Skip to main content

Characterizing the resolute part of monotonic social choice correspondences

Abstract

The resolute domain of a social choice correspondence is the set of profiles at which only one alternative is selected. The Condorcet, plurality, approval voting, and maximin correspondences—restricted to their respective resolute domains—satisfy anonymity, neutrality, and a strong monotonicity condition, but are not characterized by those three conditions. We present examples illustrating this and provide additional conditions that do suffice for characterization.

This is a preview of subscription content, access via your institution.

Notes

  1. 1.

    For the role of monotonicity conditions in social choice theory, see Guha (1972), Blau (1976), Muller and Satterthwaite (1977), Thomson and Myerson (1980), Sengupta (1980), Peleg (1981), Fishburn (1982), Strnad (1985), Saijo (1987), Jain (1988, 1996), Blau and Brown (1989), Chen (2000), Durand (2001), Tanaka (2001a, b), Samejima (2005), Aşan and Sanver (2006), Majumdar and Sen (2006), Sethuraman et al. (2006), Bikhchandani et al. (2006), Kutlu (2007, 2009), Takamiya (2007), Yi (2007), Amorós (2009), Mizukami and Wakayama (2009), Sanver and Zwicker (2009), Ollár (2010), Özkal-Sanver and Sanver (2010), Perry and Powers (2010), Powers (2010), Quesada (2010), Barberà and Moreno (2011), Carbajal et al. (2013).

References

  1. Amorós, P.: Unequivocal majority and Maskin-monotonicity. Soc. Choice Welf. 33, 521–532 (2009). doi:10.1007/s00355-009-0377-5

    Article  Google Scholar 

  2. Aşan, G., Sanver, M.R.: Maskin monotonic aggregation rules. Econ. Lett. 91, 179–183 (2006). doi:10.1016/j.econlet.2005.11.013

    Article  Google Scholar 

  3. Barberà, S., Moreno, B.: Top monotonicity: A common root for single-peakedness, single crossing, and the median voter result. Games Econ. Behav. 73, 345–359 (2011). doi:10.1016/j.geb.2011.02.004

    Article  Google Scholar 

  4. Basu, K., López-Calva, L.F.: Functionings and capabilities. In: Arrow, K.J., Sen, A.K., Suzumura, K. (eds.) Handbook of Social Choice and Welfare, chap. 16, vol. 2, pp. 153–187. Elsevier, Amsterdam (2010)

    Google Scholar 

  5. Bikhchandani, S., Chatterji, S., Lavi, R., Mu’alem, A., Nisan, N., Sen, A.: Weak monotonicity characterizes deterministic dominant-strategy implementation. Econometrica 74, 1109–1132 (2006). doi:10.1111/j.1468-0262.2006.00695.x

    Article  Google Scholar 

  6. Black, D.: On the rationale of group decision-making. J. Polit. Econ. 56, 23–24 (1948)

    Article  Google Scholar 

  7. Blau, J.H.: Neutrality, monotonicity and the right of veto: comment. Econometrica 44, 603 (1976). doi:10.2307/1913987

    Article  Google Scholar 

  8. Blau, J.H., Brown, D.J.: The structure of neutral monotonic social functions. Soc. Choice Welf. 6, 51–61 (1989). doi:10.1007/BF00433363

    Article  Google Scholar 

  9. Brams, S.J., Fishburn, P.C.: Approval Voting. Birkhauser, Boston (1983)

    Google Scholar 

  10. Campbell, D.E., Kelly, J.S.: A strategy-proofness characterization of plurality rule. J. Public. Econ. Theory (2015) (forthcoming)

  11. Carbajal, J.C., McLennan, A., Tourky, R.: Truthful implementation and preference aggregation in restricted domains. J. Econ. Theory 148, 1074–1101 (2013). doi:10.1016/j.jet.2012.11.001

    Article  Google Scholar 

  12. Chen, M.A.: Individual monotonicity and the leximin solution. Econ. Theory 15, 353–365 (2000)

    Google Scholar 

  13. Ching, S.: A simple characterization of plurality rule. J. Econ. Theory 71, 298–302 (1996). doi:10.1006/jeth.1996.0119

    Article  Google Scholar 

  14. Dasgupta, P.S., Maskin, E.S.: On the robustness of majority rule. J. Eur. Econ. Assoc. 6, 949–973 (2008). doi:10.1162/JEEA.2008.6.5.949

    Article  Google Scholar 

  15. d’Aspremont, C., Gevers, L.: Equity and the informational basis of collective choice. Rev. Econ. Stud. 44, 199–209 (1977). doi:10.2307/2297061

    Article  Google Scholar 

  16. d’Aspremont, C., Gevers, L.: Social welfare functionals and interpersonal comparability. In: Arrow, K.J., Sen, A.K., Suzumura, K. (eds.) Handbook of Social Choice and Welfare, vol. 1, pp. 459–541. Elsevier Science, Amsterdam (2002)

    Chapter  Google Scholar 

  17. Durand, S.J.: A note on monotonicity in iterated choice functions. Soc. Choice Welf. 18, 129–134 (2001). doi:10.1007/s003550000064

    Article  Google Scholar 

  18. Fishburn, P.C.: The Theory of Social Choice. Princeton University Press, Princeton (1973)

    Google Scholar 

  19. Fishburn, P.C.: Monotonicity paradoxes in the theory of elections. Discrete Appl. Math. 4, 119–134 (1982). doi:10.1016/0166-218X(82)90070-1

    Article  Google Scholar 

  20. Gaertner, W.: Domain Conditions in Social Choice Theory. Cambridge University Press, Cambridge (2006)

    Google Scholar 

  21. Guha, A.: Neutrality, monotonicity and the right of veto. Econometrica 40, 821–826 (1972). doi:10.2307/1912070

    Article  Google Scholar 

  22. Hammond, P.J.: Equity, Arrow’s conditions, and Rawls’ difference principle. Econometrica 44, 793–804 (1976). doi:10.2307/1913445

    Article  Google Scholar 

  23. Herrero, C.: Capabilities and utilities. Econ. Des. 2, 69–88 (1996). doi:10.1007/BF02499125

    Google Scholar 

  24. Jain, S.K.: Characterization of monotonicity and neutrality for binary Paretian social decision rules. Math. Soc. Sci. 15, 307–312 (1988). doi:10.1016/0165-4896(88)90015-7

    Article  Google Scholar 

  25. Jain, S.K.: Structure of neutral and monotonic binary social decision rules with quasi-transitive individual preferences. J. Econ. 64, 195–212 (1996). doi:10.1007/BF01250115

    Article  Google Scholar 

  26. Ju, B.-G.: An efficiency characterization of plurality social choice on simple preference domains. Econ. Theory 26, 115–128 (2005a). doi:10.1007/s00199-004-0533-1

    Article  Google Scholar 

  27. Ju, B.-G.: A characterization of plurality-like rules based on non-manipulability, restricted efficiency, and anonymity. Int. J. Game Theory 33, 335–354 (2005b)

    Article  Google Scholar 

  28. Kutlu, L.: Superdictatorial domains for monotonic social choice functions. Econ. Lett. 97(151–154), 2007 (2007). doi:10.1016/j.econlet.02.035

    Google Scholar 

  29. Kutlu, L.: A dictatorial domain for monotone social choice functions. Econ. Lett. 105, 14–16 (2009). doi:10.1016/j.econlet.2009.05.004

    Article  Google Scholar 

  30. Majumdar, D., Sen, A.: Top-pair and top-triple monotonicity. Soc. Choice Welf. 27, 175–187 (2006). doi:10.1007/s00355-006-0119-x

    Article  Google Scholar 

  31. May, K.O.: A set of necessary and sufficient conditions for simple majority decisions. Econometrica 20, 680–684 (1952). doi:10.2307/1907651

    Article  Google Scholar 

  32. Mizukami, H., Wakayama, T.: The relation between non-bossiness and monotonicity. Math. Soc. Sci. 58, 256–264 (2009). doi:10.1016/j.mathsocsci.2009.02.005

    Article  Google Scholar 

  33. Muller, E., Satterthwaite, M.A.: The equivalence of strong positive association and strategy-proofness. J. Econ. Theory 14, 412–418 (1977). doi:10.1016/0022-0531(77)90140-5

    Article  Google Scholar 

  34. Ollár, M.: Monotonicity and robustness of majority rule. Econ. Lett. 107, 288–290 (2010). doi:10.1016/j.econlet.2010.02.013

    Article  Google Scholar 

  35. Özkal-Sanver, I., Sanver, M.R.: A new monotonicity condition for tournament solutions. Theory Dec. 69, 439–452 (2010). doi:10.1007/s11238-009-9159-x

    Article  Google Scholar 

  36. Pattanaik, P.K.: Positional rules of collective decision-making. In: Arrow, K.J., Sen, A.K., Suzumura, K. (eds.) Handbook of Social Choice and Welfare, vol. 1, pp. 361–394. Elsevier, Amsterdam (2001)

    Chapter  Google Scholar 

  37. Peleg, B.: Monotonicity properties of social choice correspondences. In: Moeschlin, O., Pallaschke, D. (eds.) Game Theory and Mathematical Economics, pp. 97–101. North-Holland, Amsterdam (1981)

    Google Scholar 

  38. Perry, J., Powers, R.C.: Anonymity, monotonicity, and quota pair systems. Math. Soc. Sci. 60(57–60), 2010 (2010). doi:10.1016/j.mathsocsci.03.005

    Google Scholar 

  39. Powers, R.C.: Maskin monotonic aggregation rules and partial anonymity. Econ. Lett. 106, 12–14 (2010)

    Article  Google Scholar 

  40. Quesada, A.: Monotonicity + efficiency + continuity = majority. Math. Soc. Sci. 60, 149–153 (2010). doi:10.1016/j.mathsocsci.2010.04.004

    Article  Google Scholar 

  41. Rawls, J.: A Theory of Justice. Harvard University Press, Cambridge (1971)

    Google Scholar 

  42. Richelson, J.T.: Characterization result for plurality rule. J. Econ. Theory 19, 548–550 (1978)

    Article  Google Scholar 

  43. Roberts, F.S.: Characterizations of the plurality function. Math. Soc. Sci. 21, 101–127 (1991). doi:10.1016/0165-4896(91)90074-2

    Article  Google Scholar 

  44. Saijo, T.: On constant Maskin monotonic social choice functions. J. Econ. Theory 42, 382–386 (1987). doi:10.1016/0022-0531(87)90094-9

    Article  Google Scholar 

  45. Samejima, Y.: Strategic candidacy, monotonicity, and strategy-proofness. Econ. Lett. 88, 190–195 (2005). doi:10.1016/j.econlet.2005.01.017

    Article  Google Scholar 

  46. Sanver, M.R., Zwicker, W.S.: One-way monotonicity as a form of strategy-proofness. Int. J. Game Theory 38, 553–574 (2009)

    Article  Google Scholar 

  47. Sanver, M.R.: Strategy-proofness of the plurality rule over restricted domains. Econ. Theory 39, 461–471 (2009). doi:10.1007/s00199-008-0358-4

    Article  Google Scholar 

  48. Sengupta, M.: Monotonicity, independence of irrelevant alternatives and strategy-proofness of social decision functions. Rev. Econ. Stud. 47, 393–407 (1980). doi:10.2307/2297000

    Article  Google Scholar 

  49. Sethuraman, J., Teo, C.-P., Vohra, R.V.: Anonymous monotonic social welfare functions. J. Econ. Theory 128, 232–254 (2006). doi:10.1016/j.jet.2004.11.006

    Article  Google Scholar 

  50. Smith, J.H.: Aggregation of preferences with variable electorate. Econometrica 41, 1027–1041 (1973). doi:10.2307/1914033

    Article  Google Scholar 

  51. Strasnick, S.: Social choice and the derivation of Rawls’ difference principle. J. Philos. 73, 85–99 (1976). doi:10.2307/2025509

    Article  Google Scholar 

  52. Strnad, J.: The structure of continuous-valued neutral monotonic social functions. Soc. Choice Welf. 2, 181–195 (1985). doi:10.1007/BF00433528

    Article  Google Scholar 

  53. Takamiya, K.: Domains of social choice functions on which coalition strategy-proofness and Maskin monotonicity are equivalent. Econ. Lett. 95, 348–354 (2007). doi:10.1016/j.econlet.2006.11.005

    Article  Google Scholar 

  54. Tanaka, Y.: Generalized monotonicity and strategy-proofness: a note. Econ. Bull. 4, 1–6 (2001a)

    Google Scholar 

  55. Tanaka, Y.: Generalized monotonicity and strategy-proofness for non-resolute social choice correspondences. Econ. Bull. 4, 1–8 (2001b)

    Google Scholar 

  56. Thomson, W., Myerson, R.B.: Monotonicity and independence axioms. Int. J. Game Theory 9, 37–49 (1980)

    Article  Google Scholar 

  57. Yeh, C.-H.: An efficiency characterization of plurality rule in collective choice problems. Econ. Theory 34, 575–583 (2008). doi:10.1007/s00199-006-0193-4

    Article  Google Scholar 

  58. Yi, J.: Monotonicity, implementation and generalized strategy-proofness. Econ. Lett. 97, 124–127 (2007). doi:10.1016/j.econlet.2007.02.030

    Article  Google Scholar 

  59. Young, H.P.: Social choice scoring functions. SIAM J. Appl. Math. 28, 824–838 (1975). doi:10.1137/0128067

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jerry S. Kelly.

Additional information

The authors would like to thank a referee for a very careful reading of the manuscript that prompted a number of significant improvements.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Alemante, F., Campbell, D.E. & Kelly, J.S. Characterizing the resolute part of monotonic social choice correspondences. Econ Theory 62, 765–783 (2016). https://doi.org/10.1007/s00199-015-0930-7

Download citation

Keywords

  • Social choice correspondence
  • Social choice function
  • Resolute domain
  • Anonymity
  • Neutrality
  • Monotonicity
  • Upper invariance
  • Lower invariance
  • Condorcet
  • Plurality voting
  • Approval voting
  • Maximin

JEL Classification

  • D70
  • D71