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Computing Shapley Value in Supermodular Coalitional Games

  • David Liben-Nowell
  • Alexa Sharp
  • Tom Wexler
  • Kevin Woods
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7434)

Abstract

Coalitional games allow subsets (coalitions) of players to cooperate to receive a collective payoff. This payoff is then distributed “fairly” among the members of that coalition according to some division scheme. Various solution concepts have been proposed as reasonable schemes for generating fair allocations. The Shapley value is one classic solution concept: player i’s share is precisely equal to i’s expected marginal contribution if the players join the coalition one at a time, in a uniformly random order. In this paper, we consider the class of supermodular games (sometimes called convex games), and give a fully polynomial-time randomized approximation scheme (FPRAS) to compute the Shapley value to within a (1 ±ε) factor in monotone supermodular games. We show that this result is tight in several senses: no deterministic algorithm can approximate Shapley value as well, no randomized algorithm can do better, and both monotonicity and supermodularity are required for the existence of an efficient (1 ±ε)-approximation algorithm. We also argue that, relative to supermodularity, monotonicity is a mild assumption, and we discuss how to transform supermodular games to be monotonic.

Keywords

Cooperative Game Solution Concept Multicast Tree Marginal Contribution Grand Coalition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • David Liben-Nowell
    • 1
  • Alexa Sharp
    • 2
  • Tom Wexler
    • 2
  • Kevin Woods
    • 3
  1. 1.Department of Computer ScienceCarleton CollegeUSA
  2. 2.Department of Computer ScienceOberlin CollegeUSA
  3. 3.Department of MathematicsOberlin CollegeUSA

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