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.
Arguably, the state of the art method is presented in @cite_3 . It has a worst case runtime of @math and offers no polynomial runtime guarantees, but in practice it is faster than the above methods. All these methods assume a black-box that computes the value of a coalition, while we rely on a specific representation. Another approach solves the coalition structure generation problem @cite_23 , but relies on a different representation. A fixed parameter tractable approach was proposed for typed-games @cite_34 (the running time is exponential in the number of agent types''). However, in graph games the number of agent types is unbounded, so this approach is untractable. In contrast to the above approaches, we provide polynomial algorithms and sufficient conditions that guarantee various for WGGs @cite_26 .
@cite_34 @cite_23 @cite_3 @cite_26
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We revisit the coalition structure generation problem in which the goal is to partition the players into exhaustive and disjoint coalitions so as to maximize the social welfare. One of our key results is a general polynomial-time algorithm to solve the problem for all coalitional games provided that player types are known and the number of player types is bounded by a constant. As a corollary, we obtain a polynomial-time algorithm to compute an optimal partition for weighted voting games with a constant number of weight values and for coalitional skill games with a constant number of skills. We also consider well-studied and well-motivated coalitional games defined compactly on combinatorial domains. For these games, we characterize the complexity of computing an optimal coalition structure by presenting polynomial-time algorithms, approximation algorithms, or NP-hardness and inapproximability lower bounds. We present a new Dynamic Programming (DP) formulation of the Coalition Structure Generation (CSG) problem based on imposing a hierarchical organizational structure over the agents. We show the efficiency of this formulation by deriving DyPE, a new optimal DP algorithm which significantly outperforms current DP approaches in speed and memory usage. In the classic case, in which all coalitions are feasible, DyPE has half the memory requirements of other DP approaches. On graph-restricted CSG, in which feasibility is restricted by a (synergy) graph, DyPE has either the same or lower computational complexity depending on the underlying graph structure of the problem. Our empirical evaluation shows that DyPE outperforms the state-of-the-art DP approaches by several orders of magnitude in a large range of graph structures (e.g. for certain scalefree graphs DyPE reduces the memory requirements by @math and solves problems that previously needed hours in minutes). In this paper, we introduce a simple coalition formation game in the environment of bidding, which is a special case of the weighted majority game (WMG), and is named the weighted simple-majority game (WSMG). In WSMG, payoff is allocated to the winners proportional to the players’ powers, which can be measured in various ways. We define a new kind of stability: the counteraction-stability (C-stability), where any potential deviating players will confront counteractions of the other players. We show that C-stable coalition structures in WSMG always contains a minimal winning coalition of minimum total power. For the variant where powers are measured directly by their weights, we show that it is NP-hard to find a C-stable coalition structure and design a pseudo-polynomial time algorithm. Sensitivity analysis for this variant, which shows many interesting properties, is also done. We also prove that it is NP-hard to compute the Holler-Packel indices in WSMGs, and hence in WMGs as well. We present a new approach to representing coalitional games based on rules that describe the marginal contributions of the agents. This representation scheme captures characteristics of the interactions among the agents in a natural and concise manner. We also develop efficient algorithms for two of the most important solution concepts, the Shapley value and the core, under this representation. The Shapley value can be computed in time linear in the size of the input. The emptiness of the core can be determined in time exponential only in the treewidth of a graphical interpretation of our representation.