Skip to main content
SpringerLink
Log in

The potential of iterative voting to solve the separability problem in referendum elections

  • Published:
Theory and Decision Aims and scope Submit manuscript

Abstract

In referendum elections, voters are often required to register simultaneous votes on multiple proposals. The separability problem occurs when a voter’s preferred outcome on one proposal depends on the outcomes of other proposals. This type of interdependence can lead to unsatisfactory or even paradoxical election outcomes, such as a winning outcome that is the last choice of every voter. Here we propose an iterative voting scheme that allows voters to revise their voting strategies based on the outcomes of previous iterations. Using a robust computer simulation, we investigate the potential of this approach to solve the separability problem.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. We thank the anonymous referee for raising this point.

References

  • Alvarez, R. M., & Nagler, J. (2001). The likely consequences of internet voting for political representation. Loyola of Los Angeles Law Review, 34, 1115–1153.

    Google Scholar 

  • Attar, A., Majumbar, D., el Piaser, G., & Porteiro, N. (2008). Common agency games: Indifference and separable preferences. Mathematical Social Sciences, 56, 79–95.

    Article  Google Scholar 

  • Bogomolnaia, A., & Jackson, M. O. (2002). The stability of hedonic coalition structures. Games and Economic Behavior, 38, 201–203.

    Article  Google Scholar 

  • Brams, S. J., Kilgour, D. M., & Zwicker, W. S. (1997). Voting on referenda: The separability problem and possible solutions. Electoral Studies, 16, 359–388.

    Article  Google Scholar 

  • Brams, S. J., Kilgour, D. M., & Sanver, M. R. (2007). A minimax procedure for electing committees. Public Choice, 132(3–4), 401–420.

    Article  Google Scholar 

  • Burani, N., & Zwicker, W. S. (2003). Coalition formation games with separable preferences. Mathematical Social Sciences, 45, 27–52.

    Article  Google Scholar 

  • Conitzer, V., Lang, J., & Xia, L. (2011). Hypercube preference aggregation in multi-issue domains. Proceedings of IJCAI-2011, 158–163.

  • Fargier, H., Lang, J., Mengin, J., & Schmidt, N. (2012). Issue-by-issue voting: an experimental evaluation. Proceedings of MPREF-2012.

  • Fudenberg, D., & Levine, D. K. (1998). The theory of learning in games. Cambridge: MIT Press.

    Google Scholar 

  • Fudenberg, D., & Levine, D. K. (2009). Learning and equilibrium. Annual Review of Economics, 1, 385–419.

    Article  Google Scholar 

  • Haake, C.-J., Raith, M. G., & Su, F. E. (2002). Bidding for envy-freeness: A procedural approach to \(n\)-player fair-division problems. Social Choice and Welfare, 19, 723–749.

    Article  Google Scholar 

  • Hodge, J. K. (2006). Permutations of separable preference orders. Discrete Applied Mathematics, 154, 1478–1499.

    Article  Google Scholar 

  • Hodge, J. K., & Klima, R. E. (2005). The mathematics of voting and elections: A hands-on approach. Providence: American Mathematical Society.

    Google Scholar 

  • Hodge, J. K., Krines, M., & Lahr, J. (2009). Preseparable extensions of multidimensional preferences. Order, 26, 125–147.

    Article  Google Scholar 

  • Hodge, J. K., & Schwallier, P. (2006). How does separability affect the desirability of referendum election outcomes? Theory and Decision, 61, 251–276.

    Article  Google Scholar 

  • Hodge, J. K., & TerHaar, M. (2008). Classifying interdependence in multidimensional binary preferences. Mathematical Social Sciences, 55(2), 190–204.

    Article  Google Scholar 

  • Kilgour, D. M., & Bradley, W. J. (1998). Nonseparable preferences and simultaneous elections. Washington: Paper presented at American Political Science Association.

    Google Scholar 

  • Lacy, D., & Niou, E. M. S. (1998). A problem with referendums. Journal of Theoretical Politics, 10, 5–31.

    Article  Google Scholar 

  • Lagerspetz, E. (1995). Paradoxes and representation. Electoral Studies, 15, 83–92.

    Article  Google Scholar 

  • Lang, J. (2007). Vote and aggregation in combinatorial domains with structured preferences. Proceedings of IJCAI-2007. Palo Alto: AAAI Press.

  • Lang, J., & Mengin, J. (2009). The complexity of learning separable ceteris paribus preferences. Proceedings of IJCAI-2009. Palo Alto: AAAI Press.

  • Milchtaich, I. (2009). Weighted congestion games with separable preferences. Games and Economic Behavior, 67, 750–757.

    Article  Google Scholar 

  • Mohen, J., & Glidden, J. (2001). The case for internet voting. Communications of the ACM, 44(1), 72–85.

    Article  Google Scholar 

  • Moulin, H. (1988). Axioms of cooperative decision making. New York: Cambridge University Press.

    Book  Google Scholar 

  • Nurmi, H. (1998). Voting paradoxes and referenda. Social Choice and Welfare, 15, 333–350.

    Article  Google Scholar 

  • Phillips, D. M., & Spakovsky, H. A. (2001). Gauging the risks of internet elections. Communications of the ACM, 44(1), 73–85.

    Article  Google Scholar 

  • Samuelson, L. (1997). Evolutionary games and equilibrium selection. Cambridge: MIT Press.

    Google Scholar 

  • Wagner, C. (1983). Anscombe’s paradox and the rule of three-fourths. Theory and Decision, 15, 303–308.

    Article  Google Scholar 

  • Wagner, C. (1984). Avoiding anscombe’s paradox. Theory and Decision, 15, 233–238.

    Article  Google Scholar 

  • Xia, L., Conitzer, V., & Lang, J. (2008). Voting on multiattribute domains with cyclic preferential dependencies. AAAI-2008, 202–207.

  • Xia, L., Lang, J., & Ying, M. (2007). Strongly decomposable voting rules on multiattribute domains. AAAI-2007, 776–781.

Download references

Acknowledgments

This work was partially supported by National Science Foundation Grant DMS-1003993, which funds a Research Experience for Undergraduates program at Grand Valley State University.

Author information

Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

    Clark Bowman

  2. Department of Mathematics, Grand Valley State University, Allendale, MI, 49401, USA

    Jonathan K. Hodge

  3. Marian University College of Osteopathic Medicine, Indianapolis, IN, 46222, USA

    Ada Yu

Authors
  1. Clark Bowman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jonathan K. Hodge
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ada Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jonathan K. Hodge.

Appendix: profile details

Appendix: profile details

Profile 1

Every voter ranks the ASI-optimal outcome (101) \(4^\mathrm{th}\).

$$\begin{aligned} \left( \begin{array}{ccc} 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 \\ 0 &{} 1 &{} 1 \\ 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 \\ 0 &{} 1 &{} 0 \end{array} \right) ~~~\left( \begin{array}{ccc} 1 &{} 1 &{} 0 \\ 0 &{} 1 &{} 1 \\ 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 \end{array} \right) ~~~\left( \begin{array}{ccc} 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 \\ 1 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 1 \\ 1 &{} 1 &{} 1 \end{array} \right) \end{aligned}$$

Profile 2

Simultaneous voting elects the worst possible outcome (111); 2 variations.

$$\begin{aligned}&Variation 1: \left( \begin{array}{ccc} 1 &{} 1 &{} 0 \\ 0 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 1 \end{array} \right) \left( \begin{array}{ccc} 1 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 \\ 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 \\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 1 \end{array} \right) \left( \begin{array}{ccc} 0 &{} 1 &{} 1 \\ 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 \\ 1 &{} 1 &{} 1 \end{array} \right) \\&Variation 2: \left( \begin{array}{ccc} 1 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 1 \\ 1 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 1 \end{array} \right) \left( \begin{array}{ccc} 1 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 \\ 1 &{} 1 &{} 0 \\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 1 \end{array} \right) \left( \begin{array}{ccc} 0 &{} 1 &{} 1 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 0 \\ 1 &{} 1 &{} 1 \end{array} \right) \end{aligned}$$

Profile 3

The ASI-optimal outcome (011) results from every voter voting their \(3^\text{ rd }\) choice; the \(2^\text{ nd }\) best outcome (010) results from every voter voting their \(2^\text{ nd }\) choice.

$$\begin{aligned} \left( \begin{array}{ccc} 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 1 \\ 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 1 \end{array} \right) \left( \begin{array}{ccc} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 \\ 0 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 \end{array} \right) \left( \begin{array}{ccc} 1 &{} 1 &{} 1 \\ 0 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 \end{array} \right) \end{aligned}$$

Profile 4

There is a dominating block whose \(2^\text{ nd }\) choice is ASI-optimal (3:2 ratio); 2 variations.

$$\begin{aligned}&Variation 1: \left( \begin{array}{ccc} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 0 \\ 1 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 \end{array} \right) \left( \begin{array}{ccc} 1 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 \end{array} \right) \\&Variation 2: \left( \begin{array}{ccc} 1 &{} 1 &{} 1 \\ 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 \end{array} \right) \left( \begin{array}{ccc} 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 1 \end{array} \right) \end{aligned}$$

Profile 5

All voters’ preferences are separable on the first two proposals; some are separable on other sets.

$$\begin{aligned} \left( \begin{array}{ccc} 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 1 \\ 0 &{} 1 &{} 1 \\ 1 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 \end{array} \right) \left( \begin{array}{ccc} 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 1 \\ 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 1 \\ 1 &{} 1 &{} 0 \\ 1 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 \end{array} \right) \left( \begin{array}{ccc} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 \\ 1 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 1 \\ 0 &{} 1 &{} 1 \end{array} \right) \end{aligned}$$

Profile 6

There are multiple ASI-optimal outcomes; 2 variations.

$$\begin{aligned}&Variation 1: \left( \begin{array}{ccc} 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 \\ 1 &{} 1 &{} 0 \\ 1 &{} 0 &{} 1 \end{array} \right) \left( \begin{array}{ccc} 1 &{} 1 &{} 0 \\ 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 1 \\ 0 &{} 1 &{} 1 \\ 1 &{} 1 &{} 1 \end{array} \right) \left( \begin{array}{ccc} 1 &{} 0 &{} 1 \\ 1 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 0 \end{array} \right) \\&Variation 2: \left( \begin{array}{ccc} 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 1 \\ 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 \\ 1 &{} 1 &{} 0 \\ 1 &{} 0 &{} 1 \end{array} \right) \left( \begin{array}{ccc} 1 &{} 1 &{} 0 \\ 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 1 \\ 0 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 \end{array} \right) \left( \begin{array}{ccc} 1 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 0 \end{array} \right) \end{aligned}$$

Profile 7

The Condorcet winner (010) is ASI-optimal and elected by simultaneous voting (18:14:18 ratio); a Condorcet loser (100) is also present.

$$\begin{aligned} \left( \begin{array}{ccc} 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 \\ 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 1 \end{array} \right) \left( \begin{array}{ccc} 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 1 \\ 0 &{} 1 &{} 1 \\ 1 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \end{array} \right) \left( \begin{array}{ccc} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 1 \\ 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 \\ 0 &{} 1 &{} 1 \end{array} \right) \end{aligned}$$

Profile 8

Two diametrically opposed voter blocks; 1 variation with a dominating block (2:3 ratio); 1 variation with equal numbers of voters in each block.

$$\begin{aligned}&\left( \begin{array}{ccc} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 1 \\ 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 1 \\ 1 &{} 1 &{} 0 \end{array} \right) \left( \begin{array}{ccc} 1 &{} 1 &{} 0 \\ 1 &{} 0 &{} 1 \\ 1 &{} 1 &{} 1 \\ 0 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 \end{array} \right) \\&\left( \begin{array}{ccc} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 \\ 1 &{} 0 &{} 1 \\ 0 &{} 1 &{} 1 \\ 1 &{} 1 &{} 1 \end{array} \right) \left( \begin{array}{ccc} 1 &{} 1 &{} 1 \\ 0 &{} 1 &{} 1 \\ 1 &{} 0 &{} 1 \\ 1 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 \end{array} \right) \end{aligned}$$

Profile 9

Three main blocks; the third block is divided into 6 sub-blocks, all of which agree on their first choice but differ on subsequent choices (17:17:3:3:3:3:3:3 ratio).

$$\begin{aligned}&\left( \begin{array}{ccc} 1 &{} 1 &{} 0 \\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 1 \\ 1 &{} 1 &{} 1 \end{array} \right) \left( \begin{array}{ccc} 0 &{} 1 &{} 1 \\ 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 0 \\ 1 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 \end{array} \right) \left( \begin{array}{ccc} 1 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 \\ 1 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 0 \end{array} \right) \left( \begin{array}{ccc} 1 &{} 0 &{} 1 \\ 1 &{} 1 &{} 1 \\ 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \end{array} \right) \\&\left( \begin{array}{ccc} 1 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 1 \\ 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 1 \\ 1 &{} 1 &{} 0 \\ 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \end{array} \right) \left( \begin{array}{ccc} 1 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 1 \\ 0 &{} 1 &{} 1 \\ 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 0 \end{array} \right) \left( \begin{array}{ccc} 1 &{} 0 &{} 1 \\ 1 &{} 1 &{} 1 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 1 \\ 1 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \end{array} \right) \left( \begin{array}{ccc} 1 &{} 0 &{} 1 \\ 1 &{} 1 &{} 1 \\ 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 \\ 1 &{} 1 &{} 0 \\ 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \end{array} \right) \end{aligned}$$

Profile 10

The three outcomes with the lowest ASIs (110, 101, 011) agree with the ASI-optimal outcome (111) on 2 of 3 proposals.

$$\begin{aligned} \left( \begin{array}{ccc} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 1 \\ 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 0 \\ 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 1 \\ 0 &{} 1 &{} 1 \end{array} \right) \left( \begin{array}{ccc} 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 0 \\ 0 &{} 1 &{} 1 \end{array} \right) \left( \begin{array}{ccc} 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 1 \\ 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 1 \\ 1 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 \\ 1 &{} 1 &{} 0 \end{array} \right) \end{aligned}$$

Profile 11

The ASI-optimal outcome (110) is the complement of the simultaneous voting outcome (001); 2 variations, each with 28:28:24:20 ratio.

$$\begin{aligned}&Variation 1: \left( \begin{array}{ccc} 1 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 1 \\ 1 &{} 1 &{} 0 \\ 0 &{} 1 &{} 1 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \end{array} \right) \left( \begin{array}{ccc} 0 &{} 1 &{} 1 \\ 1 &{} 1 &{} 0 \\ 1 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 \end{array} \right) \left( \begin{array}{ccc} 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 0 \\ 1 &{} 1 &{} 1 \\ 0 &{} 1 &{} 1 \\ 1 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 \end{array} \right) \left( \begin{array}{ccc} 1 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 \\ 1 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 \end{array} \right) \\&Variation 2: \left( \begin{array}{ccc} 1 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 1 \\ 1 &{} 1 &{} 0 \\ 0 &{} 1 &{} 1 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \end{array} \right) \left( \begin{array}{ccc} 0 &{} 1 &{} 1 \\ 1 &{} 1 &{} 0 \\ 1 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 \end{array} \right) \left( \begin{array}{ccc} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 0 \\ 1 &{} 1 &{} 1 \\ 0 &{} 1 &{} 1 \\ 1 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 \end{array} \right) ~~\left( \begin{array}{ccc} 1 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 \\ 1 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 1 \\ 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 \end{array} \right) \end{aligned}$$

Profile 12

There is a dominating block whose first choice (111) is ASI-optimal and whose last choice (100, a Condorcet loser) is the first choice of all other voters (26:12:12 ratio).

$$\begin{aligned} \left( \begin{array}{ccc} 1 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 1 \\ 1 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \end{array} \right) \left( \begin{array}{ccc} 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 1 \\ 0 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 \\ 1 &{} 1 &{} 1 \end{array} \right) \left( \begin{array}{ccc} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 1 &{} 1 &{} 0 \\ 1 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 1 \\ 0 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 \end{array} \right) \end{aligned}$$

Profile 13

10 randomly generated blocks with equal numbers of voters.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bowman, C., Hodge, J.K. & Yu, A. The potential of iterative voting to solve the separability problem in referendum elections. Theory Decis 77, 111–124 (2014). https://doi.org/10.1007/s11238-013-9383-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11238-013-9383-2

Keywords

Search

Navigation