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Erschienen 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

verfasst von: Maurice Salles

Erschienen in: Social Choice and Welfare | Ausgabe 1-2/2023

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Abstract

Social choice functions are generalized to handle Nash’s independence of irrelevant alternatives. Possibility and impossibility results are established.
Vorheriger Artikel Financial aid in college admissions: need-based versus merit-based
Nächster Artikel Where should your daughter go to college? An axiomatic analysis
Fußnoten
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.
 
Literatur
Arrow KJ (1948) The possibility of a universal social welfare function. Document P-41, 26 September 1948, RAND Corporation
Arrow KJ (1950) A difficulty in the concept of social welfare. J Polit Econ 58:328–346 CrossRef
Arrow KJ (1951) Social choice and individual values. Wiley, New York
Arrow KJ (1959) Rational choice functions and orderings. Economica 26:121–127 CrossRef
Arrow KJ (1963) Social choice and individual values, 2nd edn. Wiley, New York
Arrow KJ, Renwick MK, Monroe LN (2017) On ethics and economics. Conversations with Kenneth J. Arrow. Routledge, New York
Balinski M, Laraki R (2010) Majority judgment. Measuring, ranking, and electing. M.I.T. Press, Cambridge
Blair DH, Bordes G, Kelly JS, Suzumura K (1976) Impossibility theorems without collective rationality. J Econ Theory 13:361–379 CrossRef
Bordes G, Tideman N (1992) Independence of irrelevant alternatives in the theory of voting. Theor Decis 30:163–186 CrossRef
Chernoff H (1954) Rational selection of decision functions. Econometrica 22:422–443 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
Ray P (1972) Independence of irrelevant alternatives. Econometrica 41:987–991 CrossRef
Saari DG (1989) A dictionnary for voting paradoxes. J Econ Theory 48:443–475 CrossRef
Salles M (2023) Independence of irrelevant alternatives: Arrow. Nash and others, In preparation
Sen AK (1970) Collective choice and social welfare. Holden-Day, San Francisco
Sen AK (1971) Choice functions and revealed preference. Rev Econ Stud 38:307–317 CrossRef
Sen AK (1982) Choice, welfare and measurement. Basil Blackwell, Oxford
Sen AK (2017) Collective choice and social welfare: an expanded edition. Harvard University Press, Cambridge CrossRef
Suzumura K (1976) Remarks on the theory of collective choice. Economica 43:381–390 CrossRef
Suzumura K (2016) Choice, preferences, and procedures. Harvard University Press, Cambridge CrossRef
Metadaten
Titel
The possibility of generalized social choice functions and Nash’s independence of irrelevant alternatives
verfasst von
Maurice Salles
Publikationsdatum
09.11.2022
Verlag
Springer Berlin Heidelberg
Erschienen in
Social Choice and Welfare / Ausgabe 1-2/2023
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI
https://doi.org/10.1007/s00355-022-01437-z

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