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 $v: 2^N \to \mathbb{R}$ with $v(\emptyset) = 0$. Let $\pi$ be a uniformly random permutation of $N$, and $P_N(\pi, i)$ be the set of players in $N$ that appear before player $i$ in the permutation $\pi$. The Shapley value of the game is defined to be $$\phi(i) = \mathbb{E}_\pi[v(P_N(\pi, i) \cup {i}) - v(P_N(\pi, i))]$$. 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 $O(n<sup>3\log<sup>2{n})$ 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 $u\cdot v$ 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 $O(n\log<sup>2{n})$ 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 $(d - 2)$-subspace in $O(n<sup>d\log<sup>2{n})$ time and $O(n)$ space for a point set in $d$-dimensional space ($d \geq 3$).