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Weak rationalizability and Arrovian impossibility theorems for responsive social choice

  • John Duggan
Article

Abstract

This paper provides representation theorems for choice functions satisfying weak rationality conditions: a choice function satisfies \(\alpha\) if and only if it can be expressed as the union of intersections of maximal sets of a fixed collection of acyclic relations, and a choice function satisfies \(\gamma\) if and only if it consists of the maximal elements of a relation that can depend on the feasible set in a particular, well-behaved way. Other rationality conditions are investigated, and these results are applied to deduce impossibility theorems for social choice functions satisfying weak rationality conditions along with positive responsiveness conditions. For example, under standard assumptions on the set of alternatives and domain of preferences, if a social choice function satisfies Pareto optimality, independence of irrelevant alternatives, a positive responsiveness condition for revealed social preferences, and a new rationality condition \(\delta ^{*}\) (a strengthening of \(\gamma\)), then some individual must have near dictatorial power.

Keywords

Acyclicity Choice consistency Impossibility theorem Positive responsiveness Preference aggregation Rationalizability Social choice 

Notes

Acknowledgements

I’m grateful for helpful feedback from an anonymous referee and from Martin Osborne. All errors are my responsibility.

References

  1. Aizerman, M., & Malishevski, A. (1981). General theory of best variants choice: Some aspects. IEEE Transactions on Automatic Control, 26, 1030–1040.CrossRefGoogle Scholar
  2. Aizerman, M., Vol’skiy, V., & Litvakov, M. (1991). Notes about theory of pseudo-criteria and binary pseudo-relations and their application to the theory of choice and voting. Caltech working paper #766.Google Scholar
  3. Arrow, K. (1959). Rational choice functions and orderings. Economica, 26, 121–127.CrossRefGoogle Scholar
  4. Arrow, K. (1963). Social choice and individual values (2nd ed.). New Haven: Cowles Foundation.Google Scholar
  5. Blair, D., Bordes, G., Kelly, J., & Suzumura, K. (1976). Impossibility theorems without collective rationality. Journal of Economic Theory, 13, 361–379.CrossRefGoogle Scholar
  6. Bordes, G. (1983). On the possibility of reasonable consistent majoritarian choice: Some positive results. Journal of Economic Theory, 31, 122–132.CrossRefGoogle Scholar
  7. Chernoff, H. (1954) Rational selection of decision functions. Econometrica, 22, 422–443.CrossRefGoogle Scholar
  8. Denicolò, V. (1985). Independent social choice correspondences are dictatorial. Economics Letters, 19, 9–12.CrossRefGoogle Scholar
  9. Duggan, J. (2007). A systematic approach to the construction of non-empty choice sets. Social Choice and Welfare, 28, 491–506.CrossRefGoogle Scholar
  10. Duggan, J. (2013). Uncovered sets. Social Choice and Welfare, 41, 489–535.CrossRefGoogle Scholar
  11. Duggan, J. (2016). Limits of acyclic voting. Journal of Economic Theory, 163, 658–683.CrossRefGoogle Scholar
  12. Fishburn, P. (1975). Semiorders and choice functions. Econometrica, 43, 5–6.Google Scholar
  13. Gibbard, A. (1969). Social choice and the arrow conditions. Discussion paper, Department of Philosophy, University of Michigan.Google Scholar
  14. Gibbard, A. (2014). Social choice and the arrow conditions. Economics and Philosophy, 30, 269–284.CrossRefGoogle Scholar
  15. Gillies, D. (1959). Solutions to general non-zero-sum games. In A. Tucker, & R. Luce (Eds.), Contributions to the theory of games IV, Annals of mathematical studies (Vol. 40). Princeton: Princeton University Press.Google Scholar
  16. Hansson, B. (1969). Voting and group decision functions. Synthese, 20, 526–537.CrossRefGoogle Scholar
  17. Horan, S., Osborne, M., & Sanver, R. (2017). Positively responsive collective choice rules and majority rule: A generalization of May’s theorem to many alternatives. Unpublished paper.Google Scholar
  18. Houthakker, H. (1950). Revealed preference and the utility function. Economica, 17, 159–174.CrossRefGoogle Scholar
  19. Jamison, D., & Lau, L. (1973). Semiorders and the theory of choice. Econometrica, 41, 901–912.CrossRefGoogle Scholar
  20. Mas-Colell, A., & Sonnenschein, H. (1972). General possibility theorems for group decisions. Review of Economic Studies, 39, 185–192.CrossRefGoogle Scholar
  21. May, K. (1952). A set of independent necessary and sufficient conditions for simple majority decision. Econometrica, 20, 680–684.CrossRefGoogle Scholar
  22. McKelvey, R. (1986). Covering, dominance, and institution-free properties of social choice. American Journal of Political Science, 30, 283–314.CrossRefGoogle Scholar
  23. Moulin, H. (1985). Choice functions over a finite set: A summary. Social Choice and Welfare, 2, 147–160.CrossRefGoogle Scholar
  24. Plott, C. (1973). Path independence, rationality, and social choice. Econometrica, 41, 1075–1091.CrossRefGoogle Scholar
  25. Samuelson, P. (1938). A note on the pure theory of consumer’s behavior. Economica, 5, 61–71.CrossRefGoogle Scholar
  26. Schwartz, T. (1976). Choice functions, ‘rationality’ conditions, and variations on the weak axiom of revealed preference. Journal of Economic Theory, 13, 414–427.CrossRefGoogle Scholar
  27. Schwartz, T. (1986). The logic of collective choice. New York: Columbia.Google Scholar
  28. Sen, A. (1971) Choice functions and revealed preference. The Review of Economic Studies, 38, 307–317.CrossRefGoogle Scholar
  29. Weymark, J. (1983). Quasitransitive rationalization and the superset property. Mathematical Social Sciences, 6, 105–108.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Political ScienceUniversity of RochesterRochesterUSA
  2. 2.Department of EconomicsUniversity of RochesterRochesterUSA

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