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Computing Shapley Values in the Plane

Authors Sergio Cabello , Timothy M. Chan

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Author Details

Sergio Cabello
  • Department of Mathematics, FMF, University of Ljubljana, Slovenia
  • Department of Mathematics, IMFM, Ljubljana, Slovenia
Timothy M. Chan
  • Department of Computer Science, University of Illinois at Urbana-Champaign, USA

Funding

  • Cabello, Sergio: Supported by the Slovenian Research Agency (P1-0297, J1-8130, J1-8155).
  • Chan, Timothy M.: Supported in part by NSF Grant CCF-1814026.

Acknowledgements

The authors are very grateful to Sariel Har-Peled for fruitful discussions.

Cite As Get BibTex

Sergio Cabello and Timothy M. Chan. Computing Shapley Values in the Plane. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 20:1-20:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.SoCG.2019.20

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Shapley values
  • stochastic computational geometry
  • convex hull
  • minimum enclosing disk
  • bounding box
  • arrangements
  • convolutions
  • airport problem

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