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.     

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.     

Population solidarity, population fair-ranking, and the egalitarian value

We investigate the implications of two axioms specifying how a value should respond to changes in the set of players for TU games. Population solidarity requires that the arrival of new players should affect all the original players in the same direction: all gain together or all lose together. On the other hand, population fair-ranking requires that the arrival of new players should not affect the relative positions of the original players. As a result, we obtain characterizations of the egalitarian value, which assigns to each player an equal share over an individual utility level. It is the only value satisfying either one of the two axioms together with efficiency, symmetry and strategic equivalence.     

Asymptotic existence of fair divisions for groups

The problem of dividing resources fairly occurs in many practical situations and is therefore an important topic of study in economics. In this paper, we investigate envy-free divisions in the setting where there are multiple players in each interested party. While all players in a party share the same set of resources, each player has her own preferences. Under additive valuations drawn randomly from probability distributions, we show that when all groups contain an equal number of players, a welfare-maximizing allocation is likely to be envy-free if the number of items exceeds the total number of players by a logarithmic factor. On the other hand, an envy-free allocation is unlikely to exist if the number of items is less than the total number of players. In addition, we show that a simple truthful mechanism, namely the random assignment mechanism, yields an allocation that satisfies the weaker notion of approximate envy-freeness with high probability.     

连续对策上的计策问题

限定一个连续对策不是平凡地无意义（例如对某个局中人绝对有利等），我们提出了连续对策上的计策的基本概念。最后得到结论，如果局中人1使用经典对策，那么他的赢得期望必不是赢得函数的最大值。如果局中人1使用计策成功（即使得局中人2中计），那么局中人1必取得赢得函数的最大值，局中人2也有对偶的结果。     

Dynamic cost allocation for economic lot sizing games

We consider a cooperative game defined by an economic lot sizing problem with concave ordering costs over a finite time horizon, in which each player faces demand for a single product in each period and coalitions can pool orders. We show how to compute a dynamic cost allocation in the strong sequential core of this game, i.e. an allocation over time that exactly distributes costs and is stable against coalitional defections at every period of the time horizon.     

Measures on minimally generated Boolean algebras

We investigate properties of minimally generated Boolean algebras. It is shown that all measures defined on such algebras are separable but not necessarily weakly uniformly regular. On the other hand, there exist Boolean algebras small in terms of measures which are not minimally generated. We prove that under CH a measure on a retractive Boolean algebra can be nonseparable. Some relevant examples are indicated. Also, we give two examples of spaces satisfying some kind of Efimov property.     

Equal treatment, symmetry and Banzhaf value axiomatizations

The Banzhaf value is the only value satisfying the equal treatment, dummy player and marginal contributions conditions and neutrality of some linear operators on the spaces of games. Under some of these neutrality assumptions, equal treatment can be replaced by even weaker conditions. For linear values having the dummy player property, equal treatment is equivalent to symmetry. All these properties together imply the marginal contributions condition, but in some Banzhaf value axiomatizations marginal contributions cannot be a substitute for linearity. Received: December 1997/Revised version: May 2001     

基于平均树值的无圈图博弈有效解

本文对无圈图博弈进行了研究,考虑了大联盟收益不小于各分支收益之和的情况。通过引入剩余公平分配性质,也就是任意两个分支联盟的平均支付变化相等,给出了一个基于平均树值的无圈图博弈有效解。同时,结合有效性和分支公平性对该有效解进行了刻画。特别地,若无圈图博弈满足超可加性时,证明了该有效解一定是核中的元素,说明此时的解是稳定的。最后,通过一案例分析了该有效解的特点,即越大的分支分得的剩余越多,并且关键参与者,也就是具有较大度的参与者可获得相对多的支付。     

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.     

Semi-marginalistic Values for Set Games

Concerning the solution theory for set games, the paper focuses on a family of values, each of which allocates to any player some type of marginalistic contribution with respect to any coalition containing the player. For any value of the relevant family, an axiomatization is given by means of three properties, namely one type of an efficiency property, the equal treatment property and one type of a monotonicity property. We present one proof technique which is based on the decomposition of any arbitrary set game into a union of simple set games, the value of which are much easier to determine. A simple set game is associated with an arbitrary, but fixed item of the universe.     

混沌优化算法在不可分稳态大系统优化中的应用

讨论了整体目标函数关于各子系统不具有可加形式的大规模稳态系统的优化问题,将混沌优化算法应用于其最优值的求解,利用混沌运动的遍历性来得到优化问题的全局最优值.仿真结果表明,该算法简单易行,求解精度和可靠性较高,是解决不可分稳态大系统优化问题的一种有效方法.     

Weighted allocation rules for standard fixed tree games

In this paper we consider standard fixed tree games, for which each vertex unequal to the root is inhabited by exactly one player. We present two weighted allocation rules, the weighted down-home allocation and the weighted neighbour-home allocation, both inspired by the painting story in Maschler et al. (1995) . We show, in a constructive way, that the core equals both the set of weighted down-home allocations and the set of weighted neighbour allocations. Since every weighted down-home allocation specifies a weighted Shapley value (Kalai and Samet (1988)) in a natural way, and vice versa, our results provide an alternative proof of the fact that the core of a standard fixed tree game equals the set of weighted Shapley values. The class of weighted neighbour allocations is a generalization of the nucleolus, in the sense that the latter is in this class as the special member where players have all equal weights.     

Cost allocation and convex data envelopment

This paper considers allocation rules. First, we demonstrate that costs allocated by the Aumann–Shapley and the Friedman–Moulin cost allocation rules are easy to determine in practice using convex envelopment of registered cost data and parametric programming. Second, from the linear programming problems involved it becomes clear that the allocation rules, technically speaking, allocate the non-zero value of the dual variable for a convexity constraint on to the output vector. Hence, the allocation rules can also be used to allocate inefficiencies in non-parametric efficiency measurement models such as Data Envelopment Analysis (DEA). The convexity constraint of the BCC model introduces a non-zero slack in the objective function of the multiplier problem and we show that the cost allocation rules discussed in this paper can be used as candidates to allocate this slack value on to the input (or output) variables and hence enable a full allocation of the inefficiency on to the input (or output) variables as in the CCR model.     

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.     

An axiomatic approach to egalitarianism in TU-games

A core concept is a solution concept on the class of balanced games that exclusively selects core allocations. We show that every continuous core concept that satisfies both the equal treatment property and a new property called independence of irrelevant core allocations (IIC) necessarily selects egalitarian allocations. IIC requires that, if the core concept selects a certain core allocation for a given game, and this allocation is still a core allocation for a new game with a core that is contained in the core of the first game, then the core concept also chooses this allocation as the solution to the new game. When we replace the continuity requirement by a weak version of additivity we obtain an axiomatization of the egalitarian solution concept that assigns to each balanced game the core allocation minimizing the Euclidean distance to the equal share allocation.     

Processing games with shared interest

This paper introduces processing problems with shared interest as an extension of processing situations with restricted capacities (Meertens, M., et al., Processing games with restricted capacities, 2004). Next to an individual capacity to handle jobs, each player now may have interest in the completion of more than one job, and the degrees of interest may vary among players. By cooperating the players can bundle their capacities and follow an optimal processing scheme to minimize total joint costs. The resulting cost allocation problem is analyzed by considering an associated cooperative cost game. An explicit core allocation of this game is provided.     

The value of information in competitive bidding

We consider the situation where two players compete to obtain a valuable object, e.g. a stand of timber in a competitive, sealed-bid environment. Prior to submitting a bid, each player may sample the stand while incurring a common, non-zero cost for each observation. On one hand, a player wishes to take as few observations as possible due to the cost of collecting information. However, one also wishes to obtain as many observations as possible to avoid a bid that overstates the value of the resource.Given different assumptions on the sampling process, the informational structure, and underlying distribution of value, we derive equilibrium bidding strategies. We use these bidding strategies to solve for equilibrium in an information collection problem from the forest products industry.     

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     

边赋权图对策Myerson值的和分解

2003年,Gómez等在考虑社会网络中心性度量时,引入了对称对策上Myerson值的和分解概念,本文将这一概念推广到边赋权图对策上,给出了相应于边赋权图对策的组内Myerson值和组间Myerson值。其中边的权表示这条边的两个端点之间的直接通讯容量,组内Myerson值衡量了每个参与者来自它所在联盟的收益,而组间Myerson值评估了参与者作为其他参与者中介所获取的收益。本文侧重分析了边赋权图对策的组内Myerson值和组间Myerson值的权稳定性和广义稳定性, 并给出了这两类值的刻画。