Stability for Coalition Structures in Terms of the Proportional Partitional Shapley Value

Magaña A., Carreras F.

This paper aims to develop, for any cooperative game, a solution notion that enjoys stability and consists of a coalition structure and an associated payoff vector derived from the Shapley value. To this end, two concepts are combined: those of strong Nash equilibrium and Aumann–Drèze coalitional value. In particular, we are interested in conditions ensuring that the grand coalition is the best preference for all players. Monotonicity, convexity, cohesiveness and other conditions are used to provide several theoretical results that we apply to numerical examples including real-world economic situations.

Alonso-Meijide J.M., Carreras F., Costa J., García-Jurado I.

A new coalitional value is proposed under the hypothesis of isolated unions. The main difference between this value and the Aumann–Drèze value is that the allocations within each union are not given by the Shapley value of the restricted game but proportionally to the Shapley value of the original game. Axiomatic characterizations of the new value, examples illustrating its application and a comparative discussion are provided.

Amer R., Carreras F.

We introduce an allocation rule for measuring power in voting situations defined by a TU-game, a cooperation index and a coalition structure, and characterize it axiomatically. This rule is an extension of the Owen coalition value; in fact, also a variety of previously studied game situations is embodied and unified by our model. Two numerical examples illustrate the application of the new value.

Fiestras-Janeiro M.G., García-Jurado I., Mosquera M.A.

The objective of this paper is to provide a general view of the literature of applications of transferable utility cooperative games to cost allocation problems. This literature is so large that we concentrate on some relevant contributions in three specific areas: transportation, natural resources and power industry. We stress those applications dealing with costs and with problems arisen outside the academic world.

Yang Y.

Koczy and Lauwers, 2004 , Koczy and Lauwers, 2007 show that the collection of absorbing outcomes, i.e., the coalition structure core, of a TU game, if non-empty, is a minimal dominant set. The paper complements the result in two respects. First, it is shown that the coalition structure core, if non-empty, can be reached from any outcome via a sequence of successive blocks in quadratic time. Second, we observe that an analogous result holds for accessible outcomes, namely, the collection of accessible outcomes, if non-empty, is a minimal dominant set. Moreover, we give an existence theorem for accessible outcomes, which implies that the minimal dominant set of a cohesive game is exactly the coalition structure core or the collection of accessible outcomes, either of which can be reached from any outcome in linear time.

TUTIC A.

In this note we present an example of a TU game where both the value presented by Aumann and Drèze (1974) and the value introduced by Wiese (2007) do not exhibit a stable coalition structure.

Casajus A.

In this paper, we introduce a component efficient value for TU games with a coalition structure which reflects the outside options of players within the same structural coalition. It is based on the idea that splitting a coalition should affect players who stay together in the same way. We show that for all TU games there is a coalition structure that is stable with respect to this value.

Amer R., Carreras F.

The weighted Shapley values are extended to situations defined by a cooperative game and a cooperation index. Two main axiomatic characterizations for the modified values are provided. We also develop further axiomatizations of the classical values and discuss two examples.

Myerson R.B.

Graph-theoretic ideas are used to analyze cooperation structures in games. Allocation rules, selecting a payoff for every possible cooperation structure, are studied for games in characteristic function form. Fair allocation rules are defined, and these are proven to be unique, closely related to the Shapley value, and stable for a wide class of games.

Moretti S., Patrone F.

A few applications of the Shapley value are described. The main choice criterion is to look at quite diversified fields, to appreciate how wide is the terrain that has been explored and colonized using this and related tools.

WIESE H.

The paper presents a coalition-structure value that is meant to capture outside options of players in a cooperative game. It deviates from the Aumann-Drèze value by violating the null-player axiom. We use this value as a power index and apply it to weighted majority games.

Carreras F., Magaña A., Munuera C.

In secret sharing, different access structures have different difficulty degrees for acceding to the secret. We give a numerical measure of how easy or how difficult is to recover the secret, depending only on the structure itself and not on the particular scheme used for realizing it. We derive some consequences.

Carreras F.

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

Carreras F., Freixas J., Albina Puente M.

The index introduced here provides a normalized measure of the effectiveness of any semicoherent structure. We study the mathematical properties of the index and derive different axiomatic characterizations for it. Moreover, close relationships are shown to reliability functions and also to the Birnbaum structural importance measure—for which twice the performance index plays the role of potential function—that give rise to computational procedures. Some numerical examples illustrate the application of the index.