Emilio Calvo

Potentials in cooperative TU-games

Abstract This paper is devoted to the study of solutions for cooperative TU-games which admit a potential function, such as the potential associated with the Shapley value (introduced by Hart and Mas-Colell). We consider the finite case and the finite type continuum. Several characterizations of this family are offered and, as a main result, it is shown that each of these solutions can be obtained by applying the Shapley value to an appropriately modified game. We also study the relationship of the potential with the noncooperative potential games, introduced by Monderer and Shapley, for the multilinear case in the continuum finite type setting.

A value for multichoice games

Abstract A multichoice game is a generalization of a cooperative TU game in which each player has several activity levels. We study the solution for these games proposed by Van Den Nouweland et al. (1995) [Van Den Nouweland, A., Potters, J., Tijs, S., Zarzuelo, J.M., 1995. Cores and related solution concepts for multi-choice games. ZOR-Mathematical Methods of Operations Research 41, 289–311]. We show that this solution applied to the discrete cost sharing model coincides with the Aumann-Shapley method proposed by Moulin (1995) [Moulin, H., 1995. On additive methods to share joint costs. The Japanese Economic Review 46, 303–332]. Also, we show that the Aumann-Shapley value for continuum games can be obtained as the limit of multichoice values for admissible convergence sequences of multichoice games. Finally, we characterize this solution by using the axioms of balanced contributions and efficiency.

The principle of balanced contributions and hierarchies of cooperation

The principle of balanced contributions has appeared repeatedly in the literature on the Shapley value. This principle is akin to the reciprocity properties shared by almost all cooperative solution concepts. We provide a new axiomatization for the level structure value. This axiomatization has the advantage that it can be applied to many important subdomains of TU games. We use the Hart-Mas-Colell potential function as a tool to prove our main result, and establish another interesting characterization for the value as a by-product.

Values of games with probabilistic graphs

Abstract In this paper we consider games with probabilistic graphs. The model we develop is an extension of the model of games with communication restrictions by Myerson (1977) . In the Myerson model each pair of players is joined by a link in the graph if and only if these two players can communicate directly. The current paper considers a more general setting in which each pair of players has some probability of direct communication. The value is defined and characterized in this context. It is a natural extension of the Myerson value and it turns out to be the Shapley value of a modified game.

Dynamic Models of International Environmental Agreements: A Differential Game Approach

This article provides a survey of dynamic models of international environmental agreements (IEAs). The focus is on environmental problems that are caused by a stock pollutant as are the cases of the acid rain and climate change. For this reason, the survey only reviews the literature that utilizes dynamic state-space games to analyze the formation of international agreements to control pollution. The survey considers both the cooperative approach and the noncooperative approach. In the case of the latter, the survey distinguishes between the models that assume binding agreements and those that assume the contrary. An evaluation of the state of the art is presented in the conclusions along with suggestions for future research.

Weighted weak semivalues

Abstract. We introduce two new value solutions: weak semivalues and weighted weak semivalues. They are subfamilies of probabilistic values, and they appear by adding the axioms of balanced contributions and weighted balanced contributions respectively. We show that the effect of the introduction of these axioms is the appearance of consistency in the beliefs of players about the game.

Extension of the Perles-Maschler solution to N-person bargaining games

The superadditive solution for 2-person Nash bargaining games was axiomatically defined in Perles/Maschler (1981). In Perles (1982) it was shown that the axioms are incompatible even for 3-person bargaining games. In this paper we offer a generalization method of this solution concept for n-person games. In this method, the Kalai-Smorodinsky solution (1975) is revealed as the rule followed to determine the movements along the path of intermediate agreements.

Solidarity in games with a coalition structure

A new axiomatic characterization of the two-step Shapley value Kamijo (2009) is presented based on a solidarity principle of the members of any union: when the game changes due to the addition or deletion of players outside the union, all members of the union will share the same gains/losses.

Random Marginal and Random Removal values

We propose two variations of the non-cooperative bargaining model for games in coalitional form, introduced by Hart and Mas-Colell (Econometrica 64:357–380, 1996a). These strategic games implement, in the limit, two new NTU-values: the random marginal and the random removal values. Their main characteristic is that they always select a unique payoff allocation in NTU-games. The random marginal value coincides with the Consistent NTU-value (Maschler and Owen in Int J Game Theory 18:389–407, 1989) for hyperplane games, and with the Shapley value for TU games (Shapley in In: Contributions to the theory of Games II. Princeton University Press, Princeton, pp 307–317, 1953). The random removal value coincides with the solidarity value (Nowak and Radzik in Int J Game Theory 23:43–48, 1994) in TU-games. In large games we show that, in the special class of market games, the random marginal value coincides with the Shapley NTU-value (Shapley in In: La Décision. Editions du CNRS, Paris, 1969), and that the random removal value coincides with the equal split value.

THE SHAPLEY-SOLIDARITY VALUE FOR GAMES WITH A COALITION STRUCTURE

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.