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International Computing and Combinatorics Conference

COCOON 2012: Computing and Combinatorics pp 568–579Cite as

Computing Shapley Value in Supermodular Coalitional Games

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,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.

This work was supported in part by NSF grant CCF-0728779 and by grants from Oberlin College and Carleton College. Thanks to Josh Davis for helpful discussions.

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Author information

Authors and Affiliations

  1. Department of Computer Science, Carleton College, USA

    David Liben-Nowell

  2. Department of Computer Science, Oberlin College, USA

    Alexa Sharp & Tom Wexler

  3. Department of Mathematics, Oberlin College, USA

    Kevin Woods

Authors
  1. David Liben-Nowell
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  2. Alexa Sharp
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  3. Tom Wexler
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  4. Kevin Woods
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Editor information

Editors and Affiliations

  1. School of IT, University of Sydney, Building J12, 2006, Sydney, NSW, Australia

    Joachim Gudmundsson , Julián Mestre  & Taso Viglas ,  & 

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© 2012 Springer-Verlag Berlin Heidelberg

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Liben-Nowell, D., Sharp, A., Wexler, T., Woods, K. (2012). Computing Shapley Value in Supermodular Coalitional Games. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds) Computing and Combinatorics. COCOON 2012. Lecture Notes in Computer Science, vol 7434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32241-9_48

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  • DOI: https://doi.org/10.1007/978-3-642-32241-9_48

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-32240-2

  • Online ISBN: 978-3-642-32241-9

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