The Harsanyi value for nontransferable utility games with restricted cooperation

A nontransferable utility (NTU) game assigns a set of feasible pay-off vectors to each coalition. In this article, we study NTU games in situations in which there are restrictions on coalition formation. These restrictions will be modelled through interior structures, which extend some of the structures considered in the literature on transferable utility games for modelling restricted cooperation, such as permission structures or antimatroids. The Harsanyi value for NTU games is extended to the set of NTU games with interior structure.     

有限制结盟的NTU对策解

对有限制结盟的NTU对策提出一种分配形式,即RC解,研究了这个解与对应的TU对策的有限制结盟边际贡献值之间的关系,同时给出RC解的刻划公理.     

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.     

A value for multichoice games

A multichoice game is a generalization of a cooperative TU game in which each player has several activity levels. We study the solution for these games proposed by Van Den Nouweland et al. (1995) [Van Den Nouweland, A., Potters, J., Tijs, S., Zarzuelo, J.M., 1995. Cores and related solution concepts for multi-choice games. ZOR-Mathematical Methods of Operations Research 41, 289–311]. We show that this solution applied to the discrete cost sharing model coincides with the Aumann-Shapley method proposed by Moulin (1995) [Moulin, H., 1995. On additive methods to share joint costs. The Japanese Economic Review 46, 303–332]. Also, we show that the Aumann-Shapley value for continuum games can be obtained as the limit of multichoice values for admissible convergence sequences of multichoice games. Finally, we characterize this solution by using the axioms of balanced contributions and efficiency.     

多选择NTU结构对策下的社会稳定核心

本文研究了多选择情形下NTU结构对策及其社会稳定核心的理论和应用。定义了多选择NTU结构对策的转移率规则和支付依赖平衡性质,给出了K-K-M-S定理在多选择NTU结构对策下的一个扩展形式,并用扩展后的K-K-M-S定理证明了如果转移率规则包含某些力量函数值,且多选择NTU结构对策关于转移率规则是支付依赖平衡的,则多选择NTU结构对策的社会稳定核心是非空的。     

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.     

多选择NTU对策核心的一个非空条件及公理化

提出了\pi-均衡多选择NTU对策的概念,证明了\pi-均衡多选择NTU对策的核心非空, 定义了多选择NTU对策的非水平性质和缩减对策,给出了相容性和逆相容性等概念. 用个体合理性、单人合理性、相容性和逆相容性对非水平多选择NTU对策的核心进行了公理化.     

The Compromise Value for Cooperative Games with Random Payoffs

This paper introduces and studies the compromise value for cooperative games with random payoffs, that is, for cooperative games where the payoff to a coalition of players is a random variable. This value is a compromise between utopia payoffs and minimal rights and its definition is based on the compromise value for NTU games and the τ-value for TU games. It is shown that the nonempty core of a cooperative game with random payoffs is bounded by the utopia payoffs and the minimal rights. Consequently, for such games the compromise value exists. Further, we show that the compromise value of a cooperative game with random payoffs coincides with the τ-value of a related TU game if the players have a certain type of preferences. Finally, the compromise value and the marginal value, which is defined as the average of the marginal vectors, coincide on the class of two-person games. This results in a characterization of the compromise value for two-person games.I thank Peter Borm, Ruud Hendrickx and two anonymous referees for their valuable comments.     

Convex and exact games with non-transferable utility

We generalize exactness to games with non-transferable utility (NTU). A game is exact if for each coalition there is a core allocation on the boundary of its payoff set.     

Replication invariance on NTU games

Two concepts of replication (conflictual and non-conflictual) are extended from the class of pure bargaining games to the class of NTU games. The behavior of the Harsanyi, Shapley NTU, Egalitarian and Maschler-Owen solutions of the replica games is compared with that of the Nash and Egalitarian solutions in pure bargaining games. Received June 1995/Final version February 2000     

A note on NTU convexity

For cooperative games with transferable utility, convexity has turned out to be an important and widely applicable concept. Convexity can be defined in a number of ways, each having its own specific attractions. Basically, these definitions fall into two categories, namely those based on a supermodular interpretation and those based on a marginalistic interpretation. For games with nontransferable utility, however, the literature mainly focuses on two kinds of convexity, ordinal and cardinal convexity, which both extend the supermodular interpretation. In this paper, we analyse three types of convexity for NTU games that generalise the marginalistic interpretation of convexity. Received: December 2000     

宏观调控政策协调的TU和NTU合作博弈分析

宏观经济政策之间的协调及其调控收益的分配实际上可看做n人合作博弈.通过构造宏观政策组合的二人TU博弈模型,分析了政策组合的调控力度、收益及其分配比例值的相互关系,并重新构建政策组合调控总收益的特征函数,把宏观政策协调的二人TU博弈分析依次扩展至n人TU博弈情形和NTU博弈情形,给出相应的政策调控收益分配的Shapley值,提出确定宏观政策组合中各项政策调控的参与力度的具体办法,有效地解释了加强政策协调配合的宏观调控精神,对宏观调控实践具有一定的指导意义.     

The Egalitarian Solution for Multichoice Games

In a multichoice game a coalition is characterized by the level at which each player is acting, and to each coalition a real number is assigned. A multichoice solution assigns, for each multichoice game, a numerical value to each possible activity level of each player, intended to measure the contribution of each such level to reaching the grand coalition in which each player is active at the maximal level. The paper focuses on the egalitarian multichoice solution, characterized by the properties of Efficiency, Zero Contribution, Additivity, Anonymity, and Level Symmetry. The egalitarian solution is also shown to satisfy the property of marginalism: it measures the effect of lowering, ceteris paribus, a certain activity level by one. The solution is compared to a multichoice solution studied in Klijn, Slikker, and Zarzuelo (1999). Finally, it is discussed how the formalism of this paper can be applied to the different framework of multi-attribute utilities.     

Two-step coalition values for multichoice games

We introduce and compare several coalition values for multichoice games. Albizuri defined coalition structures and an extension of the Owen coalition value for multichoice games using the average marginal contribution of a player over a set of orderings of the player’s representatives. Following an approach used for cooperative games, we introduce a set of nested or two-step coalition values on multichoice games which measure the value of each coalition and then divide this among the players in the coalition using either a Shapley or Banzhaf value at each step. We show that when a Shapley value is used in both steps, the resulting coalition value coincides with that of Albizuri. We axiomatize the three new coalition values and show that each set of axioms, including that of Albizuri, is independent. Further we show how the multilinear extension can be used to compute the coalition values. We conclude with a brief discussion about the applicability of the different values.     

The multichoice coalition value

In this paper we define a solution for multichoice games which is a generalization of the Owen coalition value (Lecture Notes in Economics and Mathematical Systems: Essays in Honor of Oskar Morgenstern, Springer, New York, pp. 76–88, 1977) for transferable utility cooperative games and the Egalitarian solution (Peters and Zanks, Ann. Oper. Res. 137, 399–409, 2005) for multichoice games. We also prove that this solution can be seen as a generalization of the configuration value and the dual configuration value (Albizuri et al., Games Econ. Behav. 57, 1–17, 2006) for transferable utility cooperative games.     

The construction and characterization of egalitarian solutions for multi-choice NTU games

In this article we construct a procedure to define the egalitarian solutions in the context of multi-choice non-transferable utility (NTU) games. Also, we show that in the presence of other weak axioms the egalitarian solutions are the only monotonic ones.     

On implementation via demand commitment games

A simple version of the Demand Commitment Game is shown to implement the Shapley value as the unique subgame perfect equilibrium outcome for any n-person characteristic function game. This improves upon previous models devoted to this implementation problem in terms of one or more of the following: a) the range of characteristic function games addressed, b) the simplicity of the underlying noncooperative game (it is a finite horizon game where individuals make demands and form coalitions rather than make comprehensive allocation proposals and c) the general acceptability of the noncooperative equilibrium concept. A complete characterization of an equilibrium strategy generating the Shapley value outcomes is provided. Furthermore, for 3 player games, it is shown that the Demand Commitment Game can implement the core for games which need not be convex but have cores with nonempty interiors. Received March 1995/Final version February 1997