Balbina Casas-Méndez

Computing power indices: Multilinear extensions and new characterizations

Abstract 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.

An extension of the [tau]-value to games with coalition structures

Abstract We introduce the coalitional τ -value, which is an extension of the τ -value for TU-games to games with a coalition structure. We identify a class of TU-games that satisfy the property that for every game in this class and every coalition structure on its player set it holds that the coalitional τ -value can be defined for the corresponding game with a coalition structure. We study properties of the coalitional τ -value and provide an axiomatic characterization of this allocation rule. We use the coalitional τ -value to study bankruptcy problems and the determination of aircraft landing fees.

Characterizations of the Deegan-Packel and Johnston power indices

Abstract 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.

The two-stage constrained equal awards and losses rules for multi-issue allocation situations

This paper considers two-stage solutions for multi-issue allocation situations, which are extensions of bankruptcy problems. Characterizations are provided for the two-stage constrained equal awards and constrained equal losses rules, based on the properties of composition up and composition down.

Values for Strategic Games in Which Players Cooperate

In this paper we propose a method to associate a coalitional interval game with each strategic game. The method is based on the lower and upper values of finite two-person zero-sum games. We axiomatically characterize this new method. As an intermediate step, we provide some axiomatic characterizations of the upper value of finite two-person zero-sum games.

The Deegan-Packel Index for Simple Games with a Priori Unions

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.

Balanced contributions for TU games with awards and applications

Abstract Calleja et al. [Calleja, P., Borm, P., Hendrickx, R., 2005. Multi-issue allocation situations. European Journal of Operational Research 164, 730–747] introduced multi-issue allocation situations with awards. In this paper, we extend the classical model of cooperative games with transferable utility to the cooperative games with transferable utility and awards. We define a run-to-the-bank rule for cooperative games with transferable utility and awards and characterise it in terms of a property of balanced contributions. We apply our main result to bankruptcy problems and multi-issue allocation situations with awards.

The Constrained Equal Awards Rule for Bankruptcy Problems with a Priori Unions

In this paper, we provide two extensions of the constrained equal awards rule for bankruptcy situations to the class of bankruptcy situations with a priori unions. We present some characterisations and relations with corresponding games. The two new extensions are illustrated by a specific application.

The Shapley valuation function for strategic games in which players cooperate

In this note we use the Shapley value to define a valuation function. A valuation function associates with every non-empty coalition of players in a strategic game a vector of payoffs for the members of the coalition that provides these players’ valuations of cooperating in the coalition. The Shapley valuation function is defined using the lower-value based method to associate coalitional games with strategic games that was introduced in Carpente et al. (2005). We discuss axiomatic characterizations of the Shapley valuation function.

The truncated core for games with upper bounds

We define and study games with upper bounds. In one of these games there are upper bounds on the possible payoffs for some coalitions. These restrictions require adjustments in the definitions of solution concepts. In the current paper we study the effect of the restrictions on the core and define and study the so-called truncated core.