Coincidence of and collinearity between game theoretic solutions

Abstract

Summary. The first part is the study of several conditions which are sufficient for the coincidence of the prenucleolus concept and the egalitarian nonseparable contribution (ENSC-) method. The main sufficient condition for the coincidence involved requires that the maximal excesses at the ENSC-solution are determined by the (n-1)-person coalitions in the n-person game. The second part is the study of both a new type of games, the so-called kcoalitional n-person games, and the interrelationship between solutions on the class of those games. The main results state that the Shapley value of a k-coalitional nperson game can be written as a convex or affine combination of the ENSC-solution and the centre of the imputation set.Zusammenfassung. Im ersten Teil der Arbeit werden verschiedene hinreichende Bedingungen fiir die Koinzidenz des Prenukleolus-L6sungskonzepts und der ENSCRegel vorgestellt. Es wird dabei gezeigt, dab der Prenukleolus mit der ENSC-L6sung zusammenf~illt, falls die maximalen Exzesse der ENSC-L6sung durch die (n-1)-Personen Koalitionen des n-Personenspiels bestimmt werden. Im zweiten Teil der Arbeit untersuchen wir eine Klasse yon Spielen, die sogenannten k-Koalitions-n-Personenspiele und untersuchen die Zusammenh~inge zwischen den LSsungskonzepten ffir diesen speziellen Typ yon Spielen. Es stellt sich heraus, dab der Shapleywert eines k-Koalitions-n-Personenspiels beschrieben werden kann als eine Linearkombination der ENSC-L6sung und des Schwerpunktes der Auszahlungsmenge.

Reconciling marginalism with egalitarianism: consistency, monotonicity, and implementation of egalitarian Shapley values

“…n for all i ∈ N (see Driessen and Funaki 1991) satisfies the axioms of Theorem 4.3 except weak monotonicity. Although the four axioms in Theorem 4.3 are logically independent, we want to search for an axiomatization of the egalitarian Shapley values using weak monotonicity but without linearity.…”

“…14 The CIS-value (see Driessen and Funaki 1991), also called the equal surplus solution, is given by As mentioned at the end of Sect. 3, Joosten (1996) axiomatizes the class of egalitarian Shapley values by a parametrized standardness for two-player games, and a paramatrized Hart and Mas-Colell reduced game consistency.…”

Collinearity between the Shapley value and the egalitarian division rules for cooperative games

“…As a matter of fact, the notion of average worth is our main tool in establishing the collinearity of three of the four values on the class of PAWgames. The collinearity results concerning the k-coalitional games obtained by Driessen and Funaki (1991) are therefore reproved as special cases of the same properties for PAW-games.…”

“…The ENSC-value and the CIS-value are well known concepts in the game theoretic literature (cf. [2], [3], [4], [6], [7], [9], [11], [14]). The ENAC-value has been introduced by Driessen and Funaki (1993a), who presented three motivations for the study of this value 9 The formula (2.12) for the ENAC-value, which is the sum of the egalitarian division of the overall profits and some part of the difference between two average worth with respect to (n-2)-person In the next sections, we shall provide two types of sufficient conditions on the game so that the one-point solutions according to the Shapley value, the ENSC-value (or ENAC-value) and the CIS-value are collinear (i.e., lie on the same line).…”

“…For specific classes of cooperative games, some of the onepoint solutions possess interesting geometric relationships, for example three of the relevant one-point solutions are collinear, i.e., lie on the same line. Driessen and Funaki (1991) studied the collinearity between the Shapley value, the egalitarian non-separable contribution value (called ENSC-value) and the value representing the center of the imputation set (called C1S-value) for the class of cooperative games called the k-coalitional games. Here the collinearity of three values expresses that the corresponding one-point solutions for a certain type of a cooperative game lie on the same line.…”