Two new power indices based on winning coalitions

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

Power Indices and Minimal Winning Coalitions in Simple Games with Externalities

We propose a generalization of simple games to sit uations with coalitional externalities. The main novelty of our generalization is a monotonicity property that we define for games in partition function form. This property allows us to properly speak about minimal winning embedded coalitions. We propose and characterize two power indices based on these kind of coalitions. We provide methods based on the multilinear extension of the game to compute the indices. Finally, the new indices are used to study the distribution of power in the current Parliament of Andalusia

The Least Square Nucleolus is a Normalized Banzhaf Value

In this note we study a truncated additive normalization of the Banzhaf value. We are able to show that it corresponds to the Least Square nucleolus (LS-nucleolus), which was originally introduced as the solution of a constrained optimization problem (Ruiz et al., 1996). Thus, the main result provides an explicit expression that eases the computation and contributes to the understanding of the LS-nucleolus. Lastly, the result is extended to the broader family of Individually Rational Least Square values (Ruiz et al., 1998b)

The Shapley-Shubik Index in the Presence of Externalities

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

Power indices and minimal winning coalitions for simple games in partition function form

We propose a generalization of simple games to partition function form games based on a monotonicity property that we define in this context. This property allows us to properly speak about minimal winning embedded coalitions. We propose and characterize two power indices based on such coalitions. Finally, the new indices are used to study the distribution of power in the Parliament of Andalusia that emerged after the elections of March 22, 2015

On the externality free shapley-shubik index

We address the problem of extending the Shapley-Shubik index to the class of simple games with externalities introduced in Alonso-Meijide et al. (2017). On the one hand, we provide bounds for any efficient, symmetric, and monotonic power index. On the other hand, we characterize the restriction of the externality-free value of de Clippel and Serrano (2008) to the class of games under study by adapting well-known properties

A new order on embedded coalitions: Properties and Applications

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

Marginality and convexity in partition function form games

In this paper an order on the set of embedded coalitions is studied in detail. This allows us to define new notions of superaddivity and convexity of games in partition function form which are compared to other proposals in the literature. The main results are two characterizations of convexity. The first one uses non-decreasing contributions to coalitions of increasing size and can thus be considered parallel to the classic result for cooperative games without externalities. The second one is based on the standard convexity of associated games without externalities that we define using a partition of the player set. Using the later result, we can conclude that some of the generalizations of the Shapley value to games in partition function form lie within the cores of specific classic games when the original game is convex

Complete null agent for games with externalities

Game theory provides valuable tools to examine expert multi-agent systems. In a cooperative game, collaboration among agents leads to better outcomes. The most important solution for such games is the Shapley value, that coincides with the expected marginal contribution assuming equiprobability. This assumption is not plausible when externalities are present in an expert system. Generalizing the concept of marginal contributions, we propose a new family of Shapley values for situations with externalities. The properties of the Shapley value offer a rationale for its application. This family of values is characterized by extensions of Shapley's axioms: efficiency, additivity, symmetry, and the null player property. The first three axioms have widely accepted generalizations to the framework of games with externalities. However, different concepts of null players have been proposed in the literature and we contribute to this debate with a new one. The null player property that we use is weaker than the others. Finally, we present one particular value of the family, new in the literature, and characterize it by two additional properties

The family of lattice structure values for games with externalities

We propose and characterize a new family of Shapley values for games with coalitional externalities. To define it we generalize the concept of marginal contribution by using a lattice structure on the set of embedded coalitions. The family of lattice structure values is characterized by extensions of Shapley's axioms: efficiency, additivity, symmetry, and the null player property. The first three axioms have widely accepted generalizations to the framework of games with externalities. However, different concepts of null players have been proposed in the literature and we contribute to this debate with a new one. The null player property that we use is weaker than the others. Finally, we present one particular value of the family, new in the literature, which delivers balanced payoffs and characterize it by two additional properties