We study the cost sharing problem for cooperative games in situations where the cost function @math is not available via oracle queries, but must instead be derived from data, represented as tuples @math , for different subsets @math of players. We formalize this approach, which we call statistical cost sharing, and consider the computation of the core and the Shapley value, when the tuples are drawn from some distribution @math . Previous work by in this setting showed how to compute cost shares that satisfy the core property with high probability for limited classes of functions. We expand on their work and give an algorithm that computes such cost shares for any function with a non-empty core. We complement these results by proving an inapproximability lower bound for a weaker relaxation. We then turn our attention to the Shapley value. We first show that when cost functions come from the family of submodular functions with bounded curvature, @math , the Shapley value can be approximated from samples up to a @math factor, and that the bound is tight. We then define statistical analogues of the Shapley axioms, and derive a notion of statistical Shapley value. We show that these can always be approximated arbitrarily well for general functions over any distribution @math .
The core, introduced by , is another well-studied solution concept for cooperative games. and characterized when the core is non-empty. The core has been studied in the context of multiple combinatorial games, such as facility location @cite_34 and maximum flow @cite_36 . In cases with no solutions in the core or when it is computationally hard to find one, the balance property has been relaxed to hold approximately @cite_12 @cite_39 . In applications where players submit bids, cross-monotone cost sharing, a concept stronger than the core that satisfies the group strategy proofness property, has attracted a lot of attention @cite_26 @cite_3 @cite_21 @cite_14 . We note that these applications are sufficiently different from the ones we are studying in this work.
@cite_26 @cite_14 @cite_36 @cite_21 @cite_3 @cite_39 @cite_34 @cite_12
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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. A class of multiperson mathematical optimization problems is considered and is shown to generate cooperative games with nonempty cores. The class includes, but is not restricted to, numerous versions of network flow problems. It was shown by Owen that for games generated by linear programming optimization problems, optimal dual solutions correspond to points in the core. We identify a special class of network flow problems for which the converse is true, i.e., every point in the core corresponds to an optimal dual solution. Weighted voting games are a popular model of collaboration in multiagent systems. In such games, each agent has a weight (intuitively corresponding to resources he can contribute), and a coalition of agents wins if its total weight meets or exceeds a given threshold. Even though coalitional stability in such games is important, existing research has nonetheless only considered the stability of the grand coalition. In this paper, we introduce a model for weighted voting games with coalition structures. This is a natural extension in the context of multiagent systems, as several groups of agents may be simultaneously at work, each serving a different task. We then proceed to study stability in this context. First, we define the CS-core, a notion of the core for such settings, discuss its non-emptiness, and relate it to the traditional notion of the core in weighted voting games. We then investigate its computational properties. We show that, in contrast with the traditional setting, it is computationally hard to decide whether a game has a non-empty CS-core, or whether a given outcome is in the CS-core. However, we then provide an efficient algorithm that verifies whether an outcome is in the CS-core if all weights are small (polynomially bounded). Finally, we also suggest heuristic algorithms for checking the non-emptiness of the CS-core. 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.