The shapley value on some lattices of monotonic games

It is shown that the restriction of the Shapley value to the lattice of all monotonic games whose range is contained in an arbitrary set of non-negative real numbers (which contains 0) can be uniquely characterized by the axioms that were given by Dubey (in order to characterize the Shapley value on the class of monotonic simple games). We also derive formulas for the Shapley-Shubik and Banzhaf power indices in terms of the minimal winning coalitions of the game.     

Whose deletion does not affect your payoff? The difference between the Shapley value, the egalitarian value, the solidarity value, and the Banzhaf value

This study provides a unified axiomatic characterization method of one-point solutions for cooperative games with transferable utilities. Any one-point solution that satisfies efficiency, the balanced cycle contributions property (BCC), and the axioms related to invariance under a player deletion is characterized as a corollary of our general result. BCC is a weaker requirement than the well-known balanced contributions property. Any one-point solution that is both symmetric and linear satisfies BCC. The invariance axioms necessitate that the deletion of a specific player from games does not affect the other players’ payoffs, and this deletion is different with respect to solutions. As corollaries of the above characterization result, we are able to characterize the well-known one-point solutions, the Shapley, egalitarian, and solidarity values, in a unified manner. We also studied characterizations of an inefficient one-point solution, the Banzhaf value that is a well-known alternative to the Shapley value.     

Multilinear extensions and values for multichoice games

We define multilinear extensions for multichoice games and relate them to probabilistic values and semivalues. We apply multilinear extensions to show that the Banzhaf value for a compound multichoice game is not the product of the Banzhaf values of the component games, in contrast to the behavior in simple games. Following Owen (Manag Sci 18:64–79, 1972), we integrate the multilinear extension over a simplex to construct a version of the Shapley value for multichoice games. We compare this new Shapley value to other extensions of the Shapley value to multichoice games. We also show how the probabilistic value (resp. semivalue, Banzhaf value, Shapley value) of a multichoice game is equal to the probabilistic value (resp. semivalue, Banzhaf value, Shapley value) of an appropriately defined TU decomposition game. Finally, we explain how semivalues, probabilistic values, the Banzhaf value, and this Shapley value may be viewed as the probability that a player makes a difference to the outcome of a simple multichoice game.     

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.     

Values for Interior Operator Games

The aim of this paper is to study a new class of cooperative games called interior operator games. These games are additive games restricted by antimatroids. We consider several types of cooperative games as peer group games, big boss games, clan games and information market games and show that all of them are interior operator games. Next, we analyze the properties of these games and compute the Shapley, Banzhaf and Tijs values.     

集合对策的定量边缘解

本文提出了集合对策的两类定量边缘解,并给出了两类解的公理化特征:有效性、对称性、哑元性、Banzhaf总和性和传递性．这两类解分别与TU-对策的Banzhaf权力指数和Shapley-Shubik权力指数类似．同时,本文将Shapley解与Banzzhaf解扩展到k-维欧氏空间．     

Axiomatizations of Banzhaf permission values for games with a permission structure

In games with a permission structure it is assumed that players in a cooperative transferable utility game are hierarchically ordered in the sense that there are players that need permission from other players before they are allowed to cooperate. We provide axiomatic characterizations of Banzhaf permission values being solutions that are obtained by applying the Banzhaf value to modified TU-games. In these characterizations we use power- and player split neutrality properties. These properties state that splitting a player’s authority and/or contribution over two players does not change the sum of their payoffs.     

Axiomatic of the Shapley value of a game with a priori unions

In this paper, we define a modification of the Shapley value for the model of TU games with a priori unions. We provide two characterizations of this value and a new characterization of the Banzhaf–Owen coalitional value.     

Potentials and reduced games for share functions

A value function for cooperative games with transferable utility assigns to every game a distribution of the payoffs. A value function is efficient if for every such a game it exactly distributes the worth that can be obtained by all players cooperating together. An approach to efficiently allocate the worth of the ‘grand coalition’ is using share functions which assign to every game a vector whose components sum up to one. Every component of this vector is the corresponding players’ share in the total payoff that is to be distributed. In this paper we give characterizations of a class of share functions containing the Shapley share function and the Banzhaf share function using generalizations of potentials and of Hart and Mas-Colell's reduced game property.     

Probabilistic value pricing

In this paper we consider the class of probabilistic value pricing mechanisms for cost allocation problems, which are related to probabilistic values for finite games with transferable utility. We characterize probabilistic value pricing axiomatically, as well as several related pricing mechanisms, including semivalue pricing (symmetric pricing without cost sharing), quasivalue pricing (cost sharing pricing without symmetry), and weighted Shapley value pricing. We also describe a class of problems in which (symmetric) Shapley value pricing coincides with Aumann-Shapley pricing, and a class of problems for which every quasivalue pricing mechanism is supportable.     

全对策的边际贡献值

全对策是定义在局中人集合的所有分划集上的一类特殊合作对策.本文在效用可转移情形下研究全对策的"值"问题.定义了全对策的边际贡献值,得出全对策的Shapley值,以及具有某些性质的值是边际贡献值,并给出两种边际贡献值的具体表达式,及其一些性质.     

Random order coalition structure values

Recently, attention has been focused on generalizations of the Shapley value obtained by relaxing the symmetry postulate. Shapley defined the class of weighted values and these have been characterized by Kalai and Samet. Random order values, treated by Weber, provide the most general approach to values without symmetry. This paper extends the random order idea to games with coalition structures. The symmetric CS value was defined by Owen; axiomatic characterizations have been given by Owen and Hart and Kurz. Levy and McLean extended their work and analyzed various classes of weighted CS values. The random order CS values of this paper include all the CS values described above as special cases.     

On the symmetric and weighted shapley values

We present new axiomatic characterizations of the symmetric Shapley value and of weighted Shapley values for transferable utility coalitional form games without imposing the axiom ofadditivity (Shapley [1953a,b]). Our main condition iscoalitional strategic equivalence, introduced by Chun [1989]. We show thatcoalitional strategic equivalence, together withefficiency, andsymmetry, characterizes the symmetric Shapley value, and this axiom, together withefficiency, positivity, homogeneity, andpartnership, characterizes weighted Shapley values.     

A decisiveness index for simple games

The decisiveness index introduced in this paper is designed to provide a normalized measure of the agility of all simple games, primarily viewed as collective decision-making mechanisms. We study the mathematical properties of the index and derive different axiomatic characterizations for it. Moreover, a close relationship is shown to the Banzhaf index of power––for which twice the decisiveness index plays the role of potential function––that gives rise to an effective computational procedure. Some real-world examples illustrate the usefulness of the decisiveness index, together with the Banzhaf power index, in applications to political science.     

An axiomatization of the weighted NTU value

Non-symmetric generalizations of the non-transferable utility (NTU) are defined and characterized axiomatically. The first of these is a weighted NTU value that is identical to the (symmetric) NTU value when players have the same weights. On the class of transferable utility games, this weighted NTU value coincides with the weighted Shapley value and on the pure bargaining games it coincides with the non-symmetric Nash bargaining solution. A further extension, the random order NTU value, is also defined and axiomatized and its relationship to the core is discussed.     

The Banzhaf power index for ternary bicooperative games

In this paper we analyze ternary bicooperative games, which are a refinement of the concept of a ternary voting game introduced by Felsenthal and Machover. Furthermore, majority voting rules based on the difference of votes are simple bicooperative games. First, we define the concepts of the defender and detractor swings for a player. Next, we introduce the Banzhaf power index and the normalized Banzhaf power index. The main result of the paper is an axiomatization of the Banzhaf power index for the class of ternary bicooperative games. Moreover, we study ternary bicooperative games with two lists of weights and compute the Banzhaf power index using generating functions.     

Associated consistency and values for TU games

In the framework of the solution theory for cooperative transferable utility games, Hamiache axiomatized the well-known Shapley value as the unique one-point solution verifying the inessential game property, continuity, and associated consistency. The purpose of this paper is to extend Hamiache’s axiomatization to the class of efficient, symmetric, and linear values, of which the Shapley value is the most important representative. For this enlarged class of values, explicit relationships to the Shapley value are exploited in order to axiomatize such values with reference to a slightly adapted inessential game property, continuity, and a similar associated consistency. The latter axiom requires that the solutions of the initial game and its associated game (with the same player set, but a different characteristic function) coincide.     

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.     

Assignment games with stable core

We prove that the core of an assignment game (a two-sided matching game with transferable utility as introduced by Shapley and Shubik, 1972) is stable (i.e., it is the unique von Neumann-Morgenstern solution) if and only if there is a matching between the two types of players such that the corresponding entries in the underlying matrix are all row and column maximums. We identify other easily verifiable matrix properties and show their equivalence to various known sufficient conditions for core-stability. By these matrix characterizations we found that on the class of assignment games, largeness of the core, extendability and exactness of the game are all equivalent conditions, and strictly imply the stability of the core. In turn, convexity and subconvexity are equivalent, and strictly imply all aformentioned conditions. Final version: April 1, 2001     

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.