José Manuel Zarzuelo

Cores and related solution concepts for multi-choice games

A multi-choice game is a generalization of a cooperative game in which each player has several activity levels. Cooperative games form a subclass of the class of multi-choice games.This paper extends some solution concepts for cooperative games to multi-choice games. In particular, the notions of core, dominance core and Weber set are extended. Relations between cores and dominance cores and between cores and Weber sets are extensively studied. A class of flow games is introduced and relations with non-negative games with non-empty cores are investigated.

The least square prenucleolus and the least square nucleolus. Two values for TU games based on the excess vector

The nucleolus and the prenucleolus are solution concepts for TU games based on the excess vector that can be associated to any payoff vector. Here we explore some solution concepts resulting from a payoff vector selection based also on the excess vector but by means of an assessment of their relative fairness different from that given by the lexicographical order. We take the departure consisting of choosing the payoff vector which minimizes the variance of the resulting excesses of the coalitions. This procedure yields two interesting solution concepts, both a prenucleolus-like and a nucleolus-like notion, depending on which set is chosen to set up the minimizing problem: the set of efficient payoff vectors or the set of inputations. These solution concepts, which, paralleling the prenucleolus and the nucleolus, we call least square prenucleolus and least square nucleolus, are easy to calculate and exhibit nice properties. Different axiomatic characterizations of the former are established, some of them by means of consistency for a reasonable reduced game concept.

Configuration values: Extensions of the coalitional Owen value

Abstract It is proposed an extension of the Owens coalitional value so as to consider the possibility that players organize themselves in coalitions that are not necessarily disjoint. This extension is made through a random order approach. To complement this approach we offer a characterization of this value and a heuristic approach similar to the one given by Owen for the coalitional value. Moreover we introduce a second value that can be considered as the dual of the former one, which also extends the coalitional value. Furthermore, we study the stability of the coalition configurations.

Characterizations of a multi-choice value

Abstract. A multi-choice game is a generalization of a cooperative game in which each player has several activity levels. We study the extended Shapley value as proposed by Derks and Peters (1993). Van den Nouweland (1993) provided a characterization that is an extension of Youngs (1985) characterization of the Shapley value. Here we provide several other characterizations, one of which is the analogue of Shapleys (1953) original characterization. The three other characterizations are inspired by Myersons (1980) characterization of the Shapley value using balanced contributions.

On coalitional semivalues

In this paper we propose a characterization of the coalitional value for transferable utility games (Owen, 1977), and we define and study coalitional semivalues, which are generalizations of semivalues (Dubey, Neyman and Weber, 1981).

The Bilateral Consistent Prekernel for NTU Games

Abstract A bilateral consistency principle is used to generalize the Nash bargaining solution to a new NTU solution concept. It coincides with the prekernel for TU games and can be formulated in terms of a generalized concept of excess. Journal of Economic Literature Classification Number: C71.

The egalitarian solution for convex games: some characterizations

The egalitarian solution for TU-games as introduced by Dutta and Ray [Dutta, B., Ray, D., 1989. A concept of egalitarianism under participation constraints, Econometrica 57, 615–635] is studied. Five characterizations of the restriction of this solution to the class of convex games are given. They all involve a stability property due to the concept of the equal division core from Selten [Selten, R., 1972. Equal share analysis of characteristic function experiments. In: Sauermann, H. (Ed.), Beitrage zur experimentellen Wirtschaftsforschung, Vol. 3, J.C.B. Mohr, Tubingen, pp. 130–165] and all but the third characterization involve a property restricting maximum payoffs. The first two characterizations use in addition efficiency and the reduced game properties of Hart and Mas-Colell [Hart, S., Mas-Colell, A., 1989. Potential, value and consistency. Econometrica 57, 589–614] and Davis and Maschler [Davis, M., Maschler, M., 1965. The kernel of a cooperative game. Naval Research Logistics Quarterly 12, 223–259], respectively. The fourth and fifth characterization only need in addition weak variants of the reduced game properties mentioned above. The third characterization involves besides the stability condition, efficiency and a new consistency property.

The bilateral consistent prekernel, the core, and NTU bankruptcy problems

It is shown that the bilateral consistent prekernel, an NTU solution concept that generalizes the Nash bargaining solution by means of a principle of bilateral consistency, is nonempty and intersects the core for a sufficiently rich class of NTU games: the class of boundary separating games, also introduced here. A second contribution of this paper is the NTU extension of the bankruptcy games introduced by Aumann and Maschler (1985). These games, which provide a new approach for dealing with bargaining problems with claims, are shown to be boundary separating, so that the results apply.

A Simple Algorithm for the Nucleolus of Airport Profit Games

In this paper we present a procedure for calculating the nucleolus for airport profit games which are a generalization of the airport cost games.

Some new results on Least Square Values for TU games

In this paper we present some new results concerning the least square family of values for TU games. We first examine the stability of these values from a dynamic point of view and propose a reinteropretation of them from the standpoint of the propernsity to disrupt approach. In the second part the family of individually rational least square (IRLS) values is introduced and an alternative kernel-like formulation of them is provided. Finally we describe a natural and simple algorithm for calculating any IRLS value of a game.