What are the best quorum rules? A laboratory investigation


Many political systems with direct democracy mechanisms have adopted rules preventing decisions from being made by simple majority rule. The device added most commonly to majority rule in national referendums is a quorum requirement. The two most common are participation and approval quorums. Such rules are responses to three major concerns: the legitimacy of the referendum outcome, its representativeness, and protection of minorities regarding issues that should demand a broad consensus. Guided by a pivotal voter model, we conduct a laboratory experiment to investigate the performances of different quorums in attaining such goals. We introduce two main innovations in relation to previous work on the topic. First, part of the electorate goes to the polls out of a sense of civic duty. Second, we test the performances of a different quorum, the rejection quorum, recently proposed in the literature. We conclude that, depending on the preferred criterion, either the approval or the rejection quorum is the best.

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  1. 1.

    We use both terms interchangeably.

  2. 2.

    For a detailed treatment of existing quorum rules across the world, see, for example, Kaufmann et al. (2010).

  3. 3.

    “EU referendum petition signed by more than 2.5 m”, BBC News, 25 June 2016. Available at: https://www.bbc.com/news/uk-politics-eu-referendum-36629324.

  4. 4.

    For a variety of historical and contemporary examples, see among many, West (1985), Suksi (1993), Uleri (2002), Kaufmann et al. (2010), and International IDEA (2008: 182). See also Offe (2017) on the hypothetical consequences of a participation quorum in the Brexit referendum. For cases of demobilization under approval quorums, see Suksi (1993: 211), Svensson (1996: 38–40), and Verhulst and Nijeboer (2007: 19–21).

  5. 5.

    Take the result (5,4) under an approval quorum: Changers satisfy the quorum and, nevertheless, are defeated. Conversely, under a rejection quorum, a (4,5) result means that conservatives satisfy the rejection quorum and, by doing so, win the referendum even though changers have more votes.

  6. 6.

    Although it is, of course, impossible to determine the “true” share of the electorate for whom that is the case, the evidence suggests that 25% is not entirely implausible. On the one hand, in the United States, turnout in state, local or special elections that aren’t concurrent with national ballots have turnout rates below 30% and sometimes even below 20% (Hajnal and Lewis 2003). On the other hand, the percentage of survey respondents who disagree with the statement “If a person doesn’t care how an election comes out, then the person shouldn’t vote in it” has hovered around 50%, despite evidence of a generational decline in the attachment to a sense of duty invoting (Blais and Rubenson 2013).

  7. 7.

    With that calibration, the results of the experiment are directly comparable to Aguiar-Conraria et al. (2016): if we eliminate consumption motives for voting and maintain its total benefits, the two models will be equivalent.

  8. 8.

    Like many others, e.g., Levine and Palfrey (2007), for tractability, we focus on symmetric equilibria. That focus is probably not very restrictive, as Palfrey and Pogorelskiy (2017) and Kuzmics and Rogers (2010) point out. In particular, Alós-Ferrer and Kuzmics (2013) argue that symmetric equilibria are good candidates for becoming focal points. The literature has found asymmetric equilibria to be particularly relevant in voting games with sequential decisions (Ladha et al. 1996), with approval voting (Bouton et al. 2016) or common voting costs (Kalandrakis 2009). None of those features is present in our experiment.

  9. 9.

    With no quorum or with a rejection quorum, we always have found unique equilibria, with or without negative costs.

  10. 10.

    To check if we could narrow down, even more, the set of equilibria and to add a new layer of realism to the model, we also considered quantal response equilibria, allowing players to make errors, see Mckelvey and Palfrey (1998). However, given that the equilibria were similar to the SBNE in Table 1 we do not report those results here.

  11. 11.

    The experiment was organized and subjects recruited with the software hroot (Bock et al. 2012). The experiment was programmed using the z-Tree software (Fischbacher 2007). Students came from various disciplines (28% economics, 24% other social sciences, 15% natural sciences, 10% humanities, 23% other). More than half (59%) of our subjects were female.

  12. 12.

    Subjects were not informed explicitly that they would interact repeatedly with the same set of participants. It is important to note that despite such a “fixed matching” scheme, subjects were assigned randomly to the two “teams” at the beginning of each round.

  13. 13.

    Meaning that WTP is the probability of voting.

  14. 14.

    This method of payment was chosen as a compromise between avoiding paying for all rounds (introducing wealth effects) and paying only for one round (introducing additional risk). See Morton and Williams (2010, p. 399) for a discussion of our methodological choice.

  15. 15.

    Note that the expressions (6/3), (5/4), (4/5) and (3/6) do not represent the probabilities of favoring change (i.e., µ). For example, a clear majority for the status quo (6/3) corresponds to µ = 3/9.

  16. 16.

    In all sessions, the probability of favoring change cycled deterministically as follows: 3/9, 6/9, 4/9 and 5/9. The sequence of no quorum/quorum conditions was counterbalanced. In half of the sessions, the sequence was: 8 rounds with quorum, 4 without, 8 with, and so on. In the other sessions, we began with 4 rounds with no quorum, 8 with, and so on. (We conduct twice as many rounds with quorums because when they are in place, the game is not symmetric and so we acquire fewer observations when distinguishing between changers and conservatives.) All random team assignments were drawn once prior to the first session and kept constant in all sessions. That is, the realized numbers of changers and conservatives were the same in all treatments.

  17. 17.

    By “correct outcome” we mean the outcome that corresponds to the will of the majority of the population, i.e., the outcome that would prevail if turnout were 100%.

  18. 18.

    We also can see the same effect in Online Appendix 2.A. The standard errors of the estimated coefficients for conservatives under the participation quorum treatment are larger than for the other treatments.

  19. 19.

    Moreover, as long as coordination is hard, not only in the lab but also in real life, the multiplicity of equilibria under the participation quorum will add noise to referendum outcomes, making the participation quorum even less desirable.


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This paper is financed by National Funds of the FCT—Portuguese Foundation for Science and Technology within the projects UID/ECO/03182/2019 and UID/SOC/50013/2019, and the research grant PTDC/IVC-CPO/4925/2014, with the FCT/MEC’s (Fundação para a Ciência e a Tecnologia, I.P.) financial support through national funding and by the ERDF through the Operational Programme on “Competitiveness and Internationalization–COMPETE 2020” under the PT2020 Partnership Agreement.

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Aguiar-Conraria, L., Magalhães, P.C. & Vanberg, C.A. What are the best quorum rules? A laboratory investigation. Public Choice 185, 215–231 (2020). https://doi.org/10.1007/s11127-019-00749-6

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  • Election design
  • Participation quorum
  • Approval quorum
  • Laboratory experiment

JEL Classification

  • C91
  • D72
  • D02