A value for cooperative games with a coalition structure!

International Journal of Game Theory, 2008

Bi-cooperative games have been introduced by Bilbao et al as a generalization of TU cooperative games, where each player can participate positively to the game, negatively, or do not participate. In this paper, we propose a definition of a share of the wealth obtained by some players after they decided on their participation to the game. It turns out that the cost allocation rule does not look for a given player to her contribution at the opposite action to the one she chooses. The relevance of the value is discussed on several examples.

2009

Coalitional network games are real-valued functions defined on a set of play-ers (the society) organized into a network and a coalition structure. The network specifies the nature of the relationship each individual has with the other individ-uals and the coalition structure specifies a collection of groups among the society. Coalitional network games model situations where the total productive value of a network among players depends on the players ’ group membership. These games thus capture the public good aspect of bilateral cooperation, i.e., network games with externalities. After studying the specific structure of coalitional networks, we propose an allocation rule under the perspective that players can alter the coalitional network structure. This means that the value of all potential alternative coalitional networks can and should influence the allocation of value among players in any given coalitional network structure. JEL classification: A14, C70.

Theory and Decision, 1977

Coalitions are frequently more visible than payoffs. The theory of nperson games seeks primarily to identify stable allocations of valued resources; consequently, it gives inadequate attention to predicting which coalitions form. This paper explores a way of correcting this deficiency of game-theoretic reasoning by extending the theory of two-person cooperative games to predict both coalitions and payoffs in a three-person 'game of status' in which each player seeks to maximize the rank of his total score. 1 To accomplish this, we analyze the negotiations within each potential two-person coalition from the perspective of Nash's procedure for arbitrating two-person bargaining games, 2 then assume that players expect to achieve the arbitrated outcome selected by this procedure and use these expectations to predict achieved ranks and to identify players' preferences between alternative coalition partners in order to predict the probability that each coalition forms. 3 We test these payoff and coalition predictions with data from three laboratory studies, and compare the results with those attained in the same data by yon Neumann and Morgenstern's solution of two-person cooperative games, a Aumann and Maschler's bargaining set solution for cooperative n-person games, 5 and an alternative model of coalition behavior in three-person sequential games of status. 6

2004

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.

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.

In the usual models of cooperative game theory, the outcome of a coalition formation process is either the grand coalition or a coalition structure that consists of disjoint coalitions. However, in many domains where coalitions are associated with tasks, an agent may be involved in executing more than one task, and thus may distribute his resources among several coalitions. To tackle such scenarios, we introduce a model for cooperative games with overlapping coalitions-or overlapping coalition formation (OCF) games. We then explore the issue of stability in this setting. In particular, we introduce a notion of the core, which generalizes the corresponding notion in the traditional (non-overlapping) scenario. Then, under some quite general conditions, we characterize the elements of the core, and show that any element of the core maximizes the social welfare. We also introduce a concept of balancedness for overlapping coalitional games, and use it to characterize coalition structures that can be extended to elements of the core. Finally, we generalize the notion of convexity to our setting, and show that under some natural assumptions convex games have a non-empty core. Moreover, we introduce two alternative notions of stability in OCF that allow a wider range of deviations, and explore the relationships among the corresponding definitions of the core, as well as the classic (non-overlapping) core and the Aubin core. We illustrate the general properties of the three cores, and also study them from a computational perspective, thus obtaining additional insights into their fundamental structure.

TOP, 2008

In this paper a simple probabilistic model of coalition formation provides a uni ed interpretation for several extensions of the Shapley value. Weighted Shapley values, semivalues, and weak (weighted or not) semivalues, and the Shapley value itself appear as variations of this model. Moreover, some notions that have been introduced in the search of alternatives to Shapley's seminal characterization, as 'balanced contributions' and the 'potential' are reinterpreted from this point of view. The natural relationships of these conditions with some the mentioned families of 'values' are shown. These reinterpretations strongly suggest that these conditions are more naturally interpreted in terms of coalition formation than in terms of the classical notion of 'value'.

Annals of Operations Research

In designing solution concepts for cooperative games with transferable utilities, consolidation of marginalism and egalitarianism has been widely studied. The α-Egalitarian Shapley value is one such solution that combines the Shapley value and the Equal Division rule, the two most popular extreme instances of marginalism and egalitarianism respectively. This value gives the planner the flexibility to choose the level of marginality for the players by varying the convexity parameter α. In this paper, we define the Generalized Egalitarian Shapley value that gives the planner more flexibility in choosing the level of marginality based on the coalition size. We then provide two characterizations of the Generalized Egalitarian Shapley value.

2010

Until recently, computational aspects of the Shapley value were only studied under the assumption that there are no externalities from coalition formation, i.e., that the value of any coalition is independent of other coalitions in the system. However, externalities play a key role in many real-life situations and have been extensively studied in the game-theoretic and economic literature. In this paper, we consider the issue of computing extensions of the Shapley value to coalitional games with externalities proposed by Myerson [21], Pham Do and Norde and McQuillin [17]. To facilitate efficient computation of these extensions, we propose a new representation for coalitional games with externalities, which is based on weighted logical expressions. We demonstrate that this representation is fully expressive and, sometimes, exponentially more concise than the conventional partition function game model. Furthermore, it allows us to compute the aforementioned extensions of the Shapley value in time linear in the size of the input.