Potential approach and characterizations of a Shapley value in multi-choice games

The main focus of this paper is on the restricted Shapley value for multi-choice games introduced by Derks and Peters [Derks, J., Peters, H., 1993. A Shapley value for games with restricted coalitions. International Journal of Game Theory 21, 351–360] and studied by Klijn et al. [Klijn, F., Slikker, M., Zazuelo, J., 1999. Characterizations of a multi-choice value. International Journal of Game Theory 28, 521–532]. We adopt several characterizations from TU game theory and reinterpret them in the framework of multi-choice games. We generalize the potential approach and show that this solution can be formulated as the vector of marginal contributions of a potential function. Also, we characterize the family of all solutions for multi-choice games that admit a potential. Further, a consistency result is reported.     

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).     

Transversality of the Shapley value

A few applications of the Shapley value are described. The main choice criterion is to look at quite diversified fields, to appreciate how wide is the terrain that has been explored and colonized using this and related tools. The title is inspired by a tutorial that one of the authors planned to deliver at the 7th meeting on Game Theory and Practice (Montreal, 2007), but was unable to do it for personal reasons. Thanks to Georges Zaccour whose invitation sparked the present survey.     

The Shapley value,the Proper Shapley value,and sharing rules for cooperative ventures

In this note, we discuss two solutions for cooperative transferable utility games, namely the Shapley value and the Proper Shapley value. We characterize positive Proper Shapley values by affine invariance and by an axiom that requires proportional allocation of the surplus according to the individual singleton worths in generalized joint venture games. As a counterpart, we show that affine invariance and an axiom that requires equal allocation of the surplus in generalized joint venture games characterize the Shapley value.     

Associated consistency and Shapley value

In this work, a new axiomatization of the Shapley is presented. An associated game is constructed. We define a sequence of games, when the term of order n, in this sequence, is the associated game of the term of order (n−1). We show that the sequence converges and that the limit game is inessential. The solution is obtained using the inessential game axiom, the associated consistency axiom and the continuity axiom. As a by-product, we note that neither the additivity nor the efficiency axioms are needed. Accepted September 2001     

A new basis and the Shapley value

The purpose of this paper is to introduce a new basis of the set of all TU games. Shapley (1953) introduced the unanimity game in which cooperation of all players in a given coalition yields payoff. We introduce the commander game in which only one player in a given coalition yields payoff. The set of the commander games forms a basis and has two properties. First, when we express a game by a linear combination of the basis, the coefficients related to singletons coincide with the Shapley value. Second, the basis induces the null space of the Shapley value.     

Forming coalitions and the Shapley NTU value

A simple protocol for coalition formation is presented. First, an order of the players is randomly chosen. Then, a coalition grows by sequentially incorporating new members in this order. The protocol is studied in the context of non-transferable utility (NTU) games in characteristic function form. If (weighted) utility transfers are feasible when everybody cooperates, then the expected subgame perfect equilibrium payoff allocation anticipated before any implemented game is the Shapley NTU value.     

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.     

Cooperative grey games and the grey Shapley value

This contribution is located in the common area of operational research and economics, with a close relation and joint future potential with optimization: game theory. We focus on collaborative game theory under uncertainty. This study is on a new class of cooperative games where the set of players is finite and the coalition values are interval grey numbers. An interesting solution concept, the grey Shapley value, is introduced and characterized with the properties of additivity, efficiency, symmetry and dummy player, which are straightforward generalizations of the corresponding properties in the classical cooperative game theory. The paper ends with a conclusion and an outlook to future studies.     

The Shapley value for the probability game

The main goal of this paper is to introduce the probability game. On one hand, we analyze the Shapley value by providing an axiomatic characterization. We propose the so-called independent fairness property, meaning that for any two players, the player with larger individual value gets a larger portion of the total benefit. On the other, we use the Shapley value for studying the profitability of merging two agents.     

The Shapley value without efficiency and additivity

We provide a new characterization of the Shapley value neither using the efficiency axiom nor the additivity axiom. In this characterization, efficiency is replaced by the gain-loss axiom (Einy and Haimanko, 2011), i.e., whenever the total worth generated does not change, a player can only gain at the expense of another one. Additivity and the equal treatment axiom are substituted by fairness (van den Brink, 2001) or differential marginality (Casajus, 2011), where the latter requires equal productivity differentials of two players to translate into equal payoff differentials. The third axiom of our characterization is the standard dummy player axiom.     

A dynamic approach to the Shapley value based on associated games

We propose a dynamic process leading to the Shapley value of TU games or any solution satisfying Inessential Game (IG) and Continuity (CONT), based on a modified version of Hamiache's notion of an associated game. The authors are very grateful to William Thomson and two anonymous referees for valuable comments which much improve the paper. They provide better statements and proofs of several major results than the original ones.     

Matrix approach to dual similar associated consistency for the Shapley value

In terms of the similarity of matrices, by combining the dual operator and the linear mapping with respect to Hamiache’s associated game on the game space, the Shapley value for TU-games is axiomatized as the unique value verifying dual similar associated consistency, continuity, and the inessential game property.     

The Shapley value and average convex games

In this paper we reformulate the necessary and sufficient conditions for the Shapley value to lie in the core of the game. Two new classes of games, which strictly include convex games, are introduced: average convex games and partially average convex games. Partially average convex games, which need not be superadditive, include average convex games. The Shapley value of a game for both classes is in the core. Some Cobb Douglas production games with increasing returns to scale turn out to be average convex games. The paper concludes with a comparison between the new classes of games introduced and some previous extensions of the convexity notion.The authors thank G. Owen, S. Tijs, and J. Ostroy and two anonymous referees of the International Journal of Game Theory for their comments and suggestions. The usual disclamer applies. We are grateful to the Universidad del Pais Vasco-EHU (grant UPV 209.321-H053/90) and the Ministry of Education and Science of Spain (CICYT grant PB900654) for providing reseach support.     

The Shapley value in the non differentiate case

The Shapley value is shown to exist even when there are essential non differentiabilities on the diagonal.     

The Shapley value of exact assignment games

We prove that the Shapley value of every two-sided exact assignment game lies in the core of the game.