Solidarity in 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.

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.

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.

International Journal of Game Theory, 1990

This short study reports an application of the Shapley value axioms to a new concept of "two-stage games." In these games, the formation of a coalition in the first stage entities its members to play a prespecified cooperative game at the second stage. The original Shapley axioms have natural equivalents in the new framework, and we show the existence of (non-unique) values and semivalues for two stage games, analogous to those defined by the corresponding axioms for the conventional (one-stage) games. However, we also prove that all semivalues (hence, perforce, all values) must give patently unacceptable solutions for some "two-stage majority games" (where the members of a majority coalition play a conventional majority game). Our reservations about these prescribed values are related to Roth's (1980) criticism of Shapley's "),-transfer value" for non-transferable utility (NTU) games. But our analysis has wider scope than Roth's example, and the argument that it offers appears to be more conclusive. The study also indicates how the values and semivalues for two-stage games can be naturally generalized to apply for "multi-stage games."

2016

We introduce for any TU-game, a new TU-game referred as its associated solidarity game (ASG). The ASG gives more power to the grand coalition by reducing the payoffs of others coalitions. It comes that, the Shapley value of the ASG is the Solidarity value of the initial game.

Discussion Papers in Economic Behaviour, 2011

A value for games with a coalition structure is introduced, where the rules guiding the cooperation among the members of the same coalition are di¤erent 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 [Shapley, 1953] is therefore used to compute the aggregate payo¤s of the coalitions, and the Solidarity value [Nowak and Radzik, 1994] to obtain the payo¤s of the players inside each coalition.

2017

We present a general bargaining protocol between n players in the setting of coalitional games with transferable utility. We consider asymmetric players. They are endowed with di¤erent probabilities of being chosen as proposers and with di¤erent probabilities of leaving the game if o¤ers are rejected. Two particular speci…cations of this bargaining protocol yield equilibrium proposals that we refer to as weighted solidarity values and weighted Shapley values. We compare the behavior of these values when the players’ probabilities are changed. We supplement the analysis with axiomatic characterizations of both values.

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.

Operational Research

The most popular values in cooperative games with transferable utilities are perhaps the Shapley and the Shapley like values which are based on the notion of players' marginal productivity. The equal division rule on the other hand, is based on egalitarianism where resource is equally divided among players, no matter how productive they are. However none of these values explicitly discuss players' multilateral interactions with peers in deciding to form coalitions and generate worths. In this paper we study the effect of multilateral interactions of a player that accounts for her contributions with her peers not only at an individual level but also on a group level. Based on this idea, we propose a value called the MI k-value and characterize it by the axioms of linearity, anonymity, efficiency and a new axiom: the axiom of MN k-player. An MN kplayer is one whose average marginal contribution due to her multilateral interactions upto level k is zero and can be seen as a generalization of the standard null player axiom of the Shapley value. We have shown that the MI k-value on a variable player set is asymptotically close to the equal division rule. Thus our value realizes solidarity among players by incorporating both their individual and group contributions.

2015

We consider cooperative games in which the cooperation among players is restricted by a set system, which outlines the set of feasible coalitions that actually can be formed by players in the game. In our setting, the structure of this set system is completely free, and the only restriction is that the empty set belongs to it. An extension of the Shapley value is provided as the sum of the dividends that players obtain in the game. In this general setting, we offer two axiomatic characterizations for the value: one by means of component efficiency and fairness, and the other one with efficiency and balanced contributions.