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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

Abstract

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

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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.

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.

Suzumura (2016) includes Blair et al. (1976) and Suzumura (1976).

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).

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.’

The Nash formulation of IIA is also used in Dasgupta and Maskin (2008), with a slightly different but equivalent formalism.

In a private communication, Donald Saari gave us a clue based on his paper (Saari 1989) to prove this conjecture.

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

Nash J (1950) The bargaining problem. Econometrica 18:155–162 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

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

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