In real situations, players might represent administrative areas of different scales; players might have different activity abilities. Thus, we propose an extension of the Banzhaf-Owen index in the framework of fuzzy transferable-utility games by considering supreme-utilities and weights simultaneously, which we name the weighted fuzzy Banzhaf-Owen index. Here we adopt three existing notions from traditional game theory and reinterpret them in the framework of fuzzy transferable-utility games. The first one is that this weighted index could be represented as an alternative formulation in terms of excess functions. The second is that, based on an reduced game and related consistency, we offer an axiomatic result to present the rationality of this weighted index. Finally, we introduce two dynamic processes to illustrate that this weighted index could be reached by players who start from an arbitrary efficient payoff vector and make successive adjustments.
A class of solutions are introduced by lexicographically minimizing the complaint of coalitions for cooperative games with transferable utility. Among them, the nucleolus is an important representative. From the perspective of measuring the satisfaction of coalitions with respect to a payoff vector, we define a family of optimal satisfaction values in this paper. The proportional division value and the proportional allocation of non-separable contribution value are then obtained by lexicographically maximizing two types of satisfaction criteria, respectively, which are defined by the lower bound and the upper bound of the core from the viewpoint of optimism and pessimism respectively. Correspondingly, we characterize these two proportional values by introducing the equal minimal satisfaction property and the associated consistency property. Furthermore, we analyze the duality of these axioms and propose more approaches to characterize these two values on basis of the dual axioms.
AbstractIn this paper, I introduce and study the $\gamma$-core of a general strategic game. I first show that the $\gamma$-core of an arbitrary strategic game is smaller than the conventional $\alpha$- and $\beta$- cores. I then consider the partition function form of a general strategic game and show that a prominent class of partition function games admit nonempty $\gamma$-cores. Finally, I show that each $\gamma$-core payoff vector (a cooperative solution) can be supported as an equilibrium outcome of an intuitive non-cooperative game and the grand coalition is the unique equilibrium outcome if and only if the $\gamma$-core is non-empty.
In this paper, we propose a new single valued rule based on the concept of fair division for all cooperative transferable utility (TU) games. In any cooperative TU game, primarily the coalitions that are likely to form are identified and each such coalition is fixed with a payoff vector based on the notion of fairness. The value of the single valued rule is obtained from the collection of all coalition structures consisting of the coalitions that are likely to form. The uniqueness of the new rule is followed by its existence and computational simplicity for all TU games. Finally, a linear average rule is defined, and some of its properties are discussed.
Uncertainty accompanies almost every situation in our lives and it influences our decisions. On many occasions uncertainty is so severe that we can only predict some upper and lower bounds for the outcome of our (collaborative) actions, i.e., payoffs lie in some intervals. Cooperative interval games have been proved useful for solving reward/cost sharing problems in situations with interval data in a cooperative environment. In this paper we propose two procedures for cooperative interval games. Both transform an interval allocation, i.e., a payoff vector whose components are compact intervals of real numbers, into a payoff vector (whose components are real numbers) when the value of the grand coalition becomes known (at once or in multiple stages). The research question addressed here is: How to determine for each player his/her/its payoff generated by cooperation within the grand coalition – in the promised range of payoffs to establish such cooperation – after the uncertainty on the payoff for the grand coalition is resolved? This question is an important one that deserves attention both in the literature and in game practice.
Consider the cooperative coalition games with side payments. Bargaining sets are calculated for all possible coalition structures to obtain a collection of imputations rather than single imputation. Our aim is to obtain a single payoff vector, which is acceptable by all players of the game under grand coalition. Though Shapely value is a single imputation, it is based on fair divisions rather than bargaining considerations. So, we present a method to obtain a single imputation based on bargaining considerations.
One of the properties characterizing cooperative game solutions is consistency connecting solution vectors of a cooperative game with finite set of players and its reduced game defined by removing one or more players and by assigning them the payoffs according to some specific principle (e.g., a proposed payoff vector). Consistency of a solution means that any part (defined by a coalition of the original game) of a solution payoff vector belongs to the solution set of the corresponding reduced game. In the paper the proportional solutions for TU-games are defined as those depending only on the proportional excess vectors in the same manner as translation covariant solutions depend on the usual Davis–Maschler excess vectors. The general form of the reduced games defining consistent proportional solutions is given. The efficient, anonymous, proportional TU cooperative game solutions meeting the consistency property with respect to any reduced game are described.
We present a new NTU value, which generalises the Kalai-Smorodinsky bargaining solution to NTU games using three approaches. In the first section, we define the new NTU value as the only efficient point in the segment defined by an upper and a lower bound. Next, we define a bargaining procedure and prove that this procedure leads to a single, subgame perfect equilibrium payoff vector, which coincides with the new NTU value. Finally, we characterise the new value using several properties.
In (1), M. Davis and M. Maschler define the kernel K of a characteristic-function game; they also prove, among other theorems, that K is a subset of the bargaining set M1(i) and that it is never void, i.e. that for each coalition structure b there exists a payoff vector x such that the payoff configuration (x, b) belongs to K. The main advantage of the kernel, as it seems to us, is that it is easier to compute in many cases than the bargaining set M1(i).
Maximizing the Minimal Satisfaction—Characterizations of Two Proportional Values