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Register a new userOn the Complexity of Core, Kernel, and Bargaining Set
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Added by
Francesco Scarcello
Publication date
2010
fields
Informatics Engineering
and research's language is
English
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Coalitional games are mathematical models suited to analyze scenarios where players can collaborate by forming coalitions in order to obtain higher worths than by acting in isolation. A fundamental problem for coalitional games is to single out the most desirable outcomes in terms of appropriate notions of worth distributions, which are usually called solution concepts. Motivated by the fact that decisions taken by realistic players cannot involve unbounded resources, recent computer science literature reconsidered the definition of such concepts by advocating the relevance of assessing the amount of resources needed for their computation in terms of their computational complexity. By following this avenue of research, the paper provides a complete picture of the complexity issues arising with three prominent solution concepts for coalitional games with transferable utility, namely, the core, the kernel, and the bargaining set, whenever the game worth-function is represented in some reasonable compact form (otherwise, if the worths of all coalitions are explicitly listed, the input sizes are so large that complexity problems are---artificially---trivial). The starting investigation point is the setting of graph games, about which various open questions were stated in the literature. The paper gives an answer to these questions, and in addition provides new insights on the setting, by characterizing the computational complexity of the three concepts in some relevant generalizations and specializations.
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Bargaining networks model the behavior of a set of players that need to reach pairwise agreements for making profits. Nash bargaining solutions are special outcomes of such games that are both stable and balanced. Kleinberg and Tardos proved a sharp algorithmic characterization of such outcomes, but left open the problem of how the actual bargaining process converges to them. A partial answer was provided by Azar et al. who proposed a distributed algorithm for constructing Nash bargaining solutions, but without polynomial bounds on its convergence rate. In this paper, we introduce a simple and natural model for this process, and study its convergence rate to Nash bargaining solutions. At each time step, each player proposes a deal to each of her neighbors. The proposal consists of a share of the potential profit in case of agreement. The share is chosen to be balanced in Nashs sense as far as this is feasible (with respect to the current best alternatives for both players). We prove that, whenever the Nash bargaining solution is unique (and satisfies a positive gap condition) this dynamics converges to it in polynomial time. Our analysis is based on an approximate decoupling phenomenon between the dynamics on different substructures of the network. This approach may be of general interest for the analysis of local algorithms on networks.
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