Skip to main content
Springer Nature Link

Voting behavior in one-shot and iterative multiple referenda

  • Original Paper
  • Published:
Save article
View saved research
Social Choice and Welfare Aims and scope Submit manuscript

Abstract

We consider a set of voters making a collective decision via simultaneous vote on two binary issues. Voters’ preferences are captured by payoffs assigned to combinations of outcomes for each issue and they can be nonseparable: a voter’s preference over an issue might be dependent on the other issue. When the collective decision in this context is reached by voting on both issues at the same time, multiple election paradoxes may arise, as studied extensively in the theoretical literature. In this paper we pursue an experimental approach and investigate the impact of iterative voting, in which groups deliberate by repeating the voting process until a final outcome is reached. Our results from experiments run in the lab show that voters tend to have an optimistic rather than a pessimistic behaviour when casting a vote on a non-separable issue and that iterated voting may in fact improve the social outcome. We provide the first comprehensive empirical analysis of individual and collective behavior in the multiple referendum setting.

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

Access this article

Log in via an institution

Subscribe and save

Springer+
from €39.99 /Month
  • Starting from 10 chapters or articles per month
  • Access and download chapters and articles from more than 300k books and 2,500 journals
  • Cancel anytime
View plans

Buy Now

Price includes VAT (Germany)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Explore related subjects

Discover the latest articles, books and news in related subjects, suggested using machine learning.

Notes

  1. For a survey, see Meir (2017); note that a number of papers have appeared since then. We survey the related literature in iterative voting in Sect. 2.2.

  2. Examples from the real world are ubiquitous, including the commonly cited decision-making processes in Switzerland and California. For the latter, for instance, see: https://ballotpedia.org/California_2020_ballot_propositions.

  3. Paradoxes arise even if voters have separable preferences, as pointed out by Benoît and Kornhauser (2010) as well as by Özkal-Sanver and Sanver (2006).

  4. On the other hand, iterated bargaining has a longer history, and has been studied experimentally: especially, Diermeier and Morton (2005) test the well-known Baron–Ferejohn legislative bargaining model, by having subjects playing in several rounds until they agree on a proposal; although the iterated games in legislative bargaining and multiple referenda share some similarities, the nature of decisions (proposal by a single agent vs. voting), of the payoff (part of the proposal vs. associated with alternatives) and the underlying epistemic assumption (complete vs. partial knowledge) make however these two models quite different.

  5. If \(\succ\) is a linear order then (\(TR \succ TL\) if and only if \(BR \succ BL\)) implies (\(TL \succ TR\) if and only if \(BL \succ BR\)).

  6. The restriction to two variables is crucial: counting the number of separable preferences is an open problem (Hodge 2006).

  7. If two half rational voters have the preference relation \(TR \succ TL \succ BR \succ BL\), it makes sense to distinguish further between one who votes for L and one who votes for B—the latter being perhaps more irrational than the former—but we will neglect this distinction.

  8. Note that it is not necessary to write \({\text{ Col }}\)-optimistic, \({\text{ Row }}\)-optimistic, and so on, because there is only one variable for which this distinction applied; the other variable is separable.

  9. In elections with 5 voters (Profiles 8 and 9), one subject among the 21 present in the lab is randomly chosen to wait through the election.

  10. Another interest of Profile 7 is that if the current outcome is BR, and if the margins of victory for both B and R are large, then voters 1–6 would feel that their probability of being pivotal at the next stage is low for each of both issues, and may be tempted to vote TL to try to be pivotal in either one them (the probability of being pivotal on both being negligible); such a behaviour has been analyzed theoretically by Ahn and Oliveros (2012).

  11. We mention studies by Laury (2005) and Baltussen et al. (2012) that found no difference between pay-one and pay-all designs. We chose a pay-all design to avoid adding a level of complication to the understanding of the instructions, and we believe that the effect of this choice is negligible given the small number of elections played by each user.

  12. The binomial test yields significant difference at 5% from random choice for any value under 27 (62) for both (one of the) issues, given 147 choices. Thus, all values in Tables 9 and 10 regarding Profile 1, for instance, are statistically significantly different than random choice.

  13. Note that this finding is based only on first stages of profiles 5–9, since there are no voters with semi-separable preferences in profiles 1–4.

  14. One of the authors gave many talks for over a decade about multiple referenda in which the audience was presented with a two-issue example with ordinal preferences to make a choice (in a non-repeated setting). In this pseudo-experiments, an important difference pertained to that voters’ preferences were common knowledge. There was (almost) always a majority of optimists (about two-thirds) with a significant minority of pessimists.

  15. We thank one of the reviewers for pointing this to us, which led us to add the discussion that follows.

  16. We also note that both interpretations (optimistic/pessimistic, risk seeker/adverse) make more sense as plausible explanations when the expected payoffs of the concerned lotteries are equal or almost equal. While a voter preferring a lottery with possible payoffs 0 and 4 against one with possible payoffs 2 and 3 is definitely optimistic (or risk-seeking), one with the opposite preference perhaps simply applies the principle of insufficient reason (that is, assign probability \(\frac{1}{2}\) to each outcome in each lottery) and maximises expected payoff, rather than being pessimistic stricto sensu.

  17. We thank one of the reviewers for this remark.

  18. The choices in Profiles 2 and 4 cannot be compared with choices by any other voter in an iterated profile.

  19. The estimation is restricted to fully separable and fully non-separable preferences. Semi-separable cases are analyzed in the sequel.

  20. \(\text {Prob}[\varDelta x_{it}=1|SEP=0,PIV=0]-\text {Prob}[\varDelta x_{it}=1|SEP=1,PIV=0]=34.52--27.63\%\approx 6.9\%\).

  21. Still, there are a significant number of subjects that change their votes, without however changing the result of the election. A fine-grained analysis of this can be found in Appendix 1.

  22. Note that this slightly contrasts with the measure used by Bowman et al. (2014), who computes the social welfare as the Borda score of the election outcome divided by the number of voters. The two measures differ only on profiles with utility scales that are not linear.

  23. Figure 8 in Appendix A shows the differences between the average social welfare and minimum possible social welfare as a percentage of the range of possible values in the first and last stages of profiles 5–8.

  24. Recall that our findings in Section 5 show that there is no statistical difference between voters’ behaviour in one-shot elections and first stages of iterated elections, thus we can assume that the social welfare in the first stage is the same as if the election was held as one-shot.

  25. Detailed comparative analysis regarding the “static” and “dynamic” groups can be found in the Appendix 1. Note also that Profiles 8 and 9 do not have a very large span of social welfare between the minimum and maximum attainable outcomes, as can be seen in the Table 18 in the Appendix 1.

  26. No definitive conclusion can be drawn from these numbers, as most of our profiles were chosen so as to be at least somewhat pathological in the first place and not fit for this particular question.

References

  • Ahn DS, Oliveros S (2012) Combinatorial voting. Econometrica 80(1):89–141

    Article  Google Scholar 

  • Airiau S, Grandi U, Perotto FS (2017) Learning agents for iterative voting. In: Proceedings of the 5th International Conference on Algorithmic Decision Theory (ADT). Springer, Cham, pp 139–152

  • Baltussen G, Post GT, Van Den Assem MJ, Wakker PP (2012) Random incentive systems in a dynamic choice experiment. Exp Econ 15(3):418–443

    Article  Google Scholar 

  • Baumeister D, Selker A, Wilczynski A (2020) Manipulation of opinion polls to influence iterative elections. In: Proceedings of the 19th International Conference on Autonomous Agents and Multiagent Systems (AAMAS), pp 132–140

  • Benoît J, Kornhauser LA (2010) Only a dictatorship is efficient. Games Econ Behav 70(2):261–270

    Article  Google Scholar 

  • Bowman C, Hodge JK, Yu A (2014) The potential of iterative voting to solve the separability problem in referendum elections. Theory Decis 77(1):111–124

    Article  Google Scholar 

  • Bradley WJ, Hodge JK, Kilgour DM (2005) Separable discrete preferences. Math Soc Sci 49(3):335–353

    Article  Google Scholar 

  • Brams SJ, Kilgour DM, Zwicker WS (1997) Voting on referenda: the separability problem and possible solutions. Elect Stud 16(3):359–377

    Article  Google Scholar 

  • Brams SJ, Kilgour DM, Zwicker WS (1998) The paradox of multiple elections. Soc Choice Welf 15(2):211–236

    Article  Google Scholar 

  • Brams SJ, Kilgour DM, Sanver MR (2007) A minimax procedure for electing committees. Public Choice 132(3):401–420

    Article  Google Scholar 

  • Brânzei S, Caragiannis I, Morgenstern J, Procaccia AD (2013) How bad is selfish voting? In: Proceedings of the 27th AAAI Conference on Artificial Intelligence (AAAI), pp 138–144

  • Conitzer V, Lang J, Xia L (2011) Hypercubewise preference aggregation in multi-issue domains. In: Proceedings of the 22nd International Joint Conference on Artificial Intelligence (IJCAI), pp 158–163

  • Cornelio C, Pini MS, Rossi F, Venable KB (2019) Multi-agent soft constraint aggregation via sequential voting: theoretical and experimental results. Auton Agents Multi Agent Syst 33(1–2):159–191

    Article  Google Scholar 

  • Çuhadaroğlu T, Lainé J (2012) Pareto efficiency in multiple referendum. Theory Decis 72(4):525–536

    Article  Google Scholar 

  • Diermeier D, Morton R (2005) Experiments in majoritarian bargaining. In: Austen-Smith D, Duggan J (eds) Social choice and strategic decisions: essays in honor of Jeffrey S. Banks. Springer Berlin Heidelberg, Berlin, pp 201–226

    Chapter  Google Scholar 

  • Elkind E, Grossi D, Shapiro E, Talmon N (2020) Egalitarian deliberative decision making. CoRR abs/2001.08031, https://arxiv.org/abs/2001.08031

  • Fain B, Goel A, Munagala K, Sakshuwong S (2017) Sequential deliberation for social choice. In: Proceedings of the 13th International Conference on Web and Internet Economics (WINE). Springer, Cham, pp 177–190

  • Fairstein R, Lauz A, Meir R, Gal K (2019) Modeling people’s voting behavior with poll information. In: Proceedings of the 18th International Conference on Autonomous Agents and MultiAgent Systems, AAMAS ’19, Montreal, QC, Canada, May 13–17, 2019, pp 1422–1430

  • Farnoud F, Yaakobi E, Touri B, Milenkovic O, Bruck J (2013) Building consensus via iterative voting. In: Proceedings of the 2013 IEEE International Symposium on information theory

  • Garg N, Kamble V, Goel A, Marn D, Munagala K (2019) Iterative local voting for collective decision-making in continuous spaces. J Artif Intell Res 64:315–355

    Article  Google Scholar 

  • Goel A, Lee DT (2016) Towards large-scale deliberative decision-making: small groups and the importance of triads. In: Proceedings of the 2016 ACM Conference on Economics and Computation (EC), pp 287–303

  • Gonzales C, Perny P, Queiroz S (2008) Preference aggregation with graphical utility models. In: Proceedings of the 23rd AAAI Conference on Artifcial Intelligence (AAAI), pp 1037–1042

  • Grandi U, Loreggia A, Rossi F, Venable KB, Walsh T (2013) Restricted manipulation in iterative voting: condorcet efficiency and Borda score. In: Proceeding of the 3rd International Conference on Algorithmic Decision Theory (ADT). Springer, Berlin, Heidelberg, pp 181–192

  • Hodge JK (2006) Permutations of separable preference orders. Discrete Appl Math 154(10):1478–1499

    Article  Google Scholar 

  • Hodge JK, Krines M, Lahr J (2009) Preseparable extensions of multidimensional preferences. Order 26(2):125–147

    Article  Google Scholar 

  • Kavner J, Xia L (2021) Strategic behavior is bliss: iterative voting improves social welfare. Adv Neural Inf Proc Syst 34:19021–19032

    Google Scholar 

  • Lacy D, Niou EMS (2000) A problem with referendums. J Theor Polit 12(1):5–31

    Article  Google Scholar 

  • Laffont JJ (1987) Incentives and the allocation of public goods, chap 10. In: Alan J, Feldstein AM (ed) Handbook of public economics, vol 2. Elsevier, pp 537–569

    Google Scholar 

  • Lang J, Xia L (2009) Sequential composition of voting rules in multi-issue domains. Math Soc Sci 57(3):304–324

    Article  Google Scholar 

  • Lang J, Xia L (2016) Voting in combinatorial domains. In: Brandt F, Conitzer V, Endriss U, Lang J, Procaccia AD (Eds) Handbook of computational social choice. Cambridge University Press, pp 197–222

    Chapter  Google Scholar 

  • Lang J, Mengin J, Xia L (2018) Voting on multi-issue domains with conditionally lexicographic preferences. Artif Intell 265:18–44

    Article  Google Scholar 

  • Laury S (2005) Pay one or pay all: random selection of one choice for payment. Andrew Young School of Policy Studies Research Paper Series 06-13

  • Lev O, Rosenschein JS (2016) Convergence of iterative scoring rules. J Artif Intell Res 57:573–591

    Article  Google Scholar 

  • Liekah L (2019) Multiagent reinforcement learning for iterative voting. Master’s thesis, Université Jean Monnet and Université Toulouse 1 Capitole, France

  • Meir R (2017) Iterative voting. In: Endriss U (ed) Trends in computational social choice. AI Access, pp 69–86

    Google Scholar 

  • Meir R, Polukarov M, Rosenschein J, Jennings N (2010) Convergence to equilibria in plurality voting. In: Proceedings of the AAAI Conference on Artificial Intelligence, vol 24, no 1, pp 823–828

  • Meir R, Polukarov M, Rosenschein JS, Jennings NR (2017) Iterative voting and acyclic games. Artif Intell 252:100–122

    Article  Google Scholar 

  • Meir R, Gal K, Tal M (2020) Strategic voting in the lab: compromise and leader bias behavior. Auton Agents Multi Agent Syst 34(1):31

    Article  Google Scholar 

  • Obraztsova S, Markakis E, Polukarov M, Rabinovich Z, Jennings NR (2015) On the convergence of iterative voting: How restrictive should restricted dynamics be? In: Proceedings of the AAAI Conference on Artificial Intelligence, vol 29, no 1

  • Özkal-Sanver I, Sanver MR (2006) Ensuring pareto optimality by referendum voting. Soc Choice Welf 27(1):211–219

    Article  Google Scholar 

  • Powell O (2019) jtree—a Javascript toolbox for running economics experiments. Github https://opowell.github.io/jtree. Accessed 27 Oct 2022

  • Reijngoud A, Endriss U (2012) Voter response to iterated poll information. In: Proceedings of the 11th international conference on autonomous agents and multiagent systems, vol 2, pp 635–644

  • Van der Straeten K, Laslier J, Sauger N, Blais A (2010) Strategic, sincere, and heuristic voting under four election rules: an experimental study. Soc Choice Welf 35(3):435–472

    Article  Google Scholar 

  • Wilczynski A (2019) Poll-confident voters in iterative voting. In: The 33rd AAAI Conference on Artificial Intelligence (AAAI), pp 2205–2212

Download references

Acknowledgements

We are grateful to three anonymous reviewers whose reports improved the paper substantially. We are very much indebted to Jean-François Bonnefon who was the first to have the idea of this experiment about a decade ago and who had started to design one. We also thank Jean-François Laslier for useful suggestions about the protocol. Last but not least, we thank Maxim Frolov and Owen Powell for their huge support before and during the lab sessions. We thank the participants of the Dagstuhl seminar 19381 on “Application-Oriented Computational Social Choice,” as well as our colleagues at the Institut de Recherche en Informatique de Toulouse (IRIT) for participating to various pilot versions of our lab experiment. Umberto Grandi acknowledges the support of the ANR JCJC project SCONE (ANR 18-CE23-0009-01). Jérôme Lang acknowledges the support of French government under management of Agence Nationale de la Recherche as part of the “Investissements d’avenir” program, reference ANR-19-P3IA-0001 (PRAIRIE 3IA Institute). Ali Ozkes acknowledges the support of the ANR MRSEI project EPURAI (ANR-21-MRS2-0027-01).

Author information

Authors and Affiliations

  1. Institut de Recherche en Informatique de Toulouse (IRIT), Université Toulouse 1 Capitole, Toulouse, France

    Umberto Grandi

  2. CNRS, LAMSADE, Université Paris-Dauphine, Université PSL, Paris, France

    Jérôme Lang

  3. SKEMA Business School, Université Côte d’Azur, GREDEG, Paris, France

    Ali I. Ozkes

  4. LAMSADE, Université Paris-Dauphine, Université PSL, Paris, France

    Stéphane Airiau

Authors
  1. Umberto Grandi
    View author publications

    Search author on:PubMed Google Scholar

  2. Jérôme Lang
    View author publications

    Search author on:PubMed Google Scholar

  3. Ali I. Ozkes
    View author publications

    Search author on:PubMed Google Scholar

  4. Stéphane Airiau
    View author publications

    Search author on:PubMed Google Scholar

Corresponding author

Correspondence to Ali I. Ozkes.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix 1: further experimental findings

1.1 Dominated choices and voters’ profiles

In one-shot elections, 26 out of 87 female subjects (30%) made a choice for a dominated option (either for one issue or both), whereas 9 out of 48 male subjects (19%) made a choice for a dominated option. The difference is not statistically significant (Fisher’s test statistic value is 0.2182). Note that 12 subjects did not reveal their gender. We do not observe a significant difference in iterative elections either (see Table 15).

Table 15 The number (percentage) of subjects who made a dominated choice in one-shot elections and first stages of the iterative elections, by declared gender
Full size table

As for the effect of formal education in economics, 22 out of 74 subjects (30%) who did not have a formal education in economics made a choice for a dominated option (either for one issue or both), whereas 14 out of 72 subjects (19%) who had a formal economic education made a choice for a dominated option. The difference is not statistically significant at 5% level as the Fisher exact test statistic value is 0.4357.

1.2 Best responses

We calculated the percentage of best responses (i.e., vote for their favourite cell) among the vote changes induced by pivotality (see Table 16), finding that in a majority of the cases subjects best respond when pivotal on both issues.

Table 16 The percentage of best-response choices when subjects are pivotal on both issues
Full size table

We have also checked whether voters with fully nonseparable preferences would vote for the opposite of their favourite cells, when the perceived probability of being pivotal on both issues is very low, in line with the analysis by Ahn and Oliveros (2012). Instances of such a phenomenon occur in the following circumstances, (1) fully non separable preferences, such as voters 1-6 in Profile 7; (2) the current outcome is the second worst outcome (BR for these 6 voters); (3) the margin of victory for both issues is large (e.g., without counting the voter, 5-1 or 6-0); (4) at the next round, the voter changes their vote on both issues (TL for the latter 6 voters). However, we cannot say much: among four occurrences of (1) and (2), there are two that comply with our argument. This could be worth line of inquiry to pursue in future research with an experimental design that is aimed to focus on this phenomenon.

1.3 Dynamic and static groups

To refine our analysis of quality of the final outcome, we divided the groups of voters into two categories in terms of the evolution of election outcomes. The static are those groups who terminate at the third iteration stage, and the dynamic groups are those who go further.

We observe that for both types of groups there are significant numbers of subjects who change their votes. More precisely, the percentage of subjects in dynamic groups who update their vote in the first three rounds is in the range of 20–49%, and the same figure is in the range of 13–36% in static groups, depending on the different profiles played and the iteration stage (see Table 17).

Table 17 Percentage of subjects who change their vote from 1st to 2nd and 2nd to 3rd stages in iterated profiles
Full size table

As for the average social welfare, Tables 18 and 19 show the average social welfare and the frequency of elected Condorcet winners in the first and last stage of iteration for Profiles 5–9, in both aggregated form and divided between dynamic and static groups. The maximum and minimum possible values for the average social welfare are also included.

Table 18 Average social welfare in the first and last iteration stage
Full size table
Table 19 Frequency of election of the Condorcet winner in iterated profiles in the first and last iteration
Full size table

1.4 Relative social welfare

Figure 8 shows the differences between the average social welfare and minimum possible social welfare as a percentage of the range (maximum minus minimum) for Profiles 5 to 8, with confidence intervals, for both the first stage and last stage.

Fig. 8
Fig. 8
Full size image

Difference between the average social welfare and minimum possible social welfare as a percentage of the range (maximum minus minimum) for Profiles 5–8, with 90% confidence intervals. We excluded the results on Profile 9 as the first and last stage has the same average social welfare in this profile, which leads to (Average SW – Min SW)/(Max SW – Min SW) = 61%

Appendix 2: experimental Instructions

Authors’ note: In each session, the experiment consisted of 9 main parts, followed by a questionnaire and a recap of earnings. The first four parts consisted of 4 one-shot votes, whereas each of the remaining 5 parts consisted of iterated votes. The instructions, which were printed on screens as well, were read aloud in the beginning of the first and fifth parts. The following is the English translations of the original instructions in French as appeared on subjects’ screens.

Instructions

Part 1

You will participate in an experiment that takes place in several parts. The experiment has 9 parts.

The first four parts consist of single turns. The following parts consist of several turns. In each game, you will see a table like this:

$$\begin{aligned}\begin{array}{rcc} &{} \mathrm {LEFT} &{} \mathrm {RIGHT}\\ \mathrm {TOP} &{} 4 &{} 2\\ \mathrm {BOTTOM} &{} 1 &{} 0 \end{array}\end{aligned}$$

The number that is shown in each of the boxes is what you earn if that box wins when the game ends. For example, if the winning box when the game ends is UP-RIGHT, then for that game you earn €2. The numbers will always be at least as large as 0: you will never lose anything. Your final earning will be the sum of your earnings in the 6 games.

We will now explain what determines that a box is a winner.

You will vote for a row (UP or DOWN) and for a column (LEFT or RIGHT). In each game, there will be a number of other players who will also vote for a row and for a column. Other players may have different arrays from yours, and you cannot know the values of their arrays. However, you will know the number of players (which, depending on the case, will be 3, 5 or 7).

If a majority of players voted LEFT, then the winning row is LEFT. If a majority of players voted for RIGHT, then the winning row is RIGHT.

If a majority of players voted UP, then the winning column is UP. If a majority of players voted DOWN, then the winning column is DOWN.

The winning box is at the intersection of the winning row and the winning column.

If for example there are 5 players, three vote UP and two vote DOWN, then the winning row is UP. If four players vote for RIGHT and one for LEFT, then the winning column is RIGHT, and the winning box is UP-RIGHT.

During the first four elections, there will be only one round: you will vote for a row and a column, the other players too, and you will win the sum that corresponds to the winning box in the table.

Part 2

Part with only one round.

Part 3

Part with only one round.

Part 4

Part with only one round.

Part 5

The following parts will consist of several rounds. In each round you will vote for a row and a column.

Then it will be the second round: you will vote again, and at the end of the second round, again you will know the winning box and the number of votes for UP, for LOW, for LEFT, and for RIGHT. And so on, until the election ends. The election will end when the winning box is the same three times in a row or when 12 turns have been played.

For example:

  • first round: UP 3, DOWN 2, LEFT 1, RIGHT 4. Winning box: UP-RIGHT

  • second round: UP 2, DOWN 3, LEFT 2, RIGHT 3. Winning box: DOWN-RIGHT

  • third round: UP 2, DOWN 3, LEFT 3, RIGHT 2. Winning box: DOWN-LEFT

  • fourth round: UP 1, DOWN 4, LEFT 2, RIGHT 3. Winning box: BOTTOM-RIGHT

  • fifth round: UP 0, DOWN 5, LEFT 2, RIGHT 3. Winning box: BOTTOM-RIGHT

  • sixth round: UP 1, DOWN 4, LEFT 2, RIGHT 3. Winning box: BOTTOM-RIGHT.

The DOWN-RIGHT box has won three times in a row: the game is over, and you win the amount that corresponds to the DOWN-RIGHT box.

Please note: The number of participants may vary for each part (games with 3, 5 or 7 participants). When the number of participants in a game is 5, one person in the room will not play that game.

Please read these instructions carefully. If you have questions now is the time to ask them.

Part 6

Part with several rounds.

Part 7

Part with several rounds.

Part 8

Part with several rounds.

Part 9

Part with several rounds.

Questionnaire

Recap of earnings

1.1 Questionnaire

  1. 1.

    What is your profession?

  2. 2.

    Do you have any comments on the methods, or suggestions on how to organize such choices on several issues?

  3. 3.

    What is your age range?

    • No answer

    • 18–24

    • 25–34

    • 35–44

    • 45–54

    • 55–64

    • 65+

  4. 4.

    You are

    • No answer

    • A woman

    • A man

    • Other

  5. 5.

    What is the highest degree you have obtained?

    • No answer

    • Postgraduate degree, doctorate, grande école, engineer

    • Graduate degree

    • Undergraduate degree, BTS, DUT, or equivalent, Bac + 2 level

    • General, technological, professional or equivalent baccalaureate

    • CAP, BEP or diploma of the same level

    • College diploma, BEPC

    • Primary school certificate, no diploma

  6. 6.

    Have you ever taken a course in economics?

    • No answer

    • Yes, this is my main training

    • Yes, but this is not my main training

    • No

  7. 7.

    Have you ever participated in an economic laboratory survey?

    • No answer

    • Yes

    • No it’s the first time

  8. 8.

    You have tested two choice methods on multiple questions: one where you vote only once for each of the two questions, and one where you vote in several stages. Which do you think is the most effective?

    • No answer

    • The one-step method

    • The multi-step method

    • Both methods are equally effective

    • I don’t know / I prefer not to answer

  9. 9.

    Does the one-step method seem easy to understand and use?

    • No answer

    • Easy

    • Rather easy

    • Rather difficult

    • Difficult

    • I don’t know / I prefer not to answer

  10. 10.

    Does the multi-step method seem easy to understand and use?

    • No answer

    • Easy

    • Rather easy

    • Rather difficult

    • Difficult

    • I don’t know / I prefer not to answer

1.2 Screenshots

See Figs. 9, 10.

Fig. 9
Fig. 9
Full size image

Fourth round in the sixth election. Note that the outcomes from past rounds appear on the right in reverse order. The subjects make their decisions in the middle area by choosing a row and a column and validating

Fig. 10
Fig. 10
Full size image

Result screen after the first election

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Grandi, U., Lang, J., Ozkes, A.I. et al. Voting behavior in one-shot and iterative multiple referenda. Soc Choice Welf 63, 641–675 (2024). https://doi.org/10.1007/s00355-022-01436-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Version of record:

  • Issue date:

  • DOI: https://doi.org/10.1007/s00355-022-01436-0

Profiles

  1. Umberto Grandi View author profile
  2. Jérôme Lang View author profile
  3. Ali I. Ozkes View author profile