Cooperative games provide an appropriate framework for fair and stable profit distribution in multiagent systems. In this paper, we study the algorithmic issues on path cooperative games that arise from the situations where some commodity flows through a network. In these games, a coalition of edges or vertices is successful if it enables a path from the source to the sink in the network, and lose otherwise. Based on dual theory of linear programming and the relationship with flow games, we provide the characterizations on the CS-core, least-core and nucleolus of path cooperative games. Furthermore, we show that the least-core and nucleolus are polynomially solvable for path cooperative games defined on both directed and undirected network.
We provide an efficient algorithm for computing the nucleolus for an instance of a weighted cooperative matching game. This resolves a long-standing open question posed in [Faigle, Kern, Fekete, Hochstättler, Mathematical Programming, 1998].
This paper defines a general class of cooperative games for which the nucleolus is efficiently computable. This class includes new members for which the complexity of computing their nucleolus was not previously known. We show that when the minimum excess coalition problem of a cooperative game can be formulated as a hypergraph dynamic program its nucleolus is efficiently computable. This gives a general technique for designing efficient algorithms for computing the nucleolus of a cooperative game. This technique is inspired by a recent result of Pashkovich (2018) on weighted voting games. How…
This paper addresses one of the most challenging issues in designing an efficient and sustainable ridesharing service: ridesharing market design. We formulate it as a fair cost allocation problem through the lens of the cooperative game theory. A special property of the cooperative ridesharing game is that its characteristic function values are calculated by solving an optimization problem. Several concepts of fairness are investigated and special attention is paid to a solution concept named nucleolus, which aims to minimize the maximum dissatisfaction in the system. Due to its computational …
School of Mathematical Sciences, Ocean University of China（中国海洋大学数学科学学院）；Faculty of Information Science and Engineering, Ocean University of China（中国海洋大学信息科学与工程学院）
We provide an algorithm for computing the nucleolus for an instance of a weighted voting game in pseudo-polynomial time. This resolves an open question posed by Elkind. et.al. 2007.
加权投票博弈（WVG）是一种联盟博弈，其中代理人对联盟的贡献由他的权重给出，如果联盟的总权重达到或超过给定的配额，则联盟获胜。这些游戏模型的决策，在政治机构以及合作和盈余分割多智能体域。近年来，加权投票博弈的各种解概念的计算复杂性受到了广泛的关注。特别是，Elkind et al.（2007）研究了WVG中与稳定性相关的解决方案概念的复杂性，即核心，最小核心和核仁。虽然他们已经完全描述了核心和最小核心的算法复杂性，但对于核仁，他们只提供了NP-硬度结果。在本文中，我们解决了Elkind等人提出的一个公开问题。通过证明WVG的核仁，以及更一般地，具有固定k的k-向量加权投票游戏，可以在伪多项式时间内计算，即，存在一种算法，其正确地计算核仁并且在玩家数量和最大权重的时间多项式中运行。在这样做的时候，我们提出了一个计算核仁，这可能是适用于更广泛的一类游戏的一般框架。
The arboricity of a graph is the minimum number of forests required to cover all its edges. In this paper, we examine arboricity from a game-theoretic perspective and investigate cost-sharing in the minimum forest cover problem. We introduce the arboricity game as a cooperative cost game defined on a graph. The players are edges, and the cost of each coalition is the arboricity of the subgraph induced by the coalition. We study properties of the core and propose an efficient algorithm for computing the nucleolus when the core is not empty. In order to compute the nucleolus in the core, we intr…
The computation of a solution concept of a cooperative game usually depends on values of all coalitions. However, in some applications, values of some of the coalitions might be unknown due to various reasons. We introduce a method to approximate standard solution concepts based only on partial information given by a so called incomplete game. We demonstrate the ideas on the class of minimal incomplete games. Approximations are derived for different solution concepts including the Shapley value, the nucleolus, or the core. We show explicit formulas for approximations of some of the solution co…
本文研究了合作博弈中的PAC（probably approximately correct）学习模型。具体来说，我们给出了联盟及其价值的$m$随机样本，这些样本来自一些未知的合作博弈;我们能预测看不见的联盟的价值吗？我们研究了几个著名的合作游戏类，如网络流游戏，阈值任务游戏，诱导子图游戏的PAC学习。我们还建立了一个新的PAC学习能力和核心稳定性之间的连接：对于有效学习的游戏，有可能找到支付部门，可能是稳定的使用多项式数量的样本。
The computation of a solution concept of a cooperative game usually employs values of all coalitions. However, in some applications, the values of some of the coalitions might be unknown due to high costs associated with their determination or simply because it is not possible to determine them exactly. We introduce a method to approximate standard solution concepts based only on partial characteristic function of the cooperative game. In this paper, we build on our previous results and generalise the results of our methods to a significantly larger class of structures of incomplete informatio…
Cooperative games provide a framework to study cooperation among self-interested agents. They offer a number of solution concepts describing how the outcome of the cooperation should be shared among the players. Unfortunately, computational problems associated with many of these solution concepts tend to be intractable---NP-hard or worse. In this paper, we incorporate complexity measures recently proposed by Feige and Izsak (2013), called dependency degree and supermodular degree, into the complexity analysis of cooperative games. We show that many computational problems for cooperative games …
Coreness represents solution concepts related to core in cooperative games, which captures the stability of players. Motivated by the scale effect in social networks, economics and other scenario, we study the coreness of cooperative game with truncated submodular profit functions. Specifically, the profit function $f(\cdot)$ is defined by a truncation of a submodular function $σ(\cdot)$: $f(\cdot)=σ(\cdot)$ if $σ(\cdot)\geqη$ and $f(\cdot)=0$ otherwise, where $η$ is a given threshold. In this paper, we study the core and three core-related concepts of truncated submodular profit cooperative g…
In this paper, a multi-user cooperative computing framework is applied to enable mobile users to utilize available computing resources from other neighboring users via direct communication links. An incentive scheme based on Bertrand game is proposed for the user to determine \textit{who} and \textit{how} to cooperate. We model the resource demand users as \textit{buyers} who aim to use minimal payments to maximize energy savings, whereas resource supply users as \textit{sellers} who aim to earn payments for their computing resource provision. A Bertrand game against \textit{buyer's market} is…
Cooperative interval game is a cooperative game in which every coalition gets assigned some closed real interval. This models uncertainty about how much the members of a coalition get for cooperating together. In this paper we study convexity, core and the Shapley value of games with interval uncertainty. Our motivation to do so is twofold. First, we want to capture which properties are preserved when we generalize concepts from classical cooperative game theory to interval games. Second, since these generalizations can be done in different ways, mainly with regard to the resulting level of un…
Cooperative interval games are a generalized model of cooperative games in which the worth of every coalition corresponds to a closed interval representing the possible outcomes of its cooperation. Selections are all possible outcomes of the interval game with no additional uncertainty. We introduce new selection-based classes of interval games and prove their characterization theorems and relations to existing classes based on the interval weakly better operator. We show new results regarding the core and imputations and examine a problem of equivalence for two different versions of the core,…
We generalize the notion of convexity and average-convexity to the notion of weighted average-convexity. We show several results on the relation between weighted average-convexity and cooperative games. First, we prove that if a game is weighted average-convex, then the corresponding weighted Shapley value is in the core. Second, we exhibit necessary conditions for a communication TU-game to preserve the weighted average-convexity. Finally, we provide a complete characterization when the underlying graph is a priority decreasing tree.
