The Shapley Value in Knapsack Budgeted Games
Smriti Bhagat, Anthony Kim, S. Muthukrishnan, Udi Weinsberg

TL;DR
This paper introduces methods for efficiently computing and approximating the Shapley value in a new class of cooperative games called knapsack budgeted games, with implications for broader classes of such games.
Contribution
It presents algorithms for faster and approximate computation of the Shapley value in knapsack budgeted games and introduces a general framework for efficient computation in similar cooperative games.
Findings
Shapley value can be computed faster than exponential time for large agent sets.
An additive approximation algorithm for the Shapley value is proposed.
Pseudo-polynomial time algorithms are developed for related greedy heuristic games.
Abstract
We propose the study of computing the Shapley value for a new class of cooperative games that we call budgeted games, and investigate in particular knapsack budgeted games, a version modeled after the classical knapsack problem. In these games, the "value" of a set of agents is determined only by a critical subset of the agents and not the entirety of due to a budget constraint that limits how large can be. We show that the Shapley value can be computed in time faster than by the na\"ive exponential time algorithm when there are sufficiently many agents, and also provide an algorithm that approximates the Shapley value within an additive error. For a related budgeted game associated with a greedy heuristic, we show that the Shapley value can be computed in pseudo-polynomial time. Furthermore, we generalize our proof techniques and propose what we term…
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Taxonomy
TopicsAuction Theory and Applications · Game Theory and Voting Systems · Complexity and Algorithms in Graphs
