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In this paper, we extend the equal division and the equal surplus division values for transferable utility cooperative games to the more general setup of transferable utility cooperative games with a priori unions. In the case of the equal surplus division value we propose three possible extensions. We provide axiomatic characterizations of the new values. Furthermore, we apply the proposed modifications to a particular motivating example and compare the numerical results with those obtained with the original values.

In a cooperative game, we consider three special kinds of agents: null, nullifying, and necessary agents. A coalition with a null agent receives the same payoff if this agent leaves the coalition, a coalition with a nullifying agent receives nothing, and a coalition without a necessary agent also receives nothing. The null agent property proposes zero payoff to any null agent. We introduce new properties that propose payoffs for nullifying and necessary players. With these three properties and additivity, we obtain new characterizations of the Banzhaf and Shapley values.

Given a finite set of agents, an embedded coalition consists of a coalition and a partition of the rest of agents. We study a partial order on the set of embedded coalitions of a finite set of agents. An embedded coalition precedes another one if the first coalition is contained in the second and the second partition equals the first one after removing the agents in the second coalition. This poset is not a lattice. We describe the maximal lower bounds and minimal upper bounds of a finite subset, whenever they exist. It is a graded poset and we are able to count the number of elements at a given level as well as the total number of chains. The study of this structure allows us to derive results for games with externalities. In particular, we introduce a new concept of convexity and show that it is equivalent to having non-decreasing contributions to embedded coalitions of increasing size.

In this work, we consider games with coalitional structure. We afford two new parallel axiomatic characterizations for the well-known Owen and Banzhaf–Owen coalitional values. Two properties are common to both characterizations: a property of balanced contributions and a property of neutrality. The results prove that the main difference between these two coalitional values is that the former is efficient, while the latter verifies a property of 2-efficiency.

In this paper, we define a modification of the Shapley value for the model of TU games with a priori unions. We provide two characterizations of this value and a new characterization of the Banzhaf–Owen coalitional value.

We provide axiomatizations for six variants of the Public Good Index (PGI) for games with a priori unions. Two such coalitional PGIs have been introduced and alternatively axiomatized in Alonso-Meijide et al. (Working paper 18, 2008b). They assign power in two steps. In the first step, power is distributed between unions according to the PGI of the quotient game. In a second step, the Solidarity PGI splits power equally among union members while the Owen Extended PGI takes into account so-called essential parts. The other four coalitional PGIs have been introduced in Holler and Nohn (Homo Oeconomicus 26, 2009). The first variant elaborates the original idea of Holler (Political Studies 30:262–271, 1982) that the coalitional value is a public good and only minimal winning coalitions of the quotient game are relevant. The remaining three variants also use the two-step distribution where, however, on the member stage they take into account the possibilities of players to threaten their partners through leaving their union.

In this paper we define and axiomatically characterize an extension of the Deegan–Packel index for simple games with a priori unions. A real-world example illustrates this extension.

The paper discusses the substitution of mergeability by the weighted monotonicity property in the definition of the Public Good Index. The cardinality of sets, implicit to the measurement of power, can thus be related to comparing sets and relations of counting and set inclusion. This allows for the application of results from the measurement of power to the specification of freedom of choice and thereby to connect the ‘world of agents’ with the ‘world of opportunities’. The relationship between weighted monotonicity and constrained monotonicity is specified.

An a priori system of unions or coalition structure is a partition of a finite set of players into disjoint coalitions which have made a prior commitment to cooperate in playing a game. This paper provides “ready-to-apply” procedures based on generating functions that are easily implementable to compute coalitional power indices in weighted multiple majority games. As an application of the proposed procedures, we calculate and compare coalitional power indices under the decision rules prescribed by the Treaty of Nice and the new rules proposed by the Council of the European Union.

Two new values for transferable utility games with graph restricted communication and a priori unions are introduced and characterized. Moreover, a comparison between these and the Owen graph value is provided. These values are used to analyze the distribution of power in the Basque Parliament emerging from elections in April 2005.

The Spanish norm UNE 66020-1:2001 is equivalent to the international norm ISO 2859-1:1999 and deals with inspection by attributes of lots of material. In order to implement such a norm in a canned marine food factory (which manufactures mainly bluefin tuna in vegetal oil), some preliminary work was carried out. First, a careful study of the production process led to three possible sources of variation in the final weight of the cans produced, namely the canning machine, the cans used and the balance utilized for weighing. Afterward, each of these sources of variation was studied, which resulted in a value of the process average. This average shall enable the responsible authority of the company to set an acceptance quality limit, which should allow the application of the cited norm. PRACTICAL APPLICATION This work is expected to be considered an example of the studies that need to be carried out previously to the implementation of norm UNE 66020-1:2001 (i.e., norm ISO 2859-1:1999). Moreover, an additional point to highlight is the fact that this work is not a simulation but rather, it was developed in a real factory, devoted to the manufacturing of canned marine food. © 2008 The Author(s).

Power indices are commonly required to assign at least as much power to a player endowed with some given voting weight as to any player of the same game with smaller weight. This local monotonicity and a related global property however are frequently and for good reasons violated when indices take account of a priori unions amongst subsets of players (reflecting, e.g., ideological proximity). This paper introduces adaptations of the conventional monotonicity notions that are suitable for voting games with an exogenous coalition structure. A taxonomy of old and new monotonicity concepts is provided, and different coalitional versions of the Banzhaf and Shapley-Shubik power indices are compared accordingly.

Johnston [Johnston, R.J., 1978. On the measurement of power: some reactions to Laver. Environment and Planning A 10, 907–914], Deegan and Packel [Deegan, J., Packel, E.W., 1979. A new index of power for simple n-person games. International Journal of Game Theory 7, 113–123], and Holler [Holler, M.J., 1982. Forming coalitions and measuring voting power. Political Studies 30, 262–271] proposed three power indices for simple games: Johnston index, Deegan–Packel index, and the Public Good Index. In this paper, methods to compute these indices by means of the multilinear extension of the game are presented. Furthermore, a new characterization of the Public Good Index is given. Our methods are applied to two real-world examples taken from the political field.

In this note we characterize the restriction of the externality-free value of de Clippel and Serrano (2008) to the class of simple games with externalities introduced in Alonso-Meijide et al. (2015).

A compact axiomatic characterization of the modified Banzhaf value for games with a coalition structure (Banzhaf-Owen value, for short) is provided. The axiomatic system used here can be compared with parallel axiomatizations of other coalitional values such as the Owen value or the Alonso-Fiestras value, thus giving arguments to defend the use of one of them that will depend on the context where they are to be applied.

In this paper, we provide a new characterization of the Deegan–Packel power index, which uses a new monotonicity property, and two characterizations of the Johnston power index by employing some usual properties of the literature and two new transfer properties. Finally, a real-world numerical example taken from the political field illustrates the main differences between these two indices.

Myerson 3 and 4 proposed and characterized a modification of the Shapley value in the context of communication situations. In this paper we propose several characterizations of the extension of the Banzhaf value proposed by Owen 7 to communication situations. In particular we characterize this value with the properties of fairness, isolation, and pairwise merging. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006

Alonso-Meijide and Fiestras-Janeiro (2002, Annals of Operations Research 109, 213–227) proposed a modification of the Banzhaf value for games where a coalition structure is given. In this paper we present a method to compute this value by means of the multilinear extension of the game. A real-world numerical example illustrates the application procedure.

In this paper, we characterize two power indices introduced in [1] using two different modifications of the monotonicity property first stated by [2]. The sets of properties are easily comparable among them and with previous characterizations of other power indices., A. A. Pinto acknowledges the financial support of LIAAD INESC TEC through program PEst,USP-UP project, Faculty of Sciences, University of Porto, Calouste Gulbenkian Foundation,FEDER, POCI 2010 and COMPETE Programmes, PTDC/MAT/121107/2010 and Fundação para a Ciência e a Tecnologia (FCT). J. M. Alonso-Meijide and M. Álvarez-Mozos acknowledge the financial support of the Spanish Ministry of Economy and Competitiveness through Projects ECO2008-03484-C02-02/ECO and MTM2011-27731-C03-03 and of Xunta de Galicia through Project INCITE09-207-064-PR.

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.

In this chapter, we provide a review of some recent results on power indices related to work done by the research group SaGaTh. We review axiomatic characterizations of several of the most important indices in the literature. Furthermore, we include a power index based on a desirability relation and two new power indices based on quasi-minimal winning coalitions. Finally, we summarize some results in other areas related with power indices.

In games with a coalition structure, players organize themselves in coalitions that form a partition of the set of players. In games with a coalition configuration, the union of the coalitions is the whole set of players but the coalitions are not necessarily disjoint. We introduce two variations of the Public Good Index for games with a coalition configuration, provide axiomatic characterizations of them, and apply them to a real world example.

Deegan and Packel (1979) and Holler (1982) proposed two power indices for simple games: the Deegan-Packel index and the Public Good Index. In the definition of these indices, only minimal winning coalitions are taken into account. Using similar arguments, we define two new power indices. These new indices are defined taking into account only those winning coalitions that do not contain null players. The results obtained with the different power indices are compared by means of two real-world examples taken from the political field. Deegan and Packel (1979) and Holler (1982) proposed two power indices for simple games: the Deegan-Packel index and the Public Good Index. In the definition of these indices, only minimal winning coalitions are taken into account. Using similar arguments, we define two new power indices. These new indices are defined taking into account only those winning coalitions that do not contain null players. The results obtained with the different power indices are compared by means of two real-world examples taken from the political field.

In this paper, we apply the following four power indices to the Portuguese Parliament: Shapley–Shubik index, Banzhaf index, Deegan–Packel index and Public Good Index. We also present the main notions related with simple games and discuss the features of each power index by means of their axiomatic characterizations.

In this paper we present a new power index for spatial games. We study some of its properties and indicate its advantages in comparison with the other existing indexes. Finally we illustrate our results with an example taken from a Spanish regional Parliament.

In this paper we propose a new power index useful for the evaluation of each member in a committee, or democratic institution, and the degree of influence over the voting decision making system. The proposed solution is based on the observation that democratic organizations not only tend to form coalitions which can by themselves guarantee the control of the organization, but that they also do it in an extremely efficient way that avoids the inclusion of powerful members if they can be replaced by weaker ones. The mathematical foundation of the new measure is based on two different axiomatizations. A comparison with other well-known measures is also done.

In this paper we define and axiomatically characterize an extension of the Deegan-Packel index for simple games with a priori unions. A real world example illustrates this extension.

This paper provides “ready-to-apply” procedures, based on generating functions, which allow to compute power indices in weighted majority games restricted by an a priori system of unions. We illustrate these methods by an application to the International Monetary Fund. We compare the empirical properties of the coalitional and traditional power indices keeping the game fixed or allowing for variations in its set of parameters.

In this paper we introduce a property of balanced contributions in the context of multi-issue allocation situations.Using this property, we characterise the run-to-thebank rule for multi-issue allocation situations.