模糊联盟合作对策的平均树解

给出图对策中平均树解的拓展形式, 证明其是满足分支有效性和分支公平性的唯一解. 针对具有模糊联盟的图对策, 提出了一种模糊分配, 即模糊平均树解. 当模糊联盟图对策为完全图对策时, 模糊平均树解等于模糊Shapley值. 最后, 讨论了模糊平均树解与模糊联盟核心之间的关系, 并进行了实例论证.     

On the restricted cores and the bounded core of games on distributive lattices

A game with precedence constraints is a TU game with restricted cooperation, where the set of feasible coalitions is a distributive lattice, hence generated by a partial order on the set of players. Its core may be unbounded, and the bounded core, which is the union of all bounded faces of the core, proves to be a useful solution concept in the framework of games with precedence constraints. Replacing the inequalities that define the core by equations for a collection of coalitions results in a face of the core. A collection of coalitions is called normal if its resulting face is bounded. The bounded core is the union of all faces corresponding to minimal normal collections. We show that two faces corresponding to distinct normal collections may be distinct. Moreover, we prove that for superadditive games and convex games only intersecting and nested minimal collection, respectively, are necessary. Finally, it is shown that the faces corresponding to pairwise distinct nested normal collections may be pairwise distinct, and we provide a means to generate all such collections.     

Cooperation in dividing the cake

This paper defines models of cooperation among players partitioning a completely divisible good (such as a cake or a piece of land). The novelty of our approach lies in the players’ ability to form coalitions before the actual division of the good with the aim to maximize the average utility of the coalition. A social welfare function which takes into account coalitions drives the division. In addition, we derive a cooperative game which measures the performance of each coalition. This game is compared with the game in which players start cooperating only after the good has been portioned and has been allocated among the players. We show that a modified version of the game played before the division outperforms the game played after the division.     

基于个人超出值的模糊联盟合作博弈最小二乘预核仁

本文给出了基于个人超出值的无限模糊联盟合作博弈最小二乘预核仁的求解模型,得到该模型的显式解析解,并研究该解的若干重要性质。证明了:本文给出的无限模糊联盟合作博弈的最小二乘预核仁与基于个人超出值的相等解(The equalizer solution),基于个人超出值的字典序解三者相等。进一步证明了:基于Owen线性多维扩展的无限模糊联盟合作博弈的最小二乘预核仁与基于个人超出值的经典合作博弈最小二乘预核仁相等。最后,通过数值实例说明本文提出的无限模糊联盟合作博弈求解模型的实用性与有效性。     

Dynamic resource allocation in fuzzy coalitions: a game theoretic model

We introduce an efficient and dynamic resource allocation mechanism within the framework of a cooperative game with fuzzy coalitions (cooperative fuzzy game). A fuzzy coalition in a resource allocation problem can be so defined that membership grades of the players in it are proportional to the fractions of their total resources. We call any distribution of the resources possessed by the players, among a prescribed number of coalitions, a fuzzy coalition structure and every membership grade (equivalently fraction of the total resources), a resource investment. It is shown that this resource investment is influenced by the satisfaction of the players in regard to better performance under a cooperative setup. Our model is based on the real life situations, where possibly one or more players compromise on their resource investments in order to help forming coalitions.     

Hierarchical organization structures and constraints on coalition formation

This paper studies the constraints in coalition formation that result from a hierarchical organization structure on the class of players in a cooperative game with transferable utilities. If one assumes that the superiors of a certain individual have to give permission to the actions undertaken by the individual, then one arrives at a limited collection of formable orautonomous coalitions. This resulting collection is a lattice of subsets on the player set. We show that if the collection of formable coalitions is limited to a lattice, the core allows for (infinite) exploitation of subordinates. For discerning lattices we are able to generalize the results of Weber (1988), namely the core is a subset of the convex hull of the collection of all attainable marginal contribution vectors plus a fixed cone. This relation is an equality if and only if the game is convex. This extends the results of Shapley (1971) and Ichiishi (1981).     

A minimax theorem for zero-sum stopping games in dynamic fuzzy systems

This paper deals with a stopping game in dynamic fuzzy systems with fuzzy rewards. We show that the optimal fuzzy reward is a unique solution of a fuzzy relational equation, and we estimate fuzzy rewards, by introducing a fuzzy expectation with a density given by fuzzy goals. We prove a minimax theorem for fuzzy expected values and show the existence of the value of the game.     

Cores of convex games

The core of ann-person game is the set of feasible outcomes that cannot be improved upon by any coalition of players. A convex game is defined as one that is based on a convex set function. In this paper it is shown that the core of a convex game is not empty and that it has an especially regular structure. It is further shown that certain other cooperative solution concepts are related in a simple way to the core: The value of a convex game is the center of gravity of the extreme points of the core, and the von Neumann-Morgenstern stable set solution of a convex game is unique and coincides with the core.     

The restricted core of games on distributive lattices: how to share benefits in a hierarchy

Finding a solution concept is one of the central problems in cooperative game theory, and the notion of core is the most popular solution concept since it is based on some rationality condition. In many real situations, not all possible coalitions can form, so that classical TU-games cannot be used. An interesting case is when possible coalitions are defined through a partial ordering of the players (or hierarchy). Then feasible coalitions correspond to teams of players, that is, one or several players with all their subordinates. In these situations, the core in its usual formulation may be unbounded, making its use difficult in practice. We propose a new notion of core, called the restricted core, which imposes efficiency of the allocation at each level of the hierarchy, is always bounded, and answers the problem of sharing benefits in a hierarchy. We show that the core we defined has properties very close to the classical case, with respect to marginal vectors, the Weber set, and balancedness.     

The core and the Weber set of games on augmenting systems

This paper deals with cooperative games in which only certain coalitions are allowed to form. There have been previous models developed to confront the problem of unallowable coalitions. Games restricted by a communication graph were introduced by Myerson and Owen. In their model, the feasible coalitions are those that induce connected subgraphs. Another type of model is introduced in Gilles, Owen and van den Brink. In their model, the possibilities of coalition formation are determined by the positions of the players in a so-called permission structure. Faigle proposed a general model for cooperative games defined on lattice structures. In this paper, the restrictions to the cooperation are given by a combinatorial structure called augmenting system which generalizes antimatroid structure and the system of connected subgraphs of a graph. In this framework, the core and the Weber set of games on augmenting systems are introduced and it is proved that monotone convex games have a non-empty core. Moreover, we obtain a characterization of the convexity of these games in terms of the core of the game and the Weber set of the extended game.     

Rankings and values for team games

In this article we attack several problems that arise when a group of individuals is organized in several teams with equal number of players in each one (e.g., for company work, in sports leagues, etc). We define a team game as a cooperative game v that can have non-zero values only on coalitions of a given cardinality; it is further shown that, for such games, there is essentially a unique ranking among the players. We also study the way the ranking changes after one or more players retire. Also, we characterize axiomatically different ways of ranking the players that intervene in a cooperative game.     

C-complete sets for compromise stable games

The core cover of a TU-game is a superset of the core and equals the convex hull of its larginal vectors. A larginal vector corresponds to an ordering of the players and describes the efficient payoff vector giving the first players in the ordering their utopia demand as long as it is still possible to assign the remaining players at least their minimum right. A game is called compromise stable if the core is equal to the core cover, i.e. the core is the convex hull of the larginal vectors. This paper analyzes the structure of orderings corresponding to larginal vectors of the core cover and conditions ensuring equality between core cover and core. We introduce compromise complete (or c-complete) sets that satisfy the condition that if every larginal vector corresponding to an ordering of the set is a core element, then the game is compromise stable. We use combinatorial arguments to give a complete characterization of these sets. More specifically, we find c-complete sets of minimum cardinality and a closed formula for the minimum number of orderings in c-complete sets. Furthermore, we discuss the number of different larginal vectors corresponding to a c-complete set of orderings.     

On some families of cooperative fuzzy games

In a fuzzy cooperative game the players may choose to partially participate in a coalition. A fuzzy coalition consists of a group of participating players along with their participation level. The characteristic function of a fuzzy game specifies the worth of each such coalition. This paper introduces well-known properties of classical cooperative games to the theory of fuzzy games, and studies their interrelations. It deals with convex games, exact games, games with a large core, extendable games and games with a stable core.     

On the core of multiple longest traveling salesman games

In this paper we introduce multiple longest traveling salesman (MLTS) games. An MLTS game arises from a network in which a salesman has to visit each node (player) precisely once, except to his home location, in such an order that maximizes the total reward. First it is shown that the value of a coalition of an MLTS game is determined by taking the maximum of suitable combinations of one and two person coalitions. Secondly it is shown that MLTS games with five or less players have a nonempty core. However, a six player MLTS game may have an empty core. For the special instance in which the reward between a pair of nodes is equal to 0 or 1, we provide relations between the structure of the core and the underlying network.     

The core of games on ordered structures and graphs

In cooperative games, the core is the most popular solution concept, and its properties are well known. In the classical setting of cooperative games, it is generally assumed that all coalitions can form, i.e., they are all feasible. In many situations, this assumption is too strong and one has to deal with some unfeasible coalitions. Defining a game on a subcollection of the power set of the set of players has many implications on the mathematical structure of the core, depending on the precise structure of the subcollection of feasible coalitions. Many authors have contributed to this topic, and we give a unified view of these different results.     

The core of games on ordered structures and graphs

In cooperative games, the core is the most popular solution concept, and its properties are well known. In the classical setting of cooperative games, it is generally assumed that all coalitions can form, i.e., they are all feasible. In many situations, this assumption is too strong and one has to deal with some unfeasible coalitions. Defining a game on a subcollection of the power set of the set of players has many implications on the mathematical structure of the core, depending on the precise structure of the subcollection of feasible coalitions. Many authors have contributed to this topic, and we give a unified view of these different results.     

The Harsanyi paradox and the “right to talk” in bargaining among coalitions

We describe a coalitional value from a non-cooperative point of view, assuming coalitions are formed for the purpose of bargaining. The idea is that all the players have the same chances to make proposals. This means that players maintain their own “right to talk” when joining a coalition. The resulting value coincides with the weighted Shapley value in the game between coalitions, with weights given by the size of the coalitions. Moreover, the Harsanyi paradox (forming a coalition may be disadvantageous) disappears for convex games.     

基于图对策的收益分配

基于具有交流结构的合作对策,即图对策,对平均树解拓展形式的特征进行刻画,提出此解满足可加性公理。进一步地,分析了对于无圈图对策此解是分支有效的。并且当连通分支中两个局中人相关联的边删掉后,此连通分支的收益变化情况可用平均树解表示。这一性质是Shapley值和Myerson值所不具有的。最后,我们给出了模糊联盟图对策中模糊平均树解的可加性和分支有效性。     

A short proof of convexity of step-out–step-in sequencing games

The Step out–Step in sequencing game is a particular example of a game from the sequencing game framework of Curiel, Perderzoli, and Tijs, where coalitions of players in a queue may reorder themselves to improve the their overall cost, under some restrictions. Musegaas, Borm and Quant proved, in two papers, that a simple greedy algorithm correctly computes the valuation of a coalition, and that the game is convex. These proofs entail rather involved case analyses; in this note, we give short proofs of both results.