Computing Shapley Values for Mean Width in 3-D

Abstract

The Shapley value is a common tool in game theory to evaluate the importance of a player in a cooperative setting. In a geometric context, it provides a way to measure the contribution of a geometric object in a set towards some function on the set. Recently, Cabello and Chan (SoCG 2019) presented algorithms for computing Shapley values for a number of functions for point sets in the plane. More formally, a coalition game consists of a set of players N and a characteristic function v : 2 N → R with v(∅) = 0. Let π be a uniformly random permutation of N , and PN (π, i) be the set of players in N that appear before player i in the permutation π. The Shapley value of the game is defined to be φMore intuitively, the Shapley value represents the impact of player i's appearance over all insertion orders. We present an algorithm to compute Shapley values in 3-D, where we treat points as players and use the mean width of the convex hull as the characteristic function. Our algorithm runs in O(n 3 log 2 n) time and O(n) space. Our approach is based on a new data structure for a variant of the dynamic convolution problem (u, v, p), where we want to answer u • v dynamically. Our data structure supports updating u at position p, incrementing and decrementing p and rotating v by 1. We present a data structure that supports n operations in O(n log 2 n) time and O(n) space. Moreover, the same approach can be used to compute the Shapley values for the mean volume of the convex hull projection onto a uniformly random ACM Subject Classification Theory of computation → Computational geometry; Theory of computation → Data structures design and analysis Keywords and phrases Shapley value, mean width, dynamic convolutionAcknowledgements The author is very grateful to David Mount for discussions and advice for the paper. The author is also thankful to Geng Lin for proofreading and helping improving the readability. Computing Shapley Values for Mean Width in 3-DN that appear before player i in the permutation π. The Shapley value of player i ∈ N is defined to be(1)Intuitively, the Shapley value represents the expected marginal contribution of i to the objective function over all permutations of N . There are wide applications of Shapley values.A survey by Winter [27] and a book dedicated to this topic [24] provide insights on how the concept can be interpreted and applied in multiple ways, such as utility of players, allocation of resources of the grand coalition and measure of power in a voting system. Moreover, the values can be characterized axiomatically, making it the only natural quantity that satisfies certain properties. More details can be found in standard game theory textbooks ([14], [8, Chapter 5], [22, Section 9.4]).In convex geometry and measure theory, intrinsic volumes are a key concept to characterize the "size" and "shape" of a convex body regardless of the translation and rotation in its underlying space. For example, Steiner's formula [12] relates intrinsic volumes to the volume of the Minkowski-sum of a convex body ...