Juan Carlos Santos

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.

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.

On the serial cost sharing rule

Abstract. In this paper we study a solution for discrete cost allocation problems, namely, the serial cost sharing method. We show that this solution can be computed by applying the Shapley value to an appropriate TU game and we present a probabilistic formula. We also define for cost allocation problems a multilinear function in order to obtain the serial cost sharing method as Owen (1972) did for the Shapley value in the cooperative TU context. Moreover we show that the pseudo average cost method is equivalent to an extended Shapley value.

The Serial Property and Restricted Balanced Contributions in Discrete Cost Sharing Problems

We show that the Serial Poperty and Restricted Balanced Contributions characterize the subsidy-free serial cost sharing method (Moulin (1995)) in discrete cost allocation problems.

Cost allocation schemes: An asymptotic approach

Abstract We use the asymptotic approach to the cost allocation problem. In this way, we find the limit of linear solutions on discrete cost allocation problems which define solutions on continuum cost allocation problems. Particularly we analyze the Shapley–Shubik method, the discrete Aumann–Shapley method, the serial cost sharing method and the pseudo-average cost method.

The multichoice consistent value

Abstract. We consider multichoice NTU games, i.e., cooperative NTU games in which players can participate in the game with several levels of activity. For these games, we define and characterize axiomatically the multichoice consistent value, which is a generalization of the consistent NTU value for NTU games and of the multichoice value for multichoice TU games. Moreover, we show that this value coincides with the consistent NTU value of a replicated NTU game and we provide a probabilistic interpretation.

Mixing weighted values of non-atomic games

Abstract. Weighted values of non-atomic games were introduced by Hart and Monderer (1997). They study these values by using two approaches: the potential approach and the asymptotic approach. In this study we develop the random order approach (the mixing value, Aumann and Shapley, 1974) to weighted values and prove that these values coincide with the asymptotic weighted values of Hart and Monderer in pNA.

A characterization of the pseudo-average cost method

Abstract We study the pseudo-average cost method in discrete and continuum cost allocation problems. This solution was introduced, in the discrete case, by Moulin (Moulin, H., 1995. On additive methods to share joint costs. The Japanese Economic Review 46, 303-332). In the continuous case, this solution has been introduced, asymptotically, by Larrea and Santos (Larrea, C., Santos, J.C., 2006. Cost allocation schemes: an asymptotic approach. Games and Economic Behavior 57, 63–72). Here we propose a characterization of this solution using an adaptation of the balanced contributions axioms, used by Myerson (Myerson, R., 1980. Conference structures and fair allocation rules. International Journal of Game Theory 9, 169-182) to characterize the Shapley value in TU games. The new axiom, in the continuous case, is the natural extension of the axiom used in the discrete case.

Weighted values for non-atomic games: an axiomatic approach

Abstract. Weighted values of non-atomic games were introduced by Hart and Monderer. These values have been studied by using three approaches: the potential, the asymptotic and the random order approach. In this study we analyze the axiomatic approach for one class of weight functions: the set of players is partitioned into a finite number of types, with each type having different weight.