Computing Shapley Value in Supermodular Coalitional Games
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), define and survey computational results on other standard solution concepts, and contrast these results with new results regarding the Shapley value. In particular, we give a fully polynomial-time randomized approximation scheme (FPRAS) to compute the Shapley value to within a (1 plus/minus epsilon) 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 plus/minus epsilon)-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.
Liben-Nowell, David, Alexa Sharp, Tom Wexler, and Kevin Woods. 2012. "Computing Shapley Value in Supermodular Coalitional Games," in Proceedings of the International Meeting of the Computing and Combinatorics Conference.
Algorithmic game theory