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.     

A general procedure to compute mixed modified semivalues for cooperative games with structure of coalition blocks

Semivalues are solution concepts for cooperative games that assign to each player a weighted sum of his/her marginal contributions to the coalitions, where the weights only depend on the coalition size. The Shapley value and the Banzhaf value are semivalues. Mixed modified semivalues are solutions for cooperative games when we consider a priori coalition blocks in the player set. For all these solutions, a computational procedure is offered in this paper.     

The collective value: a new solution for games with coalition structures

In this study, we provide a new solution for cooperative games with coalition structures. The collective value of a player is defined as the sum of the equal division of the pure surplus obtained by his coalition from the coalitional bargaining and of his Shapley value for the internal coalition. The weighted Shapley value applied to a game played by coalitions with coalition-size weights is assigned to each coalition, reflecting the size asymmetries among coalitions. We show that the collective value matches exogenous interpretations of coalition structures and provide an axiomatic foundation of this value. A noncooperative mechanism that implements the collective value is also presented.     

Egalitarian allocation and players of certain type

In cooperative game theory the Shapley value is different from the egalitarian value, the latter of which allocates payoffs equally. The null player property and the nullifying player property assign zero payoff to each null player and each nullifying player, respectively. It is known that if the null player property for characterizing the Shapley value is replaced by the nullifying player property, then the egalitarian value is determined uniquely. We propose several properties to replace the nullifying player property to characterize the egalitarian value. Roughly speaking, the results in this note hint that equal division for players of certain types may lead to the egalitarian allocation.     

Characterizations of a multi-choice value

A multi-choice game is a generalization of a cooperative game in which each player has several activity levels. We study the extended Shapley value as proposed by Derks and Peters (1993). Van den Nouweland (1993) provided a characterization that is an extension of Young's (1985) characterization of the Shapley value. Here we provide several other characterizations, one of which is the analogue of Shapley's (1953) original characterization. The three other characterizations are inspired by Myerson's (1980) characterization of the Shapley value using balanced contributions. Received: November 1997/final version: February 1999     

An axiomatization of the Shapley value using a fairness property

In this paper we provide an axiomatization of the Shapley value for TU-games using a fairness property. This property states that if to a game we add another game in which two players are symmetric then their payoffs change by the same amount. We show that the Shapley value is characterized by this fairness property, efficiency and the null player property. These three axioms also characterize the Shapley value on the class of simple games. Revised August 2001     

Preserving or removing special players: What keeps your payoff unchanged in TU-games?

If a player is removed from a game, what keeps the payoff of the remaining players unchanged? Is it the removal of a special player or its presence among the remaining players? This article answers this question in a complement study to Kamijo and Kongo (2012). We introduce axioms of invariance from player deletion in presence of a special player. In particular, if the special player is a nullifying player (resp. dummifying player), then the equal division value (resp. equal surplus division value) is characterized by the associated axiom of invariance plus efficiency and balanced cycle contributions. There is no type of special player from such a combination of axioms that characterizes the Shapley value.     

Equivalence theorem,consistency and axiomatizations of a multi-choice value

This paper is devoted to the study of solutions for multi-choice games which admit a potential, such as the potential associated with the extended Shapley value proposed by Hsiao and Raghavan (Int J Game Theory 21:301–302, 1992; Games Econ Behav 5:240–256, 1993). Several axiomatizations of the family of all solutions that admit a potential are offered and, as a main result, it is shown that each of these solutions can be obtained by applying the extended Shapley value to an appropriately modified game. In the framework of multi-choice games, we also provide an extension of the reduced game introduced by Hart and Mas-Colell (Econometrica 57:589–614, 1989). Different from the works of Hsiao and Raghavan (1992, 1993), we provide two types of axiomatizations, one is the analogue of Myerson’s (Int J Game Theory 9:169–182, 1980) axiomatization of the Shapley value based on the property of balanced contributions. The other axiomatization is obtained by means of the property of consistency.     

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.     

“Procedural” values for cooperative games

This paper introduces a new notion of a “procedural” value for cooperative TU games. A procedural value is determined by an underlying procedure of sharing marginal contributions to coalitions formed by players joining in random order. We consider procedures under which players can only share their marginal contributions with their predecessors in the ordering, and study the set of all resulting values. The most prominent procedural value is, of course, the Shapley value obtaining under the simplest procedure of every player just retaining his entire marginal contribution. But different sharing rules lead to other interesting values, including the “egalitarian solution” and the Nowak and Radzik “solidarity value”. All procedural values are efficient, symmetric and linear. Moreover, it is shown that these properties together with two very natural monotonicity postulates characterize the class of procedural values. Some possible modifications and generalizations are also discussed. In particular, it is shown that dropping one of monotonicity axioms is equivalent to allowing for sharing marginal contributions with both predecessors and successors in the ordering.     

A shapley value for games with restricted coalitions

A restriction is a monotonic projection assigning to each coalition of a finite player setN a subcoalition. On the class of transferable utility games with player setN, a Shapley value is associated with each restriction by replacing, in the familiar probabilistic formula, each coalition by the subcoalition assigned to it. Alternatively, such a Shapley value can be characterized by restricted dividends. This method generalizes several other approaches known in literature. The main result is an axiomatic characterization with the property that the restriction is determined endogenously by the axioms.     

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.     

Players indifferent to cooperate and characterizations of the Shapley value

In this paper we provide new axiomatizations of the Shapley value for TU-games using axioms that are based on relational aspects in the interactions among players. Some of these relational aspects, in particular the economic or social interest of each player in cooperating with each other, can be found embedded in the characteristic function. We define a particular relation among the players that it is based on mutual indifference. The first new axiom expresses that the payoffs of two players who are not indifferent to each other are affected in the same way if they become enemies and do not cooperate with each other anymore. The second new axiom expresses that the payoff of a player is not affected if players to whom it is indifferent leave the game. We show that the Shapley value is characterized by these two axioms together with the well-known efficiency axiom. Further, we show that another axiomatization of the Shapley value is obtained if we replace the second axiom and efficiency by the axiom which applies the efficiency condition to every class of indifferent players. Finally, we extend the previous results to the case of weighted Shapley values.     

Weighted Shapley hierarchy levels values

A new class of values for cooperative games with level structure is introduced. We apply a multi-step proceeding to the weighted Shapley values. For characterization, two well-known axiomatizations of the weighted Shapley values are extended, the first one by efficiency and weighted balanced contributions and the second one by weighted standardness for two-player games and consistency. We get a new axiomatization of the Shapley levels value too.     

Single-suit two-person card play

We consider a two-person constant sum perfect information game, which we call theEnd Play Game, which arises from an abstraction of simple end play positions in card games of the whist family, including bridge. This game was described in 1929 by Emanuel Lasker, the mathematician and world chess champion, who called itwhistette. The game uses a deck of cards that consists of a single totally ordered suit of 2n cards. To begin play the deck is divided into two handsA andB ofn cards each, held by players Left and Right, and one player is designated as having thelead. The player on lead chooses one of his cards, and the other player after seeing this card selects one of his own to play. The player with the higher card wins a “trick” and obtains the lead. The cards in the trick are removed from each hand, and play then continues until all cards are exhausted. Each player strives to maximize his trick total, and thevalue of the game to each player is the number of tricks he takes. Despite its simple appearance, this game is quite complicated, and finding an optimal strategy seems difficult. This paper derives basic properties of the game, gives some criteria under which one hand is guaranteed to be better than another, and determines the optimal strategies and value functions for the game in several special cases.     

收益模糊合作对策Shapley值的公理化

研究一类收益模糊的合作对策,这类对策联盟的模糊收益值可以用一个闭区间的形式来表示,本文定义了一个拓展的闭区间空间和一些闭区间线性运算算子,证明了这类对策的Shapley值可以用承载性、可替代性和可加性进行了公理化.     

Bargaining with commitments

We study a simple bargaining mechanism in which, given an order of players, the first n–1 players sequentially announce their reservation price. Once these prices are given, the last player may choose a coalition to cooperate with, and pay each member of this coalition his reservation price. The only expected final equilibrium payoff is a new solution concept, the selective value, which can be defined by means of marginal contributions vectors of a reduced game. The selective value coincides with the Shapley value for convex games. Moreover, for 3-player games the vectors of marginal contributions determine the core when it is nonempty.A previous version of this paper has benefited from helpful comments from Gustavo Bergantiños. Numerous suggestions of two anonymous referees, the Associate Editor, and William Thomson, Editor, have led to significant improvements of the final version. Financial support by the Spanish Ministerio de Ciencia y Tecnología and FEDER through grant BEC2002-04102-C02-01 and Xunta de Galicia through grant PGIDIT03PXIC30002PN is gratefully acknowledged.     

Compensations in the Shapley value and the compensation solutions for graph games

We consider an alternative expression of the Shapley value that reveals a system of compensations: each player receives an equal share of the worth of each coalition he belongs to, and has to compensate an equal share of the worth of any coalition he does not belong to. We give a representation in terms of formation of the grand coalition according to an ordering of the players and define the corresponding compensation vector. Then, we generalize this idea to cooperative games with a communication graph in order to construct new allocation rules called the compensation solutions. Firstly, we consider cooperative games with arbitrary graphs and construct rooted spanning trees (see Demange, J Political Econ 112:754–778, 2004) instead of orderings of the players by using the classical algorithms DFS and BFS. If the graph is complete, we show that the compensation solutions associated with DFS and BFS coincide with the Shapley value and the equal surplus division respectively. Secondly, we consider cooperative games with a forest (cycle-free graph) and all its rooted spanning trees. The compensation solution is characterized by component efficiency and relative fairness. The latter axiom takes into account the relative position of a player with respect to his component in the communication graph.     

Value of games with two-layered hypergraphs

This paper studies cooperative games with restricted cooperation among players. We define situations in which a priori unions and hypergraphs coexist simultaneously and mutually depend on each other. We call such structures two-layered hypergraphs. Using a two-step approach, we define a value of the games with two-layered hypergraphs. The value is characterized by Owen’s coalitional value of hypergraph-restricted games and in terms of weighted Myerson value. Further, our value is axiomatically characterized by component efficiency and a coalition size normalized balanced contributions property.