International Journal of Game Theory

, Volume 38, Issue 4, pp 553–574 | Cite as

One-way monotonicity as a form of strategy-proofness

  • M. Remzi Sanver
  • William S. Zwicker


Suppose that a vote consists of a linear ranking of alternatives, and that in a certain profile some single pivotal voter v is able to change the outcome of an election from s alone to t alone, by changing her vote from P v to \({P^\prime_{v}}\) . A voting rule \({\mathcal{F}}\) is two-way monotonic if such an effect is only possible when v moves t from below s (according to P v to above s (according to \({P^\prime_{v}}\) . One-way monotonicity is the strictly weaker requirement forbidding this effect when v makes the opposite switch, by moving s from below t to above t. Two-way monotonicity is very strong—equivalent over any domain to strategy proofness. One-way monotonicity holds for all sensible voting rules, a broad class including the scoring rules, but no Condorcet extension for four or more alternatives is one-way monotonic. These monotonicities have interpretations in terms of strategy-proofness. For a one-way monotonic rule \({\mathcal{F}}\) , each manipulation is paired with a positive response, in which \({\mathcal{F}}\) offers the pivotal voter a strictly better result when she votes sincerely.


One-way monotonicity Monotonicity Participation No-show paradox Strategy-proofness Manipulation Scoring rule Sensible virtue Condorcet extension 

JEL Classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aleskerov F, Kurbanov E (1999) A degree of manipulability of known social choice procedures. In: Aliprantis S, Alkan A, Yannelis N (eds) Studies in Economic Theory, vol 8, Current Trends in Economics: Theory and Applications. Springer Verlag, Berlin, pp 13–28Google Scholar
  2. Black D (1958) The theory of committees and elections. Cambridge University Press, CambridgeGoogle Scholar
  3. Brams SJ, Fishburn PC (1983) Paradoxes of preferential voting. Math Mag 56: 207–214CrossRefGoogle Scholar
  4. Brams SJ, Fishburn PC (2002) Voting procedures, Chap 4. In: Arrow JA, Sen AJ, Suzumura K (eds) Handbook of social choice and welfare, vol 1. Handbooks in economics 19. Elsevier, AmsterdamGoogle Scholar
  5. Campbell DE, Kelly JS (2002) Non-monotonicity does not imply the no-show paradox. Soc Choice Welf 19: 513–515CrossRefGoogle Scholar
  6. Danilov V (1992) Implementation via Nash equilibria. Econometrica 60: 43–56CrossRefGoogle Scholar
  7. Doğan O, Giritligil AE (2007) Existence of anonymous, neutral, Pareto-optimal, single-valued and monotonic social choice functions—a characterization (preprint)Google Scholar
  8. Erdem O, Sanver MR (2005) Minimal monotonic extensions of scoring rules. Soc Choice Welf 25: 31–42CrossRefGoogle Scholar
  9. Favardin P, Lepelley D, Serais J (2002) Borda rule, Copeland method, and strategic manipulation. Rev Econ Design 7: 213–228CrossRefGoogle Scholar
  10. Favardin P, Lepelley D (2006) Some further results on the manipulability of social choice rules. Soc Choice Welf 26: 485–509CrossRefGoogle Scholar
  11. Fishburn PC (1977) Condorcet social choice functions. SIAM J Appl Math 33: 469–489CrossRefGoogle Scholar
  12. Fishburn PC (1982) Monotonicity paradoxes in the theory of elections. Discrete Appl Math 4: 119–134CrossRefGoogle Scholar
  13. Gärdenfors P (1979) On definitions of manipulation of social choice functions, Chap 2. In: Lafont J (ed) Aggregation and revelation of preferences. North Holland, AmsterdamGoogle Scholar
  14. Gibbard A (1973) Manipulation of voting schemes: a general result. Econometrica 41: 587–601CrossRefGoogle Scholar
  15. Maskin E (1977) Nash equilibrium and welfare optimality. mimeoGoogle Scholar
  16. Maskin E (1999) Nash equilibrium and welfare optimality. Rev Econ Stud 66: 23–38CrossRefGoogle Scholar
  17. Merlin VR, Saari DG (1997) Copeland method II: manipulation, monotonicity, and paradoxes. J Econ Theory 72: 148–172CrossRefGoogle Scholar
  18. Moulin H (1988a) Axioms of cooperative decision making. Econometric Soc Monograph #15. Cambridge University Press, LondonGoogle Scholar
  19. Moulin H (1988b) Condorcet’s principle implies the no-show paradox. J Econ Theory 45: 53–64CrossRefGoogle Scholar
  20. Muller E, Satterthwaite M (1977) The equivalence of strong positive association and incentive compatibility. J Econ Theory 14: 412–418CrossRefGoogle Scholar
  21. Saari D (1990) Consistency of decision processes. Ann Oper Res 23: 103–137CrossRefGoogle Scholar
  22. Saari D (1994) Geometry of voting. Aliprantis C, Yannellis NC (eds) Studies in economics 3. Springer, BerlinGoogle Scholar
  23. Saari D (1999) Explaining all three-alternative voting outcomes. J Econ Theory 87: 313–355CrossRefGoogle Scholar
  24. Saari D, Barney S (2003) Consequences of reversing preferences. Math Intell 25: 17–31CrossRefGoogle Scholar
  25. Sanver MR, Zwicker WS (2009) Monotonicity properties for irresolute voting rules (working paper)Google Scholar
  26. Satterthwaite M (1975) Strategy-proofness and Arrow’s conditions: existence and correspondence theorems for voting procedures and social welfare functions. J Econ Theory 10: 187–217CrossRefGoogle Scholar
  27. Smith D (1999) Manipulability measures of common social choice functions. Soc Choice Welf 16: 639–661CrossRefGoogle Scholar
  28. Smith JH (1973) Aggregation of preferences with variable electorate. Econometrica 41: 1027–1041CrossRefGoogle Scholar
  29. Taylor AD (2005) Social choice and the mathematics of manipulation. Cambridge University Press and Mathematical Association of America, CambridgeGoogle Scholar
  30. Zwicker WS (1991) The voters’ paradox, spin, and the Borda count. Math Soc Sci 22: 187–227CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of EconomicsIstanbul Bilgi UniversityIstanbulTurkey
  2. 2.Mathematics DepartmentUnion CollegeSchenectadyUSA

Personalised recommendations