Two extensions of the Shapley value for cooperative games

Two extensions of the Shapley value are given. First we consider a probabilistic framework in which certain consistent allocation rules such as the Shapley value are characterized. The second generalization of the Shapley value is an extension to the structure of posets by means of a recursive form. In the latter setting, the Shapley value for quasi-concave games is shown to be a core-allocation. \u

Steady Marginality: A Uniform Approach to Shapley Value for Games with Externalities

The Shapley value is one of the most important solution concepts in cooperative game theory. In coalitional games without externalities, it allows to compute a unique payoff division that meets certain desirable fairness axioms. However, in many realistic applications where externalities are present, Shapley's axioms fail to indicate such a unique division. Consequently, there are many extensions of Shapley value to the environment with externalities proposed in the literature built upon additional axioms. Two important such extensions are "externality-free" value by Pham Do and Norde and value that "absorbed all externalities" by McQuillin. They are good reference points in a space of potential payoff divisions for coalitional games with externalities as they limit the space at two opposite extremes. In a recent, important publication, De Clippel and Serrano presented a marginality-based axiomatization of the value by Pham Do Norde. In this paper, we propose a dual approach to marginality which allows us to derive the value of McQuillin. Thus, we close the picture outlined by De Clippel and Serrano

Solutions for cooperative games with restricted coalition formation and almost core allocations

The thesis focuses on cooperative games with transferable utility and incorporates two topics: Solutions for TU-games with restricted coalition formation and the stability of the grand coalition. Chapters 3 is devoted to a new solution for cooperative games with coalition structures, called the α-egalitarian Owen value, and this coalitional value is characterized by three approaches. Firstly, we provide two axiomatizations by introducing the α-indemnificatory null player axiom, and the (intra) coalitional quasi-balanced contributions axiom. Secondly, we characterize the coalitional value by introducing an α-guarantee potential function. Finally, the coalitional value is implemented by a punishment-reward bidding mechanism. In Chapter 4, we continue to work with TU-games restricted by coalition structures and propose a coalitional value called the two-step Shapley-solidarity value. A procedural interpretation is provided for this coalitional value, and we introduce a new axiom called the coalitional A-null player axiom to axiomatize the value based on additivity. Moreover, two other axiomatizations on the basis of quasi-balanced contributions for the grand coalition are also provided. In Chapter 5, we focus on cooperative games with communication structures and provide efficient extensions of the Myerson value (Myerson, 1977). The idea lies in introducing the Shapley payoffs of the underlying game as players' claims to derive a graph-induced bankruptcy problem. Then, two efficient extensions of the Myerson value are achieved through bankruptcy rules, including the CEA rule and the CEL rule (Aumann &amp; Maschler, 1985).Moreover, corresponding axiomatizations are also provided.Chapter 6 proceeds with studying the stability of the grand coalition for cost TU-games by addressing an optimization problem to maximize the total shareable cost over what we called the almost core. We analyze the computational complexity of this optimization problem, in relation to the computational complexity of related problems for the core. In particular, we consider a special class of games, i.e., the minimum cost spanning tree games. We show that maximizing the total shareable costs over the (non-negative) almost core is NP-hard for mcst games, and we provide a tight 2-approximation algorithm for this almost core optimization problem with the additional non-negative constraint. <br/

Complete null agent for games with externalities

Game theory provides valuable tools to examine expert multi-agent systems. In a cooperative game, collaboration among agents leads to better outcomes. The most important solution for such games is the Shapley value, that coincides with the expected marginal contribution assuming equiprobability. This assumption is not plausible when externalities are present in an expert system. Generalizing the concept of marginal contributions, we propose a new family of Shapley values for situations with externalities. The properties of the Shapley value offer a rationale for its application. This family of values is characterized by extensions of Shapley's axioms: efficiency, additivity, symmetry, and the null player property. The first three axioms have widely accepted generalizations to the framework of games with externalities. However, different concepts of null players have been proposed in the literature and we contribute to this debate with a new one. The null player property that we use is weaker than the others. Finally, we present one particular value of the family, new in the literature, and characterize it by two additional properties

Applying Abstract Argumentation Theory to Cooperative Game Theory

We apply ideas from abstract argumentation theory to study cooperative game theory. Building on Dung's results in his seminal paper, we further the correspondence between Dung's four argumentation semantics and solution concepts in cooperative game theory by showing that complete extensions (the grounded extension) correspond to Roth's subsolutions (respectively, the supercore). We then investigate the relationship between well-founded argumentation frameworks and convex games, where in each case the semantics (respectively, solution concepts) coincide; we prove that three-player convex games do not in general have well-founded argumentation frameworks.Comment: 15 pages, 1 tabl

Bisemivalues for bicooperative games

We introduce bisemivalues for bicooperative games and we also provide an interesting characterization of this kind of values by means of weighting coefficients in a similar way as it was given for semivalues in the context of cooperative games. Moreover, the notion of induced bisemivalues on lower cardinalities also makes sense and an adaptation of Dragan’s recurrence formula is obtained. For the particular case of (p, q)-bisemivalues, a computational procedure in terms of the multilinear extension of the game is given.Peer ReviewedPostprint (author's final draft