Computing Shapley Values in the Plane

We consider the problem of computing Shapley values for points in the plane, where each point is interpreted as a player, and the value of a coalition is defined by the area of usual geometric objects, such as the convex hull or the minimum axis-parallel bounding box. For sets of n points in the plane, we show how to compute in roughly O(n^{3/2}) time the Shapley values for the area of the minimum axis-parallel bounding box and the area of the union of the rectangles spanned by the origin and the input points. When the points form an increasing or decreasing chain, the running time can be improved to near-linear. In all these cases, we use linearity of the Shapley values and algebraic methods. We also show that Shapley values for the area of the convex hull or the minimum enclosing disk can be computed in O(n^2) and O(n^3) time, respectively. These problems are closely related to the model of stochastic point sets considered in computational geometry, but here we have to consider random insertion orders of the points instead of a probabilistic existence of points

Computing Shapley Values for Mean Width in 3-D

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 $O(n^3\log^2{n})$ 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 $u\cdot v$ 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 $O(n\log^2{n})$ 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 $O(n^d\log^2{n})$ time and $O(n)$ space for a point set in $d$-dimensional space ($d \geq 3$)

On the complexity of halfspace area queries

Given a non convex simple polygon P , is it possible to construct a data structure which after preprocessing can answer halfspace area queries (i.e