Similarity Gaussian Games: A Subclass of Weighted Graph Games and Their Shapley Value as a Kernel Density Estimate

Dongshuang Hou and Aymeric Lardon
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Dongshuang Hou: School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, China
Aymeric Lardon: Université Jean Monnet Saint-Etienne, Université Lumière Lyon 2, emlyon business school, GATE UMR 5824, F-42023 Saint-Etienne, France

No 2609, Working Papers from Groupe d'Analyse et de Théorie Economique Lyon St-Etienne (GATE Lyon St-Etienne), Université de Lyon

Abstract: We introduce similarity Gaussian games, transferable-utility games in which the worth of a coalition is the total pairwise similarity among its members, measured by a Gaussian kernel. Similarity Gaussian games are convex weighted graph games, so the Shapley value is always a core allocation; it also has two complementary interpretations: it is weighted degree centrality in the associated similarity network and, up to a common factor, a Gaussian kernel density estimate evaluated at each player’s characteristic. We derive the Shapley value in closed form and study how it changes with the bandwidth. For large bandwidths, we establish an exact asymptotic affine relationship between the Shapley values of similarity Gaussian games and squared deviations games, providing a precise link between local similarity and dispersion.

Date: 2026
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