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Public Choice

, Volume 179, Issue 1–2, pp 133–144 | Cite as

Arrow, and unexpected consequences of his theorem

  • Donald G. SaariEmail author
Article

Abstract

A new way to interpret Arrow’s impossibility theorem leads to valued insights that extend beyond voting and social choice to address other mysteries ranging from the social sciences to even the “dark matter” puzzle of astronomy.

Keywords

Arrow impossibility Reductionist approach Paired comparisons 

Notes

Acknowledgements

My thanks to Santiago Guisasola, Dan Jessie, Ryan Kendall, Norm Schofield, Katri Sieberg, and June Zhao for their comments.

References

  1. Arrow, K. (1951). Social choice and individual values. New York, NY: Wiley (2nd edn. 1963).Google Scholar
  2. Arrow, K., & Debreu, G. (1954). Existence of an equilibrium for a competitive economy. Econometrica, 22(3), 265–290.CrossRefGoogle Scholar
  3. Balinski, M., & Young, P. (2001). Fair representation: Meeting the ideal of one man, one vote (2nd ed.). Washington, DC: Brooking Institution Press.Google Scholar
  4. Binney, J., & Tremaine, S. (2008). Galactic dynamics (2nd ed.). Princeton: Princeton University Press.Google Scholar
  5. Black, D. (1958). The theory of committees and elections. Cambridge, MA: Cambridge University Press.Google Scholar
  6. Borda, J. C. (1781). Memoire sur les elections au Scrutin. Histoire de l’Academie Royale des Sciences, Paris.Google Scholar
  7. Brown, J. (2009). Madoff report highlights SEC lapses in detecting fraud. PBS NewsHour.Google Scholar
  8. Condorcet, M. (1785). Éssai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix, Paris.Google Scholar
  9. Greenberg, J. (1979). Consistent majority rule over compact sets of alternatives. Econometrica, 47, 627–636.CrossRefGoogle Scholar
  10. Hazelrigg, G. (1996). The implications of Arrow’s Impossibility Theorem on approaches to optimal engineering design. Journal of Mechanical Design, 118(2), 161–164.CrossRefGoogle Scholar
  11. Huntington, E. V. (1928). The apportionment of representatives in Congress. Transactions of the American Mathematical Society, 30, 85–110.CrossRefGoogle Scholar
  12. Kearns, D. (2010). Lessons learned from the “Underwear Bomber.” Network World.Google Scholar
  13. McKenzie, L. W. (1954). On equilibrium in Graham’s model of world trade and other competitive systems. Econometrica, 22(2), 147–161.CrossRefGoogle Scholar
  14. Nakamura, K. (1975). The core of a simple game with ordinal preferences. International Journal of Game Theory, 4, 95–104.CrossRefGoogle Scholar
  15. Nakamura, K. (1978). The voters in a simple game with ordinal preferences. International Journal of Game Theory, 8, 55–61.CrossRefGoogle Scholar
  16. Nash, J. (1950). Equilibrium points in \(n\)-person games. Proceedings of the National Academy of Sciences, 36(1), 48–49.CrossRefGoogle Scholar
  17. Saari, D. G. (1978). Methods of apportionment and the House of Representatives. The American Mathematical Monthly, 85, 792–802.CrossRefGoogle Scholar
  18. Saari, D. G. (1995). Basic geometry of voting. New York, NY: Springer.CrossRefGoogle Scholar
  19. Saari, D. G. (2000). Mathematical structure of voting paradoxes 1; pairwise vote. Economic Theory, 15, 1–53.CrossRefGoogle Scholar
  20. Saari, D. G. (2001). Decisions and elections. New York, NY: Cambridge University Press.CrossRefGoogle Scholar
  21. Saari, D. G. (2008). Disposing dictators: Demystifying voting paradoxes. New York, NY: Cambridge University Press.CrossRefGoogle Scholar
  22. Saari, D. G. (2010). Aggregation and multilevel design for systems: Finding guidelines. Journal of Mechanical Design, 132, 081006-1–081006-9.CrossRefGoogle Scholar
  23. Saari, D. G. (2014a). A new way to analyze paired comparison rules. Mathematics of Operations Research, 39, 647–655.CrossRefGoogle Scholar
  24. Saari, D. G. (2014b). Unifying voting th eory from Nakamura’s to Greenberg’s Theorems. Mathematical Social Sciences, 69, 1–11.CrossRefGoogle Scholar
  25. Saari, D. G. (2015). Confronting the modeling problem: Combining parts into the whole, Lecture: IMBS workshop on “Validation; What is it?” http://www.tents/events/conferencevideos.php. Accessed 23 June 2015.
  26. Saari, D. G. (2016a). From Arrow’s Theorem to “dark matter,” (Invited featured article). British Journal of Political Science, 46, 1–9.CrossRefGoogle Scholar
  27. Saari, D. G. (2016b). Dynamics and the dark matter mystery, Invited 12/01/2016 article, SIAM News.Google Scholar
  28. Saari, D. G. (2018). Mathematics motivated by the social and behavioral sciences. Philadelphia, PA: SIAM.CrossRefGoogle Scholar
  29. Taylor, P. (2010). Preventing another underwear bomber. http://www.rollcall.com/news/44393-1.html. Accessed 24 March 2017.
  30. US Securities and Exchange Commission. (2009). Office of Inspector General, Case number OIG-509, Investigation of Failure of the SEC to uncover Bernard Madoff’s Ponzi Scheme.Google Scholar
  31. Ward, B. (1965). Majority voting and the alternative forms of public enterprise. In J. Margolis (Ed.), The public economy of urban communities (pp. 112–126). Baltimore, MD: Johns Hopkins University Press.Google Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Institute for Mathematical Behavioral SciencesUniversity of CaliforniaIrvineUSA

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