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Abstract:The nucleolus is a central concept of solution in the theory of cooperative n person games with side payments; it has been introduced and studied by Sehmeidler [1969] and several methods for finding the nucleolus have been proposed by Kopelowitz [1967 ], Bruyneel [1979 ], Stearns [1968] and Justman [ 1977 ], respectively. The aim of the present paper is that of giving a new algorithm for finding the nucleolus and to discuss the relationship of this algorithm with those given by Kopelowitz and Bruyneel.The algorithm is based upon the concept of minimal balanced set of a finite set;this last concept has been introduced for other purposes by Shapley [1967]. The relationship between the nucleolus and the balanced sets has been studied by Kohlberg [ 1971 ], where it has been shown that the so-called coalition array of an imputation is the coalition array of the nucleolus iff some parts of it are balanced sets. Our algorithm computes such a coalition array by finding a sequence of minimal balanced sets. Any element of the sequence can be found be solving a LP problem, then the nucleolus is easily found from the coalition array. The algorithm is in some sense a dual of the Kopelowitz' algorithm. It clarifies completely the relationship between the nueleolus and the minimal balanced sets, that allowed the statement of the Bruyneel's algorithm; moreover, our algorithm doesn't assume the knowledge of the list of weight vectors associated to the set of minimal balanced sets, but constructs only the part of the list needed for finding the nucleolus.Zusammenfassung: Ein kooperatives n-Personen-Spiel wird dutch eine endliche Menge N (die Spielermenge) und eine nicht additive Mengenfunktion v, definiert auf der Potenzmenge yon N (d.h. auf den Koalitionen), charakterisiert. Auf Schmeidler geht der Begriff des Nukleolus als eines ftir ein kooperatives Spiel geeigneten L6sungskonzeptes zurtick. Bruyneel und Kopelowitz haben j ewefts Algorithmen zur Berechnung des Nukleolus eines vorgegebenen kooperativen Spieles angegeben. Das vorliegende Papier gibt einen weiteren Algorithmus an. Dieser ist -~ihnlich wie der von Bruyneel entwickelte -begriffiich gestiitzt auf das Konzept der minimal balancierten Koalitionssysteme (eingef'tihrt yon Shapley). In seiner direkten Form ben6tigt der Algorithmus die Liste aller zu minimal balancierten Mengensystemen geh6renden Gewichtsvektoren, jedoch wird in Abschnitt 2 eine Methode angegeben, diese Liste mit Hilfe einer Folge linearer Programme zu vermei-1 ) Prof. Dr. Irinel Dragan,

In the theory of cooperative transferable utilities games, (TU games), the Efficient Values, that is those which show how the win of the grand coalition is shared by the players, may not be a good solution to give a fair outcome to each player. In an earlier work of the author, the Inverse Problem has been stated and explicitely solved for the Shapley Value and for the Least Square Values. In the present paper, for a given vector, which is the Shapley Value of a game, but it is not coalitional rational, that is it does not belong to the Core of the game, we would like to find out a new game with the Shapley Value equal to the a priori given vector and for which this vector is also in the Core of the game. In other words, in the Inverse Set relative to the Shapley Value, we want to find out a new game, for which the Shapley Value is coalitional rational. The results show how such a game may be obtained, and some examples are illustrating the technique. Moreover, it is shown that beside the original game, there are always other games for which the given vector is not in the Core. The similar problem is solved for the Least Square Values.

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The Semivalues were introduced by Dubey, Neiman and Weber (1981). as thc values of TU games satisfying a set of axioms, precisely: linearity, symmetry, monotonicity and projection axioms. In thin paper, a potential approach is used to prove a characterization of Semivalues as the unique values satisfying a new type of consistency relative to a new implicitly defined reduced game of HartIMas CoIelI type and weighted standardness for all two person games. The definition of the reduced game requires some combinatorial results on the auxiliary game of the given game. As a byproduct, we derive from these results two other combinatorial properties of the Semivalues: (i) any Semivalue is the Shapley value of the auxiliary game of the given game, and (ii) any Semivalue satisfies the fairness principle introduced by Myerson (1977) as the principle of balanced contributions.