Sharing Supermodular Costs

Andreas S. Schulz (schulzmit.edu)
Nelson A. Uhan (nuhanpurdue.edu)

Abstract: In this work, we apply ideas from cooperative game theory to study situations in which a set of agents face supermodular costs. These situations appear in a variety of scheduling contexts, as well as in some settings related to facility location and network design. Intuitively, cooperation amongst rational agents who face supermodular costs is unlikely. However, in circumstances where the failure to cooperate may lead to negative externalities, one might be interested in methods of encouraging cooperation. In cooperative game theory, one indication that cooperation is achievable is the existence of an efficient and stable cost allocation. The least core value of a cooperative game is the minimum penalty we need to charge a coalition for acting independently that ensures the existence of an efficient and stable cost allocation. The set of all such cost allocations is called the least core. In this paper, we study the computational complexity and approximability of computing the least core value of supermodular cost cooperative games. We show that computing the least core value of supermodular cost cooperative games is strongly NP-hard, and build a framework to approximate the least core value of these games using oracles that approximately determine maximally violated constraints. This framework yields a $(3+\epsilon)$-approximation algorithm for computing the least core value of supermodular cost cooperative games. As a by-product, we show how to compute accompanying approximate least core cost allocations for these games. We also apply our approximation framework to obtain better results for two particular classes of supermodular cost cooperative games that arise from scheduling and matroid optimization.

Keywords:

Category 1: Other Topics (Game Theory )

Category 2: Combinatorial Optimization (Approximation Algorithms )

Category 3: Combinatorial Optimization (Graphs and Matroids )

Citation: Working Paper, Massachusetts Institute of Technology, August 2007. Revised April 2008, February 2009.

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Entry Submitted: 08/28/2007
Entry Accepted: 08/28/2007
Entry Last Modified: 03/22/2009

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