Skip to main content
Top
Published in: Social Choice and Welfare 1-2/2023

09-11-2022 | Original Paper

The possibility of generalized social choice functions and Nash’s independence of irrelevant alternatives

Author: Maurice Salles

Published in: Social Choice and Welfare | Issue 1-2/2023

Log in

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

search-config
loading …

Abstract

Social choice functions are generalized to handle Nash’s independence of irrelevant alternatives. Possibility and impossibility results are established.

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!

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!

Footnotes
1
In Arrow’s original version, the unanimity (Pareto) property is a consequence of three properties: non-imposition (or in the 1948 version the procedure has to be non conventional)–meaning essentially that the procedure is not a sort of constant function; monotonicity (in Arrow 1948) later called positive association of social and individual values (Arrow 1950, 1951); and independence of irrelevant alternatives.
 
2
Finiteness of N is crucial to get Arrow’s theorem. However, in Arrow’s framework, finiteness of X is not important since what is needed is to get a transitive social preference. Here, with the choice-theoretic setting, finiteness of X is important. With an infinite set of alternatives we would need specific mathematical properties, for instance a topological structure, to guarantee the non-emptiness of choice sets.
 
3
Suzumura (2016) includes Blair et al. (1976) and Suzumura (1976).
 
4
A weaker property would be the following: if an alternative is chosen in the larger set and still belongs to the smaller set, it must be chosen in the smaller set–a property often attributed to Herman Chernoff (1954) and developed by Sen (1970, 2017).
 
5
In Arrow (2017), one can still read: ‘For example, if you have a three-person election and one is chosen, suppose one of the losers drops out. Now compare that situation when one of the losers never even ran. You should get the same outcome, no matter what system you have anyway.’
 
6
The Nash formulation of IIA is also used in Dasgupta and Maskin (2008), with a slightly different but equivalent formalism.
 
7
In a private communication, Donald Saari gave us a clue based on his paper (Saari 1989) to prove this conjecture.
 
Literature
go back to reference Arrow KJ (1948) The possibility of a universal social welfare function. Document P-41, 26 September 1948, RAND Corporation Arrow KJ (1948) The possibility of a universal social welfare function. Document P-41, 26 September 1948, RAND Corporation
go back to reference Arrow KJ (1950) A difficulty in the concept of social welfare. J Polit Econ 58:328–346CrossRef Arrow KJ (1950) A difficulty in the concept of social welfare. J Polit Econ 58:328–346CrossRef
go back to reference Arrow KJ (1951) Social choice and individual values. Wiley, New York Arrow KJ (1951) Social choice and individual values. Wiley, New York
go back to reference Arrow KJ (1959) Rational choice functions and orderings. Economica 26:121–127CrossRef Arrow KJ (1959) Rational choice functions and orderings. Economica 26:121–127CrossRef
go back to reference Arrow KJ (1963) Social choice and individual values, 2nd edn. Wiley, New York Arrow KJ (1963) Social choice and individual values, 2nd edn. Wiley, New York
go back to reference Arrow KJ, Renwick MK, Monroe LN (2017) On ethics and economics. Conversations with Kenneth J. Arrow. Routledge, New York Arrow KJ, Renwick MK, Monroe LN (2017) On ethics and economics. Conversations with Kenneth J. Arrow. Routledge, New York
go back to reference Balinski M, Laraki R (2010) Majority judgment. Measuring, ranking, and electing. M.I.T. Press, Cambridge Balinski M, Laraki R (2010) Majority judgment. Measuring, ranking, and electing. M.I.T. Press, Cambridge
go back to reference Blair DH, Bordes G, Kelly JS, Suzumura K (1976) Impossibility theorems without collective rationality. J Econ Theory 13:361–379CrossRef Blair DH, Bordes G, Kelly JS, Suzumura K (1976) Impossibility theorems without collective rationality. J Econ Theory 13:361–379CrossRef
go back to reference Bordes G, Tideman N (1992) Independence of irrelevant alternatives in the theory of voting. Theor Decis 30:163–186CrossRef Bordes G, Tideman N (1992) Independence of irrelevant alternatives in the theory of voting. Theor Decis 30:163–186CrossRef
go back to reference Chernoff H (1954) Rational selection of decision functions. Econometrica 22:422–443CrossRef Chernoff H (1954) Rational selection of decision functions. Econometrica 22:422–443CrossRef
go back to reference Dasgupta P, Maskin E (2008) On the robustness of majority rule. J Eur Econ Assoc 6:949–973 (Also in Maskin E. and Sen A (2014) The Arrow impossibility theorem. Columbia University Press, New York, pp 101–142)CrossRef Dasgupta P, Maskin E (2008) On the robustness of majority rule. J Eur Econ Assoc 6:949–973 (Also in Maskin E. and Sen A (2014) The Arrow impossibility theorem. Columbia University Press, New York, pp 101–142)CrossRef
go back to reference Ray P (1972) Independence of irrelevant alternatives. Econometrica 41:987–991CrossRef Ray P (1972) Independence of irrelevant alternatives. Econometrica 41:987–991CrossRef
go back to reference Saari DG (1989) A dictionnary for voting paradoxes. J Econ Theory 48:443–475CrossRef Saari DG (1989) A dictionnary for voting paradoxes. J Econ Theory 48:443–475CrossRef
go back to reference Salles M (2023) Independence of irrelevant alternatives: Arrow. Nash and others, In preparation Salles M (2023) Independence of irrelevant alternatives: Arrow. Nash and others, In preparation
go back to reference Sen AK (1970) Collective choice and social welfare. Holden-Day, San Francisco Sen AK (1970) Collective choice and social welfare. Holden-Day, San Francisco
go back to reference Sen AK (1971) Choice functions and revealed preference. Rev Econ Stud 38:307–317CrossRef Sen AK (1971) Choice functions and revealed preference. Rev Econ Stud 38:307–317CrossRef
go back to reference Sen AK (1982) Choice, welfare and measurement. Basil Blackwell, Oxford Sen AK (1982) Choice, welfare and measurement. Basil Blackwell, Oxford
go back to reference Sen AK (2017) Collective choice and social welfare: an expanded edition. Harvard University Press, CambridgeCrossRef Sen AK (2017) Collective choice and social welfare: an expanded edition. Harvard University Press, CambridgeCrossRef
go back to reference Suzumura K (1976) Remarks on the theory of collective choice. Economica 43:381–390CrossRef Suzumura K (1976) Remarks on the theory of collective choice. Economica 43:381–390CrossRef
go back to reference Suzumura K (2016) Choice, preferences, and procedures. Harvard University Press, CambridgeCrossRef Suzumura K (2016) Choice, preferences, and procedures. Harvard University Press, CambridgeCrossRef
Metadata
Title
The possibility of generalized social choice functions and Nash’s independence of irrelevant alternatives
Author
Maurice Salles
Publication date
09-11-2022
Publisher
Springer Berlin Heidelberg
Published in
Social Choice and Welfare / Issue 1-2/2023
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI
https://doi.org/10.1007/s00355-022-01437-z

Other articles of this Issue 1-2/2023

Social Choice and Welfare 1-2/2023 Go to the issue

Premium Partner