Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Social Choice and Welfare
Published in: Social Choice and Welfare 4/2015

01-04-2015

A note on the Sobolev consistency of linear symmetric values

Author: Norman L. Kleinberg

Published in: Social Choice and Welfare | Issue 4/2015

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In van den Brink et al. (Soc Choice Welf 40:693–714, 2013) it is demonstrated that every \(\alpha \)-egalitarian value (Joosten in Dynamics, equilibria and values dissertation, Maastricht University, Maastricht, 1996) is consistent with respect to the well-known Sobolev reduced game function. However, the authors did not consider the question of whether or not these were the only Sobolev-consistent values. In this note we delineate, in the context of cooperative games with transferable utility and linear, symmetric and efficient values, the entire set of solutions that are consistent with respect to the Sobolev function.
previous article On weighted utilitarianism and an application
next article The Banzhaf value in the presence of externalities

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
Footnotes
1
We hasten to add this is nothing new; for example, and as we note below, a number of authors have accounted for this dependence by choosing to include the grand coalition as an argument to the solution itself. It is just that, in the model we consider herein, a value family is a natural construct.
 
2
What we here call symmetry is there called “anonymity”; we stick to the former as this is the term, now common in the literature, that was originally used in Shapley (1953).
 
3
Note that (2) does not agree with the “usual” definition of permuted game, which is \(\sigma v(S) = v(\sigma (S))\). However, it is easy to show that our symmetry axiom is equivalent to the usual one. Also, (2) does agree with the definition in Shapley (1953).
 
4
We note it follows directly from (3) that every value \(\psi ^2\) formally satisfies \(\alpha \)-standardness for two-player games, with \(\alpha = c^1_1\), although of course this property is only actually defined for \(\alpha \in [0,1]\).
 
5
This is the operation of “\(\fancyscript{B}\)-scaling” as defined in Driessen (2010), which also contains, in a different guise, a result analogous to, but not the same as, our Proposition 1.
 
6
Note we are not claiming this particular perspective is novel, merely that it is one permitting us to focus on questions of marginality and egalitarianism.
 
Literature
Driessen TSH (2010) Associated consistency and values for TU games. Int J Game Theory 39:467–482CrossRef
Hart S, Mas-Colell A (1989) Potential, value and consistency. Econometrica 57:589–614CrossRef
Hernández-Lamoneda L, Juárez R, Sánchez-Sánchez F (2007) Dissection of solutions in cooperative game theory using representation techniques. Int J Game Theory 35:395–426CrossRef
Joosten R (1996) Dynamics, equilibria and values dissertation. Maastricht University, Maastricht
Kleinberg NL, Weiss JH (1986) Weak values, the core and new axioms for the Shapley value. Math Soc Sci 12:21–30CrossRef
Shapley LS (1953) A value for n-person games. Ann Math Stud 28:307–317
Sobolev AI (1973) The functional equations that give the payoffs of the players in an n-person game. In: Vilkas E (ed) Advances in game theory, ”Mintis”. Vilnius, pp 94–151. (in Russian)
van den Brink R, Funaki Y, Ju Y (2013) Reconciling marginalism with egalitarianism: consistency, monotonicity, and implementation of egalitarian Shapley values. Soc Choice Welf 40:693–714CrossRef
Metadata
Title
A note on the Sobolev consistency of linear symmetric values
Author
Norman L. Kleinberg
Publication date
01-04-2015
Publisher
Springer Berlin Heidelberg
Published in
Social Choice and Welfare / Issue 4/2015
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI
https://doi.org/10.1007/s00355-014-0859-y

Other articles of this Issue 4/2015

Social Choice and Welfare 4/2015 Go to the issue

OriginalPaper

On stable outcomes of approval, plurality, and negative plurality games

OriginalPaper

Composition properties in the river claims problem

OriginalPaper

Voting power apportionments

OriginalPaper

Super Tuesday: campaign finance and the dynamics of sequential elections

OriginalPaper

Multidimensional welfare rankings under weight imprecision: a social choice perspective

OriginalPaper

Inefficiency in the shadow of unobservable reservation payoffs