Values for two-stage games: Another view of the Shapley axioms (original) (raw)

Axiomatizations of two types of Shapley values for games on union closed systems

Economic Theory, 2011

A situation in which a finite set of players can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utility, or simply a TU-game. A (single-valued) solution for TU-games assigns a payoff distribution to every TU-game. A well-known solution is the Shapley value. In the literature various models of games with restricted cooperation can be found. So, instead of allowing all subsets of the player set N to form, it is assumed that the set of feasible coalitions is a subset of the power set of N. In this paper, we consider such sets of feasible coalitions that are closed under union, i.e. for any two feasible coalitions also their union is feasible. We consider and axiomatize two solutions or rules for these games that generalize the Shapley value: one is obtained as the conjunctive permission value using a corresponding superior graph, the other is defined as the Shapley value of a modified game similar as the Myerson value for games with limited communication.

Axiomatisation of the Shapley value and power index for bi-cooperative games

2006

Bi-cooperative games have been introduced by Bilbao as a generalization of classical cooperative games, where each player can participate positively to the game (defender), negatively (defeater), or do not participate (abstentionist). In a voting situation (simple games), they coincide with ternary voting game of Felsenthal and Mochover, where each voter can vote in favor, against or abstain. In this paper, we propose a definition of value or solution concept for bi-cooperative games, close to the Shapley value, and we give an interpretation of this value in the framework of (ternary) simple games, in the spirit of Shapley-Shubik, using the notion of swing. Lastly, we compare our definition with the one of Felsenthal and Machover, based on the notion of ternary roll-call, and the Shapley value of multi-choice games proposed by Hsiao and Ragahavan.

The axiomatic approach to three values in games with coalition structure

European Journal of Operational Research, 2009

We study three values for transferable utility games with coalition structure, including the Owen coalitional value and two weighted versions with weights given by the size of the coalitions. We provide three axiomatic characterizations using the properties of Efficiency, Linearity, Independence of Null Coalitions, and Coordination, with two versions of Balanced Contributions inside a Coalition and Weighted Sharing in Unanimity Games, respectively.

Axiomatization of values of cooperative games using a fairness property

Applicationes Mathematicae, 2005

We propose new systems of axioms which characterize four types of values of cooperative games: the Banzhaf value, the Deegan-Packel value, the least square prenucleolus and the least square nucleolus. The common element used in these axiomatizations is a fairness property. It requires that if to a cooperative game we add another game in which two given players are symmetric, then their payoffs change by the same amount. In our analysis we will use an idea applied by R. van den Brink (2001) to obtain an axiomatic characterization of the Shapley value.

The Shapley value for bicooperative games

2004

and centrA: RESUMEN El objetivo de este trabajo es analizar un concepto de solución que asigna a cada juego bicooperativo un único vector. En el contexto de los juegos bicooperativos introducidos por Bilbao (2000), definimos una solución denominada valor de Shapley porque este valor puede interpretarse de una ma nera semejante al clásico valor de Shapley para juegos cooperativos. El resultado más importante del trabajo es una caracterización axiomática de este valor.

Axiomatizations of the Shapley value for cooperative games on antimatroids

Mathematical Methods of Operations Research (ZOR), 2003

Games on antimatroids are cooperative games restricted by a combinatorial structure which generalize the permission structure. So, cooperative games on antimatroids group several well-known families of games which have important applications in economic and politic. Therefore, the study of the rectricted games by antimatroids allows to unify criteria of various lines of research. The current paper establishes axioms that determine the restricted Shapley value on antimatroids by conditions on the cooperative game v and the structure determined by the antimatroid. This axiomatization generalizes the axiomatizations of both the conjunctive and disjunctive permission value for games with a permission structure. We also provide an axiomatization of the Shapley value restricted to the smaller class of poset antimatroids.

THE SHAPLEY-SOLIDARITY VALUE FOR GAMES WITH A COALITION STRUCTURE

International Game Theory Review, 2013

A value for games with a coalition structure is introduced, where the rules guiding cooperation among the members of the same coalition are different from the interaction rules among coalitions. In particular, players inside a coalition exhibit a greater degree of solidarity than they are willing to use with players outside their coalition. The Shapley value is therefore used to compute the aggregate payoffs for the coalitions, and the solidarity value to obtain the payoffs for the players inside each coalition.

Bargaining without a planner: a non-cooperative approach to the Shapley

We present a simple mechanism of negotiation, based on offers and counteroffers. This leads to a unified solution theory for nontransferable utility (NTU) games that has as special cases the Nash bargaining solution for pure bargaining problems, the Shapley value for transfer utility (TU) games, and the Shapley NTU value for general cooperative games. These results are similar to those of the bargaining mechanism of Hart and yielding the consistent value Owen (1898, 1992)). The mechanism presented here solves some problematic issues in Hart and Mas-Colell's model. Furthermore, a natural extension to games with coalition structure, yielding the Owen value (Owen (1977)) for TU games is provided.

Axiomatizations of the Shapley value for games on augmenting systems

European Journal of Operational Research, 2009

This paper deals with cooperative games in which only certain coalitions are allowed to form. There have been previous models developed to confront the problem of unallowable coalitions. Games restricted by a communication graph were introduced by Myerson and Owen. In their model, the feasible coalitions are those that induce connected subgraphs. Another type of model is introduced in Gilles, Owen and van den Brink. In their model, the possibilities of coalition formation are determined by the positions of the players in a so-called permission structure. Faigle proposed another model for cooperative games defined on lattice structures. We introduce a combinatorial structure called augmenting system which is a generalization of the antimatroid structure and the system of connected subgraphs of a graph. In this framework, the Shapley value of games on augmenting systems is introduced and two axiomatizations of this value are showed.

Analyzing cooperative game theory solutions: core and Shapley value in cartesian product of two sets

Frontiers in applied mathematics and statistics, 2024

The core and the Shapley value stand out as the most renowned solutions for addressing sharing problems in cooperative game theory. These concepts are widely acknowledged for their e ectiveness in tackling negotiation, resource allocation, and power dynamics. The present study aims to extend various notions of cooperative games from the standard set N to a new class of cooperative games defined on the cartesian product N × N ′ (utilizing the specific coalition A * B). This extension encompasses fundamental concepts such as rationality, core, and Shapley value. The findings presented in this study demonstrate that the core concept as a solution yields a set of imputations without favoring any specific point within the set, in contrast to the Shapley value, which o ers a singular solution. Moreover, the results confirm that the Shapley value satisfies the conditions defining the core of a game. Through both theoretical analysis and numerical findings, employing a practical example, it becomes evident that the Shapley value o ers a more distinct solution to the sharing problem compared with the core solution.