Computing Shapley Values for Mean Width in 3-D
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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 P_N(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 ϕ(i) = 𝔼_π[v(P_N(π, i) ∪{i}) - v(P_N(π, i))]. More 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^3log^2n) 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(nlog^2n) 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 (d - 2)-subspace in O(n^dlog^2n) time and O(n) space for a point set in d-dimensional space (d ≥ 3).
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