On the symmetry axiom for values of nonatomic games
type: Item
instance of: scholarly article
title: On the symmetry axiom for values of nonatomic games
zbMATH Open document ID: 0707.90110
DOI: 10.1155/S0161171290000229
author: Dov Monderer
author: William H. Ruckle
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/46511
review text: A game on the unit interval I of players, is a set function v that assigns to a (measurable) coalition \(B\subset I\) a number v(B). A value on a linear space V of games is a mapping that assigns to each \(v\in V\) a measure \(\phi\) (v) on I, and such that \(\phi\) is linear on V, symmetric (i.e. \(\phi (v\circ \pi)=\phi \pi (v)\) with \(\pi\) a permutation on I, and \(v\circ \pi\) and \(\phi_{\pi}\) are defined in a natural way) and \(\phi (I)=v(I)\). Existence of a value does not always hold, and for instance, Aumann and Shapley showed that there is no value on the space of games v with bounded variation. The present paper examines the existence of a relaxed version of a value, called H-value and it is such that the symmetry property is relaxed: Instead of requiring it to hold for all permutations \(\pi\), it is required to hold for a subgroup H of permutations. Existence of an H-value depends now on the properties of the subgroups H. The paper proves that if every finitely generated subgroup of H is valuable, then so is H, and that every almost compact group is valuable. Consequences are derived for subspaces of the space of games with bounded variation.
Mathematics Subject Classification ID: 91A70
Mathematics Subject Classification ID: 91A12
Mathematics Subject Classification ID: 91A07
zbMATH DE Number: 4162696
zbMATH Keywords: Shapley value
zbMATH Keywords: continuum of players
zbMATH Keywords: set-function
zbMATH Keywords: linear projection
zbMATH Keywords: H- value
zbMATH Keywords: symmetry property
zbMATH Keywords: games with bounded variation
Wikidata QID: Q114612694
MaRDI profile type: Publication
OpenAlex ID: W2066633184
reviewed by: Zvi Artstein
description: scientific article; zbMATH DE number 4162696