Skip to main content

Measuring Nearly Single-Peakedness of an Electorate: Some New Insights

  • Conference paper
  • First Online:
Algorithmic Decision Theory (ADT 2021)


After introducing a new distance measure of a preference profile to single-peakedness, directly derived from the very definition of single-peaked preferences by Black [4], we undertake a brief comparison with other popular distance measures to single-peakedness. We then tackle the computational aspects of the optimization problem raised by the proposed measure, namely we show that the problem is NP-hard and we propose an integer programming formulation. Finally, we carry out numerical tests on real and synthetic voting data. The obtained results show the interest of the proposed measure, but also shed new light on the advantages and drawbacks of some popular distance measures.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions


  1. 1.

    As in any axiomatic approach, the specific situation considered here does not need to often happen in practice: it is a thought experiment in which one considers a hypothetical situation and examines whether the measure would behave well in such a case.

  2. 2.

    Tests performed on an Intel Core i7 (1.3 GHz base, 3.9 GHz turbo) with 8 GB RAM.

  3. 3.

  4. 4.

  5. 5.

  6. 6.

  7. 7.

    For a weak axis A, each preference relation in the profile is generated in two steps: (1) an axis \(A'\) compatible with A is generated uniformly at random; (2) a candidate \(c^*\) is drawn uniformly at random in \(\mathcal {C}\), then a preference relation is generated uniformly at random among preferences of peak \(c^*\) single-peaked w.r.t. \(A'\).


  1. Arrow, K.: Social Choice and Individual Values. Wiley, Chapman & Hall, New York (1951)

    Google Scholar 

  2. Barberà, S., Gul, F., Stacchetti, E.: Generalized median voter schemes and committees. J. Econ. Theory 61(2), 262–289 (1993)

    Google Scholar 

  3. Bartholdi III, J., Trick, M.A.: Stable matching with preferences derived from a psychological model. Oper. Res. Lett. 5(4), 165–169 (1986)

    Google Scholar 

  4. Black, D.: On the rationale of group decision-making. J. Polit. Econ. 56(1), 23–34 (1948)

    Google Scholar 

  5. Bouveret, S., et al.: Voter autrement 2017 - online experiment, July 2018.

  6. Bredereck, R., Chen, J., Woeginger, G.J.: Are there any nicely structured preference profiles nearby? Math. Soc. Sci. 79, 61–73 (2016)

    Google Scholar 

  7. Conitzer, V.: Eliciting single-peaked preferences using comparison queries. J. Artif. Intell. Res. 35, 161–191 (2009)

    Google Scholar 

  8. Coombs, C.H.: Psychological scaling without a unit of measurement. Psychol. Rev. 57(3), 145 (1950)

    Google Scholar 

  9. Cornaz, D., Galand, L., Spanjaard, O.: Bounded single-peaked width and proportional representation. In: ECAI, pp. 270–275. IOS Press (2012)

    Google Scholar 

  10. Doignon, J., Falmagne, J.: A polynomial time algorithm for unidimensional unfolding representations. J. Algorithms 16(2), 218–233 (1994)

    Google Scholar 

  11. Elkind, E., Lackner, M.: On detecting nearly structured preference profiles. In: Proceedings of the AAAI Conference on Artificial Intelligence (2014)

    Google Scholar 

  12. Elkind, E., Lackner, M., Peters, D.: Structured preferences (Chap. 10). In: Trends in Computational Social Choice, pp. 187–207. AI Access (2017)

    Google Scholar 

  13. Erdélyi, G., Lackner, M., Pfandler, A.: Computational aspects of nearly single-peaked electorates. In: Proceedings of 27th AAAI, pp. 283–289. AAAI Press (2013). (An extended version appeared in J. AI Res. in 2017)

    Google Scholar 

  14. Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L.A.: The complexity of manipulative attacks in nearly single-peaked electorates. In: TARK, pp. 228–237. ACM (2011). (An extended version appeared in Artif. Intel. in 2014)

    Google Scholar 

  15. Feld, S.L., Grofman, B.: Research note partial single-peakedness: an extension and clarification. Publ. Choice 51(1), 71–80 (1986)

    Google Scholar 

  16. Jaeckle, F., Peters, D., Elkind, E.: On recognising nearly single-crossing preferences. In: Proceedings of 32th AAAI (2018)

    Google Scholar 

  17. Mattei, N., Walsh, T.: Preflib: a library for preferences In: Perny, P., Pirlot, M., Tsoukiás, A. (eds.) ADT 2013. LNCS, vol. 8176, pp. 259–270. Springer, Heidelberg (2013).

  18. Nehring, K., Puppe, C.: The structure of strategy-proof social choice-part I: general characterization and possibility results on median spaces. J. Econ. Theory 135(1), 269–305 (2007)

    Google Scholar 

  19. Niemi, R.G.: Majority decision-making with partial unidimensionality. Am. Polit. Sci. Rev. 63(2), 488–497 (1969)

    Google Scholar 

  20. Sui, X., Francois-Nienaber, A., Boutilier, C.: Multi-dimensional single-peaked consistency and its approximations. In: Proceedings of the Twenty-Third IJCAI, pp. 375–382. AAAI Press, Beijing (2013)

    Google Scholar 

  21. Walsh, T.: Uncertainty in preference elicitation and aggregation. In: AAAI, vol. 7, pp. 3–8 (2007)

    Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Olivier Spanjaard .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Escoffier, B., Spanjaard, O., Tydrichová, M. (2021). Measuring Nearly Single-Peakedness of an Electorate: Some New Insights. In: Fotakis, D., Ríos Insua, D. (eds) Algorithmic Decision Theory. ADT 2021. Lecture Notes in Computer Science(), vol 13023. Springer, Cham.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-87755-2

  • Online ISBN: 978-3-030-87756-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics