, v, B, d) ∈ SGD and each player i ∈ N a positive real number f i (N, v, B, d) which can be seen as the power of i or her influence in (N, v, B, d). Following the literature, the Owen value restricted to SGD can be called the Owen index and the Shapley value on SGD can be called the Shapley-Shubik index (Shapley and Shubik, 1954). ...
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A cooperative game with diversity constraints is given by a cooperative game, a coalition structure which partitions the set of players into communities, and a vector of integers specifying, for each community, the minimal number of its members that a coalition must possess to be considered as diverse. We provide axioms for a value on the class of such cooperative games with diversity constraints. Some combinations of axioms characterize two values inspired by the Shapley value (Shapley, 1953) and the Owen value (Owen, 1977) for games with a coalition structure. More specifically, the Diversity Owen value is characterized as the Owen value of the diversity-restricted game with a coalition structure, where the diversity-restricted game assigns a null worth to a coalition if it does not meet the diversity requirements or its original worth otherwise. Similarly, the Diversity Shapley value is characterized as the Shapley value of the diversity-restricted game (without coalition structure). Some of our axiomatic characterizations can be adapted to the class of simple games by replacing the Additivity axiom by the Transfer axiom (Dubey, 1975).
... The Shapley value [20] is a central solution concept for cooperative games based on a system of axioms. For simple games it is also referred to as the Shapley-Shubik power index [21]. For k with 1 ≤ k ≤ n and an ordering (π(1), π(2), . . . ...
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Let f ⁣:{0,1}n{0,1}f\colon \{0,1\}^n\to \{0,1\} be a monotone Boolean functions, let ψk(f)\psi_k(f) denote the Shapley value of the kth variable and bk(f)b_k(f) denote the Banzhaf value (influence) of the kth variable. We prove that if we have ψk(f)t\psi_k(f) \le t for all k, then the threshold interval of f has length O(1log(1/t))\displaystyle O \left(\frac {1}{\log (1/t)}\right). We also prove that if f is balanced and bk(f)tb_k(f) \le t for every k, then maxkψk(f)O(loglog(1/t)log(1/t))\displaystyle \max_{k} \psi_k(f) \le O\left(\frac {\log \log (1/t)}{\log(1/t)}\right) .
... Shapley values were derived in 1952 by L. S. Shapley [42] and then later applied in the context of cooperative game theory by Shapley and Shubik [41]. Shapley values compute the average marginal contribution of a feature value across all possible feature subsets. ...
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The increasing digitalisation of multi-modal data in medicine and novel artificial intelligence (AI) algorithms opens up a large number of opportunities for predictive models. In particular, deep learning models show great performance in the medical field. A major limitation of such powerful but complex models originates from their ’black-box’ nature. Recently, a variety of explainable AI (XAI) methods have been introduced to address this lack of transparency and trust in medical AI. However, the majority of such methods have solely been evaluated on single data modalities. Meanwhile, with the increasing number of XAI methods, integrative XAI frameworks and benchmarks are essential to compare their performance on different tasks. For that reason, we developed BenchXAI, a novel XAI benchmarking package supporting comprehensive evaluation of fifteen XAI methods, investigating their robustness, suitability, and limitations in biomedical data. We employed BenchXAI to validate these methods in three common biomedical tasks, namely clinical data, medical image and signal data, and biomolecular data. Our newly designed sample-wise normalisation approach for post-hoc XAI methods enables the statistical evaluation and visualisation of performance and robustness. We found that the XAI methods Integrated Gradients, DeepLift, DeepLiftShap, and GradientShap performed well over all three tasks, while methods like Deconvolution, Guided Backpropagation, and LRP- α 1- β 0 struggled for some tasks. With acts such as the EU AI Act the application of XAI in the biomedical domain becomes more and more essential. Our evaluation study represents a first step toward verifying the suitability of different XAI methods for various medical domains.
... Introduction Shapley and Shubik (1954) advertised the Shapley value as "A method for evaluating the distribution of power in a committee system" almost immediately with the value's introduction by Lloyd S. Shapley (1953). Their motivation included not only the problem of measuring a priori voting power in a given weighted voting system or in multicameral legislatures such as the US Congress, but they explicitly referred to the design of decision-making bodies and asked: "Can a consistent criterion for 'fair representation' be found?" ...
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When delegations to an assembly or council represent differently sized constituencies, they are often allocated voting weights which increase in population numbers (EU Council, US Electoral College, etc.). The Penrose square root rule (PSRR) is the main benchmark for fair representation of all bottom-tier voters in the top-tier decision making body, but rests on the restrictive assumption of independent binary decisions. We consider intervals of alternatives with single-peaked preferences instead, and presume positive correlation of local voters. This calls for a replacement of the PSRR by a linear Shapley rule: representation is fair if the Shapley value of the delegates is proportional to their constituency sizes.
... If the weights add up to one, then we speak of relative weights. The insight that the power distribution differs from relative weights, triggered the invention of so-called power indices like the Shapley-Shubik index [40], the Penrose-Banzhaf index [3], or the nucleolus [39]. Due to the combinatorial nature of most of those indices, qualitative assessments are technically demanding and large numbers of involved parties cause computational challenges [4]. ...
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Determining the power distribution of the members of a shareholder meeting or a legislative committee is a well-known problem for many applications. In some cases it turns out that power is nearly proportional to relative voting weights, which is very beneficial for both theoretical considerations and practical computations with many members. We present quantitative approximation results with precise error bounds for several power indices as well as impossibility results for such approximations between power and weights.
... Recently, Owen (2014) established a relation between the Shapley values (Shapley and Shubik, 1954) coming from the field of game theory and Sobol' indices. Song et al. (2016) proposed an algorithm to estimate these indices. ...
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In global sensitivity analysis, the well known Sobol' sensitivity indices aim to quantify how the variance in the output of a mathematical model can be apportioned to the different variances of its input random variables. These indices are based on the functional variance decomposition and their interpretation become difficult in the presence of statistical dependence between the inputs. However, as there is dependence in many application studies, that enhances the development of interpretable sensitivity indices. Recently, the Shapley values developed in the field of cooperative games theory have been connected to global sensitivity analysis and present good properties in the presence of dependencies. Nevertheless, the available estimation methods don't always provide confidence intervals and require a large number of model evaluation. In this paper, we implement a bootstrap sampling in the existing algorithms to estimate confidence intervals of the indice estimations. We also proposed to consider a metamodel in substitution of a costly numerical model. The estimation error from the Monte-Carlo sampling is combined with the metamodel error in order to have confidence intervals on the Shapley effects. Besides, we compare for different examples with dependent random variables the results of the Shapley effects with existing extensions of the Sobol' indices.
... For any subset S of supporters v(S) ∈ {0, 1}, where v is surjective and monotone, i.e., v(S) ≤ v(T ) for all S ⊆ T . The importance, influence or power of an agent in a simple game is measured by so-called power indices like the Shapley-Shubik [41] or the Penrose-Banzhaf index [2,38], see also [14,40]. The model is appropriate to model situations as complex as networks of companies, where several agents own shares of some companies that are owning shares of other companies themselves and so are indirectly controlling each other. ...
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Given a system where the real-valued states of the agents are aggregated by a function to a real-valued state of the entire system, we are interested in the influence or importance of the different agents for that function. This generalizes the notion of power indices for binary voting systems to decisions over interval policy spaces and has applications in economics, engineering, security analysis, and other disciplines. Here, we study the question of importance in systems with interval decisions. Based on the classical Shapley-Shubik and Penrose-Banzhaf index, from binary voting, we motivate and analyze two importance measures. Additionally, we present some results for parametric classes of aggregation functions.
... The most prominent measures of a player's power, or influence, in a weighted voting game are the Shapley-Shubik and Banzhaf power indices. Merging and extending the results of [BE08] and [AP09], Aziz et al. [ABEP11] in particular study the problem of whether merging or splitting players in weighted voting games is beneficial in terms of the Shapley-Shubik index [Sha53,SS54] and the normalized Banzhaf index [Ban65] (see Section 2 for formal definitions). Rey and Rothe [RR10] extend this study for the probabilistic Banzhaf index proposed by Dubey and Shapley [DS79]. ...
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False-name manipulation refers to the question of whether a player in a weighted voting game can increase her power by splitting into several players and distributing her weight among these false identities. Analogously to this splitting problem, the beneficial merging problem asks whether a coalition of players can increase their power in a weighted voting game by merging their weights. Aziz et al. [ABEP11] analyze the problem of whether merging or splitting players in weighted voting games is beneficial in terms of the Shapley-Shubik and the normalized Banzhaf index, and so do Rey and Rothe [RR10] for the probabilistic Banzhaf index. All these results provide merely NP-hardness lower bounds for these problems, leaving the question about their exact complexity open. For the Shapley--Shubik and the probabilistic Banzhaf index, we raise these lower bounds to hardness for PP, "probabilistic polynomial time", and provide matching upper bounds for beneficial merging and, whenever the number of false identities is fixed, also for beneficial splitting, thus resolving previous conjectures in the affirmative. It follows from our results that beneficial merging and splitting for these two power indices cannot be solved in NP, unless the polynomial hierarchy collapses, which is considered highly unlikely.
... Power indices are a way to measure the relative power of the players in a simple game. The most famous of these are the Shapley-Shubik index [6] and the Banzhaf index [1]. Semivalues were introduced in 1979 by Weber [8] as a generalization of the notion of a power index to general cooperative games. ...
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In this paper we study rankings induced by power indices of players in simple game models of bicameral legislatures. For a bicameral legislature where bills are passed with a simple majority vote in each house we give a condition involving the size of each chamber which guarantees that a member of the smaller house has more power than a member of the larger house, regardless of the power index used. The only case for which this does not apply is when the smaller house has an odd number of players, the larger house has an even number of players, and the larger house is less than twice the size of the smaller house. We explore what can happen in this exceptional case. These results generalize to multi-cameral legislatures. Using a standard model of the US legislative system as a simple game, we use our results to study power index rankings of the four types of players -- the president, the vice president, senators, and representatives. We prove that a senator is always ranked above a representative and ranked the same as or above the vice president. We also show that the president is always ranked above the other players. We show that for most power index rankings, including the Banzhaf and Shapley-Shubik power indices, the vice president is ranked above a representative, however, there exist power indices ranking a representative above the vice president.
... If u ≡ [q;w 1 , w 2 , … , w n ] is a weighted majority game with integer representation, then the dual game is u * ≡ [ − q + 1;w 1 , w 2 , … , w n ] , where = ∑ i∈N w i . When it applies to simple games, the Shapley value is sometimes called the Shapley-Shubik power index since the seminal article by Shapley and Shubik (1954) and is well accepted as an individual measure of "power" (in a generic sense). For instance, it was recognized as such for reliability by Barlow and Broschan (1975). ...
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Many coalitional values have been introduced in the literature on cooperative games over the last decades, and especially since 2000. The multiplicity of options suggests the convenience of testing the existence of stable coalition structures, in the sense of Hart and Kurz (1983, Econometrica), when payments are made using some of these values. We recall their concept of γγ\gamma–stability and give results for the proportional partitional Shapley value, introduced by Alonso–Meijide et al (2015, Discrete Appl. Math.), which shares the utility of any coalition proportionally to the Shapley value of the involved players in the original game.
... The Shapley value [30] can be viewed as a "power index", a tool for measuring an individual's contribution or importance in the success of a team of agents, or for quantifying an agent's ability to influence a game's outcome [31,6]. The Shapley value was used for measuring political influence of parties forming a coalition in legislative bodies [14], analyzing network reliabilitiy [5,2,4] and fair cost allocation [27,33]. ...
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We show how the quality of decisions based on the aggregated opinions of the crowd can be conveniently studied using a sample of individual responses to a standard IQ questionnaire. We aggregated the responses to the IQ questionnaire using simple majority voting and a machine learning approach based on a probabilistic graphical model. The score for the aggregated questionnaire, Crowd IQ, serves as a quality measure of decisions based on aggregating opinions, which also allows quantifying individual and crowd performance on the same scale. We show that Crowd IQ grows quickly with the size of the crowd but saturates, and that for small homogeneous crowds the Crowd IQ significantly exceeds the IQ of even their most intelligent member. We investigate alternative ways of aggregating the responses and the impact of the aggregation method on the resulting Crowd IQ. We also discuss Contextual IQ, a method of quantifying the individual participant's contribution to the Crowd IQ based on the Shapley value from cooperative game theory.
... This is the key idea behind the notion of Banzhaf power (Penrose 1946;Banzhaf 1964Banzhaf , 1966Banzhaf , 1968. The main competing voting power index, the Shapley-Shubik power index (Shapley and Shubik 1954), also has an interpretation in terms of pivotality (Straffin 1977). ...
Article
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This paper proposes a method of evaluating elections in terms of freedom of choice. It evaluates voting institutions in terms of their allocation of control. Formally, the paper develops the symmetric power order, a measure of voting power for multicandidate elections. The measure generalizes standard pivotality-based voting power measures for binary elections, such as Banzhaf power. At the same time, the measure is not based on pivotality, but rather on a measure of freedom of choice in individual decisions. I show that pivotality only measures freedom-based voting power in monotonic elections, and is not a good measure in multicandidate elections. Pivotality only provides an upper bound on freedom-based voting power. This result establishes a relation between voting power and strategyproofness. I argue that my results are robust, and that pivotality should generally be expected to over-estimate other sensible measures of voting power in multicandidate elections.
... Thus, a set of players is said to be a winner if the sum of their individual weights is greater than or equal to the quota. In these settings, several power indices have been proposed in the literature, such as the Shapley-Shubik index (Shapley & Shubik, 1954), the Banzhaf index (Banzhaf, 1965), the Deegan-Packel index (Deegan & Packel, 1978), the Johnston index (Johnston, 1978), the Public Good index (Holler, 1982) and the Colomer-Martínez measure (Colomer & Martínez, 1995), defined specifically for weighted majority games. ...
Article
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In this paper we introduce new procedures, based on generating functions, for calculating some power measures for weighted majority games. In particular, we present methods for computing the Johnston index and the Colomer–Martínez measure. Besides, we introduce a new power measure that combines the principles underlying the Johnston index and Colomer–Martínez measure as well as a procedure for computing it using generating functions. Finally, we introduce the new R package powerindexR and describe its capabilities to compute some power measures by means of generating functions. We illustrate its performance with a real example.
... Recall that a coalitional value represents, for each TU game, an allocation that is assigned to each agent. For instance, we mention the Banzhaf index (Banzhaf, 1965) and the restriction of the Shapley value for TU games (Shapley, 1953) to simple games, namely the Shapley-Shubik power index (Shapley & Shubik, 1954). Both are based on the relative frequency of a player being pivotal. ...
... The 1970s was a period of intense work in game theory, resulting in a profound transformation of economics (documented for example in Myerson 1999). Even previously, the concept of Shapley value, another key idea from cooperative game theory (simply put, a reasonable and axiomatically justified way to allocate individual payoffs in these games), had been applied in political science (Shapley and Shubik 1954). Most members of the Leningrad group were working with this concept, but given the absence of a political science discipline and of a democratic political system, similar applications of game theory were inconceivable in the Soviet Union (see, however, Boldyrev 2020 on an exception). ...
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What are the effects of authoritarian regimes on scholarly research in economics? And how might economic theory survive ideological pressures? The article addresses these questions by focusing on the mathematization of economics over the past century and drawing on the history of Soviet science. Mathematics in the USSR remained internationally competitive and generated many ideas that were taken up and played important roles in economic theory. These same ideas, however, were disregarded or adopted only in piecemeal fashion by Soviet economists, despite the efforts of influential scholars to change the economic research agenda. The article draws this contrast into sharper focus by exploring the work of Soviet mathematicians in optimization, game theory, and probability theory that was used in Western economics. While the intellectual exchange across the Iron Curtain did help advance the formal modeling apparatus, economics could only thrive in an intellectually open environment absent under the Soviet rule.
... I show that Shapley-Shubik Power Index (Shapley and Shubik, 1954) can quantify the relative importance of each member in triggering a binary decision such as the structural presumption for merger screening. value is known as the Shapley-Shubik power index. ...
Preprint
Market definition holds significant importance in antitrust cases, yet achieving consensus on the correct approach remains elusive. As a result, analysts routinely entertain multiple market definitions to ensure the resilience of their conclusions. I propose a simple framework for conducting organized sensitivity analysis with respect to market definition. I model candidate market definitions as partially ordered and use a Hasse diagram, a directed acyclic graph representing a finite partial order, to summarize the sensitivity analysis. I use the Shapley value and the Shapley-Shubik power index to quantify the average marginal contribution of each firm in driving the conclusion. I illustrate the method's usefulness with an application to the Albertsons/Safeway (2015) merger.
... Considering the 80% success rate, for each weighted voting game Γ SS (X), we stated the quota q T (X) as the 80% of the total n presentations considered, i.e., q T (X) = 0.8 · n. From these weighted voting game Γ T (X), we are able to compute the corresponding Shapley-Shubik value of X with respect to Γ T (X), denoted by µ ss (X) (Shapley and Shubik, 1954). In essence, µ ss (X) is the number of times that X is pivot (it makes that a coalition of characters will be successfully) divided by all possible permutations. ...
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This book contains the proceedings of the 9th International Conference on Complexity, Future Information Systems and Risk (COMPLEXIS 2024). This year, COMPLEXIS was held in Angers, France, from April 28 - 29, 2024. It was sponsored by the Institute for Systems and Technologies of Information, Control and Communication (INSTICC). COMPLEXIS 2024 was also organized in cooperation with the Centre of Complex Systems IPN and the Centre for Complex Systems Studies. The International Conference on Complexity, Future Information Systems and Risk, is a yearly meeting place for presenting and discussing innovative views on all aspects of Complex Information Systems, in different areas such as Informatics, Telecommunications, Computational Intelligence, Biology, Biomedical Engineering and Social Sciences. Information is pervasive in many areas of human activity – perhaps all – and complexity is a characteristic of current Exabyte-sized, highly connected and hyper dimensional, information systems. COMPLEXIS 2024 received 19 paper submissions from 14 countries of which 32% were accepted and published as full papers. A double-blind paper review was performed for each submission by at least 2 but usually 3 or more members of the International Program Committee, which is composed of established researchers and domain experts. The high quality of the COMPLEXIS 2024 program is enhanced by the keynote lecture delivered by distinguished speakers who are renowned experts in their fields: Luigi Atzori (Università degli Studi di Cagliari, Italy) and Samuel Fosso Wamba (Toulouse Business School, France). All presented papers will be available at the SCITEPRESS Digital Library and will be submitted for evaluation for indexing by SCOPUS, Google Scholar, The DBLP Computer Science Bibliography, Semantic Scholar, Engineering Index and Web of Science / Conference Proceedings Citation Index. As recognition for the best contributions, several awards based on the combined marks of paper reviewing, as assessed by the Program Committee, and the quality of the presentation, as assessed by session chairs at the conference venue, are conferred at the closing session of the conference. A shortlist of papers presented at the conference will be selected for recommended of extended and revised versions in the special issues of the Springer Nature Computer Science Journal, Journal of Global Information Management, IMA Journal of Management Mathematics, Socio-Economic Planning Sciences and Big Data Journal, Big Data Journal and Internet of Things . The program for this conference required the dedicated effort of many people. Firstly, we must thank the authors, whose research efforts are herewith recorded. Next, we thank the members of the Program Committee and the auxiliary reviewers for their diligent and professional reviewing. We would also like to deeply thank the invited speakers for their invaluable contribution and for taking the time to prepare their talks. Finally, a word of appreciation for the hard work of the INSTICC team; organizing a conference of this level is a task that can only be achieved by the collaborative effort of a dedicated and highly capable team. We wish you all an exciting and inspiring conference. We hope to have contributed to the development of our research community, and we look forward to having additional research results presented at the next edition of COMPLEXIS, details of which are available at https://complexis.scitevents.org.
... It is hard to pin down a value to such coalitions, but in a sense they are all equally powerful: each one of them may enact laws, while no other type of coalition can. This intuitively plausible observation was utilized by Shapley and Shubik (1954) when they suggested a method for measuring power in collective decision-making bodies. The method is today known as the Shapley-Shubik index. ...
Research Proposal
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MODEL POLITICAL ECONOMY
... The Shapley-Shubik index is used as an indicator of the political power of a hypothetical coalition member, applying the notion of a pivot (i.e., a party whose participation makes a non-winning coalition into a winning one) and taking into consideration all permutations of possible coalitions (Shapley & Shubik, 1954). The index is expressed as a value between zero and one, which denotes the number of variations for which a particular party is a pivot. ...
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This study is theoretically anchored in the office-seeking approach and coalition theory of political science. It is based on the most widespread theories of portfolio distribution in executive and legislative bodies (proportionality, power index, voting weights), and it tests these theories using data related to coalition negotiations in the Czech Republic for the new 2021 government coalition, a case of a surplus majority government. The study investigates the distribution of ministerial positions in the government, the office of President (Speaker) of the Chamber of Deputies, and the chairs of permanent parliamentary committees. The analysis also explores the types of electoral coalitions formed; the SPOLU coalition was a superadditive coalition, while the PirSTAN coalition was only an additive coalition. Grounded in the theory of electoral games, the Shapley-Shubik and Banzhaf power indices and the theory of coalition formation are applied in order to analyse the possible minimal winning coalitions that could be formed following the 2021 elections to the Chamber of Deputies.
... It is hard to pin down a value to such coalitions, but in a sense they are all equally powerful: each one of them may enact laws, while no other type of coalition can. This intuitively plausible observation was utilized by Shapley and Shubik (1954) when they suggested a method for measuring power in collective decision-making bodies. The method is today known as the Shapley-Shubik index. ...
Presentation
Models of Political Economy
... The most well-known power indices in the literature are the Shapley-Shubik power index (Shapley and Shubik, 1954) and the Banzhaf power index (Banzhaf, 1964). 2 Another well-known power index is the Equal division power index (ED) defined by ...
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We consider cooperative games where the coalition structure is given by the set of winning coalitions of a simple game. This type of games models some real-life situations in which some agents have economic performances while some others are endowed with a political power. On this class of cooperative games, the Myerson value has been identified as the Harsanyi power solution associated to the Equal Division power index and has been characterized in the large class of Harsanyi power solutions with respect to the associated power index. In this paper, we provide a characterization of the Myerson value for this class of games without focusing on the whole family of Harsanyi power solutions and therefore, without taking into account any power index. We identify the Myerson value as the only allocation rule that satisfies efficiency, additivity, modularity, extra-null player property, and Equal Treatment of Veto.
... Several indices have so far been defined in the literature on simple games. Among the most studied indices are the Shapley and Shubik (1954), Banzhaf (1965), Coleman (1971) and Johnston (1978) power indices. Other less studied power indices in simple games include the Deegan-Packel (Deegan and Packel 1978), shift (Alonso Meijide and Freixas 2010), shift (Freixas and Kurz 2015) and Public Good (Holler and Packel 1983) power indices. ...
Article
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Power measures are used to quantify the influence of members of a democratic institution. We consider voting games with abstention or (3,2) games, which are decision-making processes in which voting options include yes, no and abstention. The power indices that we study are based on the notions of minimal and shift minimal winning tripartitions. We define and characterize the Deegan–Packel and shift Deegan–Packel power indices in the class of (3,2) games. Furthermore, owing to the parameterization result obtained by Freixas et al. (Discret Appl Math 255:21–39, 2019), we provide computational formulae of these indices in the class of I-complete (3,2) games. These formulae allow us to determine the power of each player in a game, regardless of the number of minimal and/or shift minimal winning tripartitions of the game.
... To this end, Shapley-Shubik Power Index (SSI) is proposed in this particular study to determine pivotal players (facilities) and identify the optimal allocation of resources in FWEIP. The SSI was first found to determine the power of each voter in affecting the outcome of the voting system and the permutations of all voters in the game are used to evaluate sequential coalitions (Shapley and Shubik 1954;Arnell et al. 2020). To win the alliance, the total number of votes supplied by each player must exceed the defined quota. ...
Article
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The tremendous production of fish has resulted in an increased fish waste generation, which ultimately led to the current triple planetary crises on climate, biodiversity, and pollution. In this study, a Fish Waste-based Eco-Industrial Park (FWEIP) model is developed in an attempt to convert the linear economy in existing fish waste management into a circular economy model. Process Graph (P-graph) is used for combinatorial optimization to synthesize optimal FWEIP with the consideration of economic and environmental aspects. The model favors the production of biofuel using the gasification process (Rank 1) with a promising economic benefit of $2.28 million/y without proposing circular synergy within the FWEIP ecosystem. On the other hand, suboptimal solutions—suboptimal 1 (black soldier fly (BSF)) and suboptimal 2 (pyrolysis and gasification) solutions—exhibit gross profit of 17.98% and 24.12% lower than that of the optimal solution. Both suboptimal solutions offer greater circularity with self-sustaining resources (e.g., fish feed, chitosan, and energy). The sensitivity analysis indicates the potential debottlenecking of suboptimal 2 with the use of a catalyst to improve the conversion of bio-oil in the pyrolysis pathway and exhibits a gross profit of 22.54% higher than that of the optimal solution. Following the Shapley-Shubik power index analysis, the hydroponics facility is identified as the pivotal player for both optimal and suboptimal 2 cases with the exception of suboptimal 1 indicating both BSF and hydroponics as a pivotal player. In brief, this research provides the fish waste-based industry with insights and strategies for the implementation of a circular economy as a step toward sustainable development.
... Nonetheless, his approach has gained more attention only after Banzhaf (1965) and Coleman (1971) have "rediscovered" it; thus, it is called the Penrose-Banzhaf-Coleman, or, simply, Banzhaf index. The other popular power measure is the Shapley-Shubik index (Shapley and Shubik, 1954). In addition, since the power indices are usually defined via a weighted voting game, the influence of the voters can be quantified by essentially any solution concept of cooperative game theory such as the nucleolus (Schmeidler, 1969;Zaporozhets et al., 2016). ...
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The Council of the European Union (EU) is one of the main decision-making bodies of the EU. Many decisions require a qualified majority: the support of 55% of the member states (currently 15) that represent at least 65% of the total population. We investigate how the power distribution, based on the Shapley-Shubik index, and the proportion of winning coalitions change if these criteria are modified within reasonable bounds. The influence of the two countries with about 4% of the total population each is found to be almost flat. The level of decisiveness decreases if the population criterion is above 68% or the states criterion is at least 17. The proportion of winning coalitions can be increased from 13.2% to 20.8% (30.1%) such that the maximal relative change in the Shapley-Shubik indices remains below 3.5% (5.5%). Our results are indispensable to evaluate any proposal for reforming the qualified majority voting system.
... (TJC) 6 logró importantes resultados en las contribuciones de Nash (1953) y Shapley (1963) sobre los juegos de negociación, y las de Gillies (1953) y Shapley (1963 sobre el núcleo de un juego. 7 Shapley, junto con otros autores posteriores, Shapley y Shubik (1954), y Banzhaf (1965), llevarían a cabo el desarrollo de soluciones distintas a la solución núcleo 8 mediante propuestas de diferentes valores de poder de decisión 9 dentro de un juego cooperativo. ...
Article
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Decisions made in different congresses are of great importance for the social and economic sphere of a country, since without cooperation between parties, the approval of reforms and laws can be stalled. The way in which decision-making’s power dynamics between the different political forces is studied can be approached from a quantitative point of view. That is why in the present investigation the decision-making power of political parties in the LXIII and LXIV Legislatures is analyzed through different indices offered by cooperative game theory and through simulations developed in Scilab. The importance of studying this type of topics from an interdisciplinary approach lies in the better understanding of political behavior within congresses, and in the knowledge of the multiple ways that can be had to approve the different agreements. It is found that in three years MORENA increased its decision-making power by more than 60% and the PRI has lost almost 50% of it. It was also possible to verify that the PRI is the party that benefits the most from making coalitions and that the PAN is the most harmed in this type of analysis.
... The restriction of the Shapley value to voting games usually is referred to as the Shapley-Shubik index (Shapley and Shubik, 1954), and payoffs are interpreted as the players' power in a voting game. We denote the Shapley-Shubik index by SSI. ...
Article
We suggest a new component efficient solution for monotonic TU games with a coalition structure, the conditional Shapley value. In contrast to other such solutions, it satisfies the null player property. Nevertheless, it accounts for the players’ outside options in productive components of coalition structures. For all monotonic games, there exist coalition structures that are stable under the conditional Shapley value. For voting games, such stable coalition structures support Gamson’s theory of coalition formation (Gamson, 1961).
... In scenarios where such variations exist, the weighted Shapley values (Shapley 1953b) offer a more suitable approach for fair allocation. By incorporating weights that account for the aforementioned asymmetries, the weighted Shapley values provide a more nuanced and balanced distribution of payoffs among the players. ...
Article
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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.
... Another prominent contribution coming from cooperative game theory is the Shapley-Shubik power index (Shapley and Shubik, 1954). The authors introduced a measure of a player's strategic influence in the coalition formation process based on the "chance they have of being critical to the success of a winning coalition". ...
Article
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We consider a set of empirical assumptions formulated by Gamson (1961), namely, Gamson’s Laws, which remain at the heart of government formation forecast in parliamentary systems. While the critical resource postulated in Gamson’s approach is the proportion of votes received by each party, other versions of Gamson’s Laws can be defined by a different choice of critical resource. We model coalition formation as a cooperative game, and provide axiomatic foundations for a version of Gamson’s Laws in which the critical resource is identified with strategic influence, as measured by the Shapley value. We compare the empirical accuracy of the resulting Gamson–Shapley theory against the original Gamson’s Laws in a panel of 33 parliamentary elections, and find that it leads to significantly more accurate predictions of both coalition structure and power distribution. Finally, we propose an extension of the Gamson–Shapley approach which also incorporates information about policy distance among coalition partners. In particular, we discuss the advantages of the extended approach in the context of the German elections in 1987 and 2017.
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Az Európai Unió Tanácsa (korábban Miniszterek Tanácsa) az Unió egyik fő döntéshozó szervezete, amelyben minden országot egyetlen fő képvisel, akinek szavazási súlya az adott ország népességével arányos. A döntések meghozatalához számos szavazás során minősített többség szükséges, amely megfelelő számú és összlakosságú tagállam támogatását jelenti. A korlátokat és súlyokat az Unió bővülésével gyakran újratárgyalták, ugyanakkor Nagy-Britannia kilépésével nem módosultak. Kutatásunkban az egyik legelterjedtebb hatalmi index, a Shapley--Shubik index segítségével elemezzük, hogy a két korlát változtatása hogyan befolyásolná az erőviszonyokat. Vizsgáljuk az Európai Unió döntésképességét és a tagállamok befolyásának egyenlőtlenségét is. Eredményeink szerint Magyarország szavazati ereje alig csökkenthető az összlakosságból való 2,17%-os részesedése alá. Ugyanakkor 15 tagállam egyetértésének megkövetelésével nem lehet magasabb 3%-nál, és kisebb tagállamkvóta esetén sem emelkedhet 3,5% fölé.
Chapter
When two players are facing each other, the question of cooperation is simple: either they cooperate or they do not cooperate. When instead, there are three players or more, players can form coalitions. This enriches considerably the analysis. The concept of coalition is indeed central in the analysis of games involving more than two players. If a group of players decides to form a coalition, it means that they dissociate themselves from the set of all players (called the “grand coalition”) in order to cooperate between themselves. However, the question that is being addressed concerns the cooperation between all players and the allocation among them of the resulting “social” outcome. Hence, the role played by coalitions remains potential. Beyond the maximum gain that the players can generate altogether by cooperating, we need to know the gain that each coalition of players could obtain through the cooperation of its members, independently of the actions of the players outside the coalition. When a coalition forms, what matters is the outcome it can achieve, without specifying the organizational details. Quoting Shapley, cooperative game theory “is concerned with things like cooperation, coalition, organizational structure, commitment, trust, compromise, threatThreat, enforceability and indeed the whole legal/social/cultural environment. It deemphasizes questions of tactical optimization, the detailed spelling out of rules and the numerical calculation of outcomes and payoffs”.1 The possibility of binding and enforceable agreements is an element that further differentiates cooperative games from non-cooperative games. In Aumann's words, “A game is called cooperative if there is available a mechanism, such as a court, to enforce agreements. In a cooperative game, any feasible outcome may be achieved if the players subscribe to the appropriate agreement”.
Chapter
“Political science, as an empirical discipline, is the study of the shaping and sharing of power” (Lasswell & Kaplan, 1950). As the quote above suggests, the question of power is central to the analysis of political processes. But what do we mean by power? Can it be measured? This is the purpose of this chapter. We will limit ourselves to the analysis of collective decisions within committees. A committee is a group of decision-makers in which decisions are made according to well-established rules. These rules translate into the list of winning coalitionsWinning coalition, these coalitions of decision-makers who, according to the rules, are able to decide. The set of decision-makers and the list of winning coalitionsWinning coalition define a voting gameVoting game. We do not intend to describe the decision procedures as such: a proposal is put on the table and the question is which subsets of decision-makers, if they agree, can together impose a decision, that is, accept or reject the proposal. Decision-makers do not necessarily have the same weight in the decision-making process. They may have a different number of votes, some may have a veto right and special restrictions may be added, such as, for example, the protection of a minority. A general meeting of shareholders, a parliament (where decision-makers are identified with the parties) and international organizations such as the European Council or the United Nations Security Council, are all examples of such committees. The problem addressed in this chapter is the measurement of the relative power of each member of a committee, beyond the apparent power conferred on them by the rules: we want to measure the ability of each decision-maker to influence the outcome of a vote given the rules in force.
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O presente trabalho objetiva fazer um esforço introdutório de caracterização dos jogos cooperativos com uma atenção especial ao conceito de solução do valor de Shapley que tem suas propriedades descritas em detalhes. Para cumprir esse objetivo são apresentados exemplos teóricos que podem ser resolvidos tanto cooperativamente quanto não cooperativamente, possibilitando uma comparação direta entre as duas abordagens da Teoria dos Jogos. Além disso, são descritas tradicionais aplicações do valor de Shapley em problemas práticos – jogos de alocação de custos e jogos de votação – de forma a aprofundar a compreensão sobre este conceito de solução. Ao mostrar os aspectos introdutórios e a praticidade do valor de Shapley acredita-se que este trabalho possa servir de ponto de partida para futuras aplicações e discussões, uma vez que existem poucas obras no Brasil que tratam dos jogos cooperativos e seus conceitos de solução.
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In the realm of critical decision-making, few scenarios carry the gravitas and potential consequences of a nuclear launch decision. "Decision Under Uncertainty: Psychological and Mathematical Analysis of a Nuclear Launch Vote" delves into the complexities of a 3 out of 5 voting system used in such high-stakes environments. This paper examines the dual aspects of psychological impact and mathematical probability that underpin these decisions. By exploring how voter anonymity and the weight of each vote contribute to both cognitive dissonance and decision-making outcomes, this analysis aims to reveal the intricate balance between collective responsibility and individual ethical dilemmas. The study utilizes theoretical models, including binomial probability and game theory power indices, to articulate how different voting thresholds influence decision robustness and fairness. Through this interdisciplinary approach, the paper seeks to provide insights that could guide the development of more effective and ethically grounded decision-making frameworks in scenarios involving monumental moral and practical implications. Keywords: critical decision-making, nuclear launch, voting systems, psychological impact, mathematical probability, binomial probability, game theory, power indices, ethical considerations, cognitive dissonance, moral uncertainty, strategic voting, decision theory.
Article
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Proponemos la aplicación del valor coalicional al estudio del comportamiento estratégico de los partidos en la formación de coaliciones parlamentarias. Del criterio de optimización del valor se deriva una noción de estabilidad equivalente a la del equilibrio fuerte de nash para los juegos no cooperativos.Las restricciones a la cooperación por razones ideológicas o tácticas se incorporan al modelo formal. como ilustración, analizamos el parlamento de Cataluña (1980-1984) y el parlamento del País Vasco (1986-1990).
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This study proposes a two-game theory model to decide on the deployment of health resources in all afflicted districts in response to the COVID-19 pandemic. First, we model the interaction of Coronaviruses with a response commander as a two-player zero-sum game and then calculate COVID-19 risk values for each district using the mixed strategy Nash equilibrium. The threat, vulnerability, and consequence of the COVID-19 epidemic on the afflicted state are represented by the risk value. Second, the risk values from all states are applied to calculate a Shapley-Shubik index (SSI) for each district. The medical resources of all districts are distributed fairly depending on their marginal contribution to a simple cooperative game. According to the experimental results, administrators can utilize this framework to quantify the risk of COVID-19 in each afflicted district, and the Shapley-Shubik index is feasible as a method for deploying medical resources. And the SSI division deploys medical resources more efficiently than a proportional (RV) division.
Article
The aim of this article is to study empirically the relationship between political governance and public debt by testing a number of hypotheses. We examine the effects of the dispersion of power on public debt with an econometric study carried out on a sample of 13 developed countries using macroeconomic and political data covering the period 1996–2012. It is found that the lack of consensus between political parties in a government coalition and the dispersion of power within the government are factors explaining the increase in public debt.
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