On weighted Shapley values

Nonsymmetric Shapley values for coalitional form games with transferable utility are studied. The nonsymmetries are modeled through nonsymmetric weight systems defined on the players of the games. It is shown axiomatically that two families of solutions of this type are possible. These families are strongly related to each other through the duality relationship on games. While the first family lends itself to applications of nonsymmetric revenue sharing problems the second family is suitable for applications of cost allocation problems. The intersection of these two families consists essentially of the symmetric Shapley value. These families are also characterized by a probabilistic arrival time to the game approach. It is also demonstrated that lack of symmetries may arise naturally when players in a game represent nonequal size constituencies.     

On ordinal equivalence of the Shapley and Banzhaf values for cooperative games

In this paper I consider the ordinal equivalence of the Shapley and Banzhaf values for TU cooperative games, i.e., cooperative games for which the preorderings on the set of players induced by these two values coincide. To this end I consider several solution concepts within semivalues and introduce three subclasses of games which are called, respectively, weakly complete, semicoherent and coherent cooperative games. A characterization theorem in terms of the ordinal equivalence of some semivalues is given for each of these three classes of cooperative games. In particular, the Shapley and Banzhaf values as well as the segment of semivalues they limit are ordinally equivalent for weakly complete, semicoherent and coherent cooperative games.     

The Owen value values friendship

This paper presents two new axiomatizations of the Owen value for games with coalition structures. Two associated games are defined and a consistency axiom is required. The construction of the associated games presupposes that coalitions behave in an aggressive manner towards players who are not members of the same unions and in a friendly manner towards players that do belong to their unions. The consistency axiom necessitates the definition of only one associated game which is not a reduced game. Received: February 1999/Revised version: January 2000     

On a result of Aumann and Shapley about values of non-atomic games

We focus our attention on the argument developed by R.J. Aumann and L.S. Shapley in a proposition used to prove the existence of a value on a certain class of games. Since we have found such an argument to be inexact, we have devised a supplementary construction which should be incorporated in the original proof to make it run in a satisfactory way.     

Alternative axiomatic characterizations of the Shapley and Banzhaf values

In a paper in 1975, Dubey characterized the Shapley-Shubik index axiomatically on the class of monotonic simple games. In 1979, Dubey and Shapley characterized the Banzhaf index in a similar way. This paper extends these characterizations to axiomatic characterizations of the Shapley and Banzhaf values on the class of control games, on the class of simple games and on the class of all transferable utility games. In particular, it is shown that the additivity axiom which is usually used to characterize these values on the class of all transferable utility games can be weakened without changing the result.This research is sponsored by the Foundation for the Promotion of Research in Economic Sciences, which is part of the Dutch Organization for Scientific Research (NWO).     

Characterizing the Banzhaf and Shapley values assuming limited linearity

This paper presents characterizations of the Banzhaf-Coleman and Shapley-Shubik indices for monotonic simple games. The characterizations are obtained without explicitly requiring that the indices satisfy the linearity assumptionψ (v∧ w) +ψ (v ∨ w) =ψ (v) + ψ (w). The ideas developed are then used to obtain a characterization of the Banzhaf value for the class of alln-person games in characteristic function form.     

Some non-superadditive games, and their Shapley values, in the Talmud

The Shapley value of certain non-superadditive games appears in the Talmudic literature, and in the Talmud itself, in several different contexts.     

On the uniqueness of the Shapley value

L.S. Shapley [1953] showed that there is a unique value defined on the classD of all superadditive cooperative games in characteristic function form (over a finite player setN) which satisfies certain intuitively plausible axioms. Moreover, he raised the question whether an axiomatic foundation could be obtained for a value (not necessarily theShapley value) in the context of the subclassC (respectivelyC′, C″) of simple (respectively simple monotonic, simple superadditive) gamesalone. This paper shows that it is possible to do this. Theorem I gives a new simple proof ofShapley's theorem for the classG ofall games (not necessarily superadditive) overN. The proof contains a procedure for showing that the axioms also uniquely specify theShapley value when they are restricted to certain subclasses ofG, e.g.,C. In addition it provides insight intoShapley's theorem forD itself. Restricted toC′ orC″, Shapley's axioms donot specify a unique value. However it is shown in theorem II that, with a reasonable variant of one of his axioms, a unique value is obtained and, fortunately, it is just theShapley value again.     

On the Asymmetric Shapley Value for Cooperative Games

We discover an interesting relationship between the Shapley value and the asymmetric Shapley value.AMS Subject Classification (2000): Primary 91A06, 91A12, 91B12, 91B74, Secondary 91A80     

On the relationship between modular and hypergeometric functions

We study the relationship between two Hecke theta series, the Dedekind function, and the Gauss hypergeometric function. The main result of the present paper is given by formulas for the representation of the theta series in the form of compositions of the squared Dedekind function, a power of the absolute invariant, and canonical integrals of the second-order hypergeometric differential equation with special values of the three parameters. The proofs of these representations are based on the properties of the matrix transforming the canonical integrals of the Gauss equation in a neighborhood of zero into canonical integrals of the same equation in a neighborhood of unity.     

On the relationship between the Kolmogorov and Henstock integrals

The Henstock integrals are proved to be narrower than the Kolmogorov integral. Translated fromMatematicheskie Zametki, Vol. 60, No. 6, pp. 832–844, December, 1996. In conclusion, it should be mentioned that this work is stimulated by the author's personal conversation with Proffessor V. Skvortsov, who pointed out the significance of studying the relationship between the Kolmogorov and Henstok integrals.     

On stochastic orderings between residual record values

In this paper we establish some stochastic ordering results among residual record values in two sample problems. We also discuss some applications.     

On the relationship between two three-critical-point theorems

In this paper we analyze the hypotheses of two theorems which have proven to be useful tools for obtaining the existence of three critical points in nonlinear problems. In both cases the hypotheses can be stated in terms of some elementary inequalities involving “coercive” (in a suitable sense) sets of affine functions. We prove that the two resulting inequalities are in fact equivalent.     

On the relationship between continuous- and discrete-time control systems

Building on previous results of the author this paper presents two new error estimates for the reachable set of an affine control system if only piece-wise constant admissible controls on a uniform mesh are used instead of all measurable admissible controls. It is natural to expect that the resulting “shrinkage” of the reachable set is of the order of the mesh size. In this paper it is proved that under certain reasonable conditions the error is of higher than first order.     

On the relationship between combinatorial functions and representation theory

The paper is devoted to the study of well-known combinatorial functions on the symmetric group S _n —the major index maj, the descent number des, and the inversion number inv—from the representation-theoretic point of view. We show that these functions generate the same ideal in the group algebra C[S _n ], and the restriction of the left regular representation of the group S _n to this ideal is isomorphic to its representation in the space of n×n skew-symmetric matrices. This allows us to obtain formulas for the functions maj, des, and inv in terms of matrices of an exceptionally simple form. These formulas are applied to find the spectra of the elements under study in the regular representation, as well as derive a series of identities relating these functions to one another and to the number fix of fixed points.