Advertisement

Public Choice

, Volume 179, Issue 1–2, pp 113–124 | Cite as

Reflections on Arrow’s theorem and voting rules

  • Nicholas R. MillerEmail author
Article

Abstract

These reflections, written in honor of Kenneth Arrow, sketch out how one political scientist thinks about Arrow’s theorem and its implications for voting rules. The basic claim is that Arrow’s theorem means that all real-world voting rules are problematic in two quite specific ways—namely, they can be neither ‘strategyproof’ nor ‘spoilerproof’. However, Condorcet’s pairwise version of majority rule, while not a fully specified voting rule because of the cyclical majorities problem, is itself both strategyproof and spoilerproof. Moreover, the cycling problem seems to occur only rarely in practice.

Keywords

Kenneth Arrow Arrow’s theorem May’s theorem Voting rules Majority rule Borda rule 

Notes

Acknowledgements

For helpful comments, I thank Jac Heckelman, Dan Felsenthal, and Michel Le Breton.

References

  1. Achen, C. H., & Bartels, L. M. (2016). Democracy for realists: Why elections do not produce responsive government. Princeton: Princeton University Press.Google Scholar
  2. Arrow, K. A. (1951). Social choice and individual values. New York, NY: Wiley.Google Scholar
  3. Arrow, K. A. (1963). Social choice and individual values (2nd ed.). New York, NY: Wiley.Google Scholar
  4. Barberá, S. (1980). Pivotal voters: A new proof of Arrow’s theorem. Economics Letters, 6(1), 13–16.Google Scholar
  5. Black, D. (1948). On the rationale of group decision-making. Journal of Political Economy, 56(1), 23–34.Google Scholar
  6. Black, D. (1958). The theory of committees and elections. Cambridge: Cambridge University Press.Google Scholar
  7. Borda, J.-C. de ([1784] 1995). Mémoire sur les élections au scrutin. Translation in I. McLean & A. B. Urken (Eds.), Classics of social choice (pp. 83–89). Ann Arbor, MI: University of Michigan Press.Google Scholar
  8. Bordes, G., & Tideman, N. (1991). Independence of irrelevant alternatives in the theory of voting. Theory and Decision, 30(2), 163–186.Google Scholar
  9. Chamberlin, J. R., Cohen, J. L., & Coombs, C. H. (1984). Social choice observed: Five presidential elections of the American Psychological Association. Journal of Politics, 46(2), 479–502.Google Scholar
  10. Condorcet, Marquis de ([1785] 1995). Essai sur l’application de l’analyse à la probabilité des decisions rendues à la pluralité des voix. Translation in I. McLean & A. B. Urken (Eds.), Classics of social choice (pp. 91–112). Ann Arbor, MI: University of Michigan Press, 1995.Google Scholar
  11. Dahl, R. A., & Lindbloom, C. E. (1953). Politics, economics, and welfare. New York, NY: Harper and Row.Google Scholar
  12. Dasgupta, P., & Maskin, E. (2008). On the robustness of majority rule. Journal of the European Economic Association, 6(5), 949–973.Google Scholar
  13. Dodgson, C. ([1876] 1995). A method of taking votes on more than two issues. Reprinted in I. McLean & A. B. Urken (Eds.), Classics of social choice (pp. 288–297). Ann Arbor, MI: University of Michigan Press, 1995.Google Scholar
  14. Dougherty, K. L., & Heckelman, J. C. (2017). The probability of violating Arrow’s conditions. Working paper, Department of Political Science, University of Georgia.Google Scholar
  15. Dowding, K., & Van Hees, M. (2008). In praise of manipulation. British Journal of Political Science, 38(1), 1–15.Google Scholar
  16. Dutta, B., Jackson, M. O., & Le Breton, M. (2001). Strategic candidacy and voting procedures. Econometrica, 69(4), 1013–1037.Google Scholar
  17. Feld, S. L., & Grofman, B. (1986). Partial single-peakedness: An extension and clarification. Public Choice, 51(1), 71–80.Google Scholar
  18. Feld, S. L., & Grofman, B. (1988). Ideological consistency as a collective phenomenon. American Political Science Review, 82(3), 773–788.Google Scholar
  19. Feld, S. L., & Grofman, B. (1992). Who’s afraid of the big bad cycle? Evidence from 36 elections. Journal of Theoretical Politics, 4(2), 231–237.Google Scholar
  20. Felsenthal, D. S. (2012). Review of paradoxes afflicting procedures for electing a single candidate. In D. S. Felsenthal & M. Machover (Eds.), Electoral systems: Paradoxes, assumptions, and procedures (pp. 19–91). Heidelberg: Springer.Google Scholar
  21. Fey, M. (2014). A straightforward proof of Arrow’s theorem. Economics Bulletin, 34(3), 1792–1797.Google Scholar
  22. Geanakoplos, J. (2005). Three brief proofs of Arrow’s impossibility theorem. Economic Theory, 26(1), 211–215.Google Scholar
  23. Gehrlein, W. V. (2006). Condorcet’s paradox. Berlin: Springer.Google Scholar
  24. Gehrlein, W. V., & Lepelley, D. (2011). Voting paradoxes and group coherence. Berlin: Springer.Google Scholar
  25. Gibbard, A. (1973). Manipulation of voting schemes: A general result. Econometrica, 41(4), 587–601.Google Scholar
  26. Mackie, G. (2003). Democracy defended. Cambridge: Cambridge University Press.Google Scholar
  27. Maskin, E., & Sen, A. (2014). The Arrow impossibility theorem. New York, NY: Columbia University Press. ReviewGoogle Scholar
  28. Maskin, E. & Sen, A. (2017). A new electoral system? In The New York Review of Books (pp. 8–10).Google Scholar
  29. May, K. O. (1952). A set of independent necessary and sufficient conditions for simple majority decision. Econometrica, 20(4), 680–684.Google Scholar
  30. McKelvey, R. D. (1976). Intransitivities in multidimensional voting models and some implications for agenda control. Journal of Economic Theory, 12(3), 472–482.Google Scholar
  31. McKelvey, R. D. (1979). General conditions for global intransitivities in formal voting models. Econometrica, 47(5), 1085–1112.Google Scholar
  32. McLean, I. (1995). Independence of irrelevant alternatives before Arrow. Mathematical Social Sciences., 30(2), 107–126.Google Scholar
  33. McLean, I. (2003). The reasonableness of independence: A conversation from Condorcet and Borda to the present day. Nuffield College Politics Working Paper 2003-W6, University of Oxford.Google Scholar
  34. Muller, E., & Satterthwaite, M. A. (1977). The equivalence of strong positive association and strategy-proofness. Journal of Economic Theory, 14(2), 412–418.Google Scholar
  35. Nanson, E. J. ([1882] 1995). Methods of election. Reprinted in I. McLean & A. B. Urken (Eds.), Classics of social choice (pp. 321–359). Ann Arbor, MI: University of Michigan Press, 1995.Google Scholar
  36. Penn, E. M. (2015). Arrow’s theorem and its descendants. In J. C. Heckelman & N. R. Miller (Eds.), Handbook of social choice and voting (pp. 237–262). Cheltenham, UK; Northhampton, MA: Edward Elgar.Google Scholar
  37. Poundstone, W. (2008). Gaming the vote: Why elections aren’t fair. New York, NY: Hill and Wang.Google Scholar
  38. Ray, P. (1973). Independence of irrelevant alternatives. Econometrica, 41(5), 987–991.Google Scholar
  39. Reny, P. J. (2010). Arrow’s theorem and the Gibbard–Satterthwaite theorem: A unified approach. Economics Letters, 70(1), 99–105.Google Scholar
  40. Riker, W. H. (1953). Democracy in the United States. New York, NY: Macmillan.Google Scholar
  41. Riker, W. H. (1982). Liberalism against populism: A confrontation between the theory of democracy and the theory of social choice. San Francisco, CA: W. H. Freeman and Company.Google Scholar
  42. Samuelson, P. (1967). Arrow’s mathematical politics. In S. Hook (Ed.), Human values and economic policy. New York, NY: New York University Press.Google Scholar
  43. Satterthwaite, M. A. (1975). Strategy-proofness and Arrow’s conditions: Existence and correspondence theorems for voting procedures and social welfare functions. Journal of Economic Theory, 10(2), 187–217.Google Scholar
  44. Sen, A. K. (1970). Collective choice and social welfare. San Francisco, CA: Holden-Day Inc.Google Scholar
  45. Sen, A. (2014). Arrow and the impossibility theorem. In E. Maskin & A. Sen (Eds.), The Arrow impossibility theorem (pp. 29–55). New York, NY: Columbia University Press.Google Scholar
  46. Tideman, T. N. (1987). Independence of clones as a criterion for voting rules. Social Choice and Welfare, 4(3), 185–206.Google Scholar
  47. Tideman, N. (2006). Collective decisions and voting: The potential for public choice. Aldershot: Ashgate.Google Scholar
  48. Tsetlin, I., Regenwetter, M., & Grofman, B. (2003). The impartial culture maximizes the probability of majority cycles. Social Choice and Welfare, 21(3), 387–398.Google Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Political ScienceUniversity of Maryland Baltimore County (UMBC)BaltimoreUSA

Personalised recommendations