The Shapley Value in Knapsack Budgeted Games

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 $S$ of agents is determined only by a critical subset $T\subseteq S$ of the agents and not the entirety of $S$ due to a budget constraint that limits how large $T$ 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 algorithmic representation framework that captures a broad class of cooperative games with the property of efficient computation of the Shapley value. The main idea is that the problem of determining the efficient computation can be reduced to that of finding an alternative representation of the games and an associated algorithm for computing the underlying value function with small time and space complexities in the representation size.