Representation languages for coalitional games are a key research area in algorithmic game theory. There is an inherent tradeoff between how general a language is, allowing it to capture more elaborate games, and how hard it is computationally to optimize and solve such games. One prominent such language is the simple yet expressive Weighted Graph Games (WGGs) representation [14], which maintains knowledge about synergies between agents in the form of an edge weighted graph. We consider the problem of finding the optimal coalition structure in WGGs. The agents in such games are vertices in a graph, and the value of a coalition is the sum of the weights of the edges present between coalition members. The optimal coalition structure is a partition of the agents to coalitions, that maximizes the sum of utilities obtained by the coalitions. We show that finding the optimal coalition structure is not only hard for general graphs, but is also intractable for restricted families such as planar graphs which are amenable for many other combinatorial problems. We then provide algorithms with constant factor approximations for planar, minor-free and bounded degree graphs.
Much work in algorithmic game theory has been dedicated to team formation, cooperative game representations and methods for finding optimal teams game theoretic solutions. Several papers describe representations of cooperative domains based on combinatorial structures @cite_26 @cite_20 @cite_23 @cite_24 @cite_5 and a survey in @cite_7 . A detailed presentation of such languages is given in @cite_1 @cite_22 . Generation of the optimal coalition structure received much attention @cite_9 @cite_10 @cite_21 @cite_3 due to its applications, such as vehicle routing and multi-sensor networks. An early approach @cite_9 focused on overlapping coalitions and gave a loose approximation algorithm. Another early approach @cite_10 has a worst case complexity of @math , whereas dynamic programming approaches @cite_30 have a worst case guarantee of @math . Such algorithms were examined empirically in @cite_25 .
@cite_30 @cite_26 @cite_22 @cite_7 @cite_9 @cite_21 @cite_1 @cite_3 @cite_24 @cite_23 @cite_5 @cite_10 @cite_25 @cite_20
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The complete set partitioning (CSP) problem is a special case of the set partitioning problem where the coefficient matrix has 2 m −1 columns, each column being a binary representation of a unique integer between 1 and 2 m −1,m⩾1. It has wide applications in the area of corporate tax structuring in operations research. In this paper we propose a dynamic programming approach to solve the CSP problem, which has time complexityO(3 m ), wheren=2 m −1 is the size of the problem space. We study from a complexity theoretic standpoint the various solution concepts arising in cooperative game theory. We use as a vehicle for this study a game in which the players are nodes of a graph with weights on the edges, and the value of a coalition is determined by the total weight of the edges contained in it. The Shapley value is always easy to compute. The core is easy to characterize when the game is convex, and is intractable (NP-complete) otherwise. Similar results are shown for the kernel, the nucleolus, the e-core, and the bargaining set. As for the von Neumann-Morgenstern solution, we point out that its existence may not even be decidable. Many of these results generalize to the case in which the game is presented by a hypergraph with edges of size k > 2. This exciting and pioneering new overview of multiagent systems, which are online systems composed of multiple interacting intelligent agents, i.e., online trading, offers a newly seen computer science perspective on multiagent systems, while integrating ideas from operations research, game theory, economics, logic, and even philosophy and linguistics. The authors emphasize foundations to create a broad and rigorous treatment of their subject, with thorough presentations of distributed problem solving, game theory, multiagent communication and learning, social choice, mechanism design, auctions, cooperative game theory, and modal logics of knowledge and belief. For each topic, basic concepts are introduced, examples are given, proofs of key results are offered, and algorithmic considerations are examined. An appendix covers background material in probability theory, classical logic, Markov decision processes and mathematical programming. Written by two of the leading researchers of this engaging field, this book will surely serve as THE reference for researchers in the fastest-growing area of computer science, and be used as a text for advanced undergraduate or graduate courses. Preface. 1. Structures. 2. Linear optimization methods. 3. Discrete convex analysis. 4. Computational complexity. 5. Restricted games by partition systems. 6. Restricted games by union stable systems. 7. Values for games on convex geometries. 8. Values for games on matroids. 9. The core, the selectope and the Weber set. 10. Simple games on closure spaces. 11. Voting power. 12. Computing values with Mathematica. Bibliography. Index. Task execution in multi-agent environments may require cooperation among agents. Given a set of agents and a set of tasks which they have to satisfy, we consider situations where each task should be attached to a group of agents that will perform the task. Task allocation to groups of agents is necessary when tasks cannot be performed by a single agent. However it may also be beneficial when groups perform more efficiently with respect to the single agents' performance. In this paper we present several solutions to the problem of task allocation among autonomous agents, and suggest that the agents form coalitions in order to perform tasks or improve the efficiency of their performance. We present efficient distributed algorithms with low ratio bounds and with low computational complexities. These properties are proven theoretically and supported by simulations and an implementation in an agent system. Our methods are based on both the algorithmic aspects of combinatorics and approximation algorithms for NP-hard problems. We first present an approach to agent coalition formation where each agent must be a member of only one coalition. Next, we present the domain of overlapping coalitions. We proceed with a discussion of the domain where tasks may have a precedence order. Finally, we discuss the case of implementation in an open, dynamic agent system. For each case we provide an algorithm that will lead agents to the formation of coalitions, where each coalition is assigned a task. Our algorithms are any-time algorithms, they are simple, efficient and easy to implement. A key problem when forming effective coalitions of autonomous agents is determining the best groupings, or the optimal coalition structure, to select to achieve some goal. To this end, we present a novel, anytime algorithm for this task that is significantly faster than current solutions. Specifically, we empirically show that we are able to find solutions that are optimal in 0.082 of the time taken by the state of the art dynamic programming algorithm (for 27 agents), using much less memory (O(2n) instead of O(3n) for n agents). Moreover, our algorithm is the first to be able to find solutions for more than 17 agents in reasonable time (less than 90 minutes for 27 agents, as opposed to around 2 months for the best previous solution). The theory of cooperative games provides a rich mathematical framework with which to understand the interactions between self-interested agents in settings where they can benefit from cooperation, and where binding agreements between agents can be made. Our aim in this talk is to describe the issues that arise when we consider cooperative game theory through a computational lens. We begin by introducing basic concepts from cooperative game theory, and in particular the key solution concepts: the core and the Shapley value. We then introduce the key issues that arise if one is to consider the cooperative games in a computational setting: in particular, the issue of representing games, and the computational complexity of cooperative solution concepts. Forming effective coalitions is a major research challenge in the field of multi-agent systems. Central to this endeavour is the problem of partitioning the set of agents into exhaustive and disjoint coalitions such that the social welfare is maximized. This coalition structure generation problem is extremely challenging due to the exponential number of partitions that need to be examined. Specifically, given n agents, there are O(nn) possible partitions. To date, the only algorithm that can find an optimal solution in O(3n) is the Dynamic Programming (DP) algorithm, due to However, one of the main limitations of DP is that it requires a significant amount of memory. In this paper, we devise an Improved Dynamic Programming algorithm (IDP) that is proved to perform fewer operations than DP (e.g. 38.7 of the operations given 25 agents), and is shown to use only 33.3 of the memory in the best case, and 66.6 in the worst. Coalition formation is an important capability of automated negotiation among self-interested agents. In order for coalitions to be stable, a key question that must be answered is how the gains from cooperation are to be distributed. Recent research has revealed that traditional solution concepts, such as the Shapley value, core, least core, and nucleolus, are vulnerable to various manipulations in open anonymous environments such as the Internet. These manipulations include submitting false names, collusion, and hiding some skills. To address this, a solution concept called the anonymity-proof core, which is robust against such manipulations, was developed. However, the representation size of the outcome function in the anonymity-proof core (and similar concepts) requires space exponential in the number of agents skills. This paper proposes a compact representation of the outcome function, given that the characteristic function is represented using a recently introduced compact language that explicitly specifies only coalitions that introduce synergy. This compact representation scheme can successfully express the outcome function in the anonymity-proof core. Furthermore, this paper develops a new solution concept, the anonymity-proof nucleolus, that is also expressible in this compact representation. We show that the anonymity-proof nucleolus always exists, is unique, and is in the anonymity-proof core (if the latter is nonempty). and assigns the same value to symmetric skills. We consider optimizing the coalition structure in Coalitional Skill Games (CSGs), a succinct representation of coalitional games (Bachrach and Rosenschein 2008). In CSGs, the value of a coalition depends on the tasks its members can achieve. The tasks require various skills to complete them, and agents may have different skill sets. The optimal coalition structure is a partition of the agents to coalitions, that maximizes the sum of utilities obtained by the coalitions. We show that CSGs can represent any characteristic function, and consider optimal coalition structure generation in this representation. We provide hardness results, showing that in general CSGs, as well as in very restricted versions of them, computing the optimal coalition structure is hard. On the positive side, we show that the problem can be reformulated as constraint satisfaction on a hyper graph, and present an algorithm that finds the optimal coalition structure in polynomial time for instances with bounded tree-width and number of tasks. Preference aggregation is used in a variety of multiagent applications, and as a result, voting theory has become an important topic in multiagent system research. However, power indices (which reflect how much "real power" a voter has in a weighted voting system) have received relatively little attention, although they have long been studied in political science and economics. We consider a particular multiagent domain, a threshold network flow game. Agents control the edges of a graph; a coalition wins if it can send a flow that exceeds a given threshold from a source vertex to a target vertex. The relative power of each edge agent reflects its significance in enabling such a flow, and in real-world networks could be used, for example, to allocate resources for maintaining parts of the network. We examine the computational complexity of calculating two prominent power indices, the Banzhaf index and the Shapley-Shubik index, in this network flow domain. We also consider the complexity of calculating the core in this domain. The core can be used to allocate, in a stable manner, the gains of the coalition that is established. We show that calculating the Shapley-Shubik index in this network flow domain is NP-hard, and that calculating the Banzhaf index is #P-complete. Despite these negative results, we show that for some restricted network flow domains there exists a polynomial algorithm for calculating agents' Banzhaf power indices. We also show that computing the core in this game can be performed in polynomial time. Coalition formation is a key topic in multiagent systems. One may prefer a coalition structure that maximizes the sum of the values of the coalitions, but often the number of coalition structures is too large to allow exhaustive search for the optimal one. Furthermore, finding the optimal coalition structure is NP-complete. But then, can the coalition structure found via a partial search be guaranteed to be within a bound from optimum? We show that none of the previous coalition structure generation algorithms can establish any bound because they search fewer nodes than a threshold that we show necessary for establishing a bound. We present an algorithm that establishes a tight bound within this minimal amount of search, and show that any other algorithm would have to search strictly more. The fraction of nodes needed to be searched approaches zero as the number of agents grows. If additional time remains, our anytime algorithm searches further, and establishes a progressively lower tight bound. Surprisingly, just searching one more node drops the bound in half. As desired, our algorithm lowers the bound rapidly early on, and exhibits diminishing returns to computation. It also significantly outperforms its obvious contenders. Finally, we show how to distribute the desired search across self-interested manipulative agents. © 1999 Elsevier Science B.V. All rights reserved. Coalition formation is a key topic in multiagent systems. One would prefer a coalition structure that maximizes the sum of the values of the coalitions, but often the number of coalition structures is too large to allow for exhaustive search for the optimal one. We present experimental results for three anytime algorithms that search the space of coalition structures. We show that, in the average case, all three algorithms do much better than the recently established theoretical worst case results in (1999a). We also show that no one algorithm is dominant. Each algorithm's performance is influenced by the particular instance distribution, with each algorithm outperforming the others for different instances. We present a possible explanation for the behaviour of the algorithms and support our hypothesis with data collected from a controlled experimental run.