Approval quorums dominate participation quorums

See e.g. Matsusaka (2005a, b) for an account of the increasing use of direct democracy around the world.

See also LeDuc (2003) for a discussion of the fear to have a minority of the population prevail over a passive majority.

Among the examples of participation quorums used in reality, the Italian example is the most used, even if similar quorums exist in other countries. For the approval quorum type of rules, on the other hand, Germany is the most recognized example. See Corte-Real and Pereira (2004) for a description of the various types of turnout requirements used in the world and for an axiomatic discussion.

In Italian politics, from 1996 until 2011, all popular referenda have failed because of low voter turnout (30 % or lower). More specifically, referenda in 1997 (seven bills, various topics including hunting, abolishing agrarian ministry, and conscientious objectors), 1999 (elimination of PR), 2000 (seven bills, dealing with electoral law, unions, and judges), 2003 (rehiring illegally fired workers and electricity on private property), 2005 (stem cells and IVF), 2009 (electoral law) all failed. The clearest case of strategic abstention was for the 2005 referendum: overall turnout was only 24–26 %, and is thought to be low due to encouragement from Vatican and Catholic bishops to abstain. For the four initiatives, of those voting approximately 88 %were for eliminating research limitations and 78 % for allowing IVF. In 2009 the question was on eliminating connection between lists and giving majority prize to coalition of lists in the Chamber of Deputies and Senate, respectively. Turnout was 23 % for both (and of those voters, 22 % voted against); and for eliminating the ability of a candidate to stand for election in more than one constituency, turnout was 23 % (with 13 % voting against). See http://​www.​economist.​com/​blogs/​newsbook/​2011/​06/​italys-referendums. See also Herrera and Mattozzi (2010), Hizen and Shinmyo (2011), and Aguiar-Conraria and Magalhaes (2010a), who find evidence from cross country data from 1970 to 2007. See also Zwart (2009) for a discussion of how “high” a participation quorum should be, conditional on having chosen to have one.

See Aguiar-Conraria and Magalhaes (2010b) for a different analysis of this second type of mistake.

If one believes that elections should be studied focusing on parties’ “mobilization” efforts rather than on individual strategic voting, then the key difference between approval and participation quorums that we emphasize becomes undiscernable. This is why Herrera and Mattozzi (2010), using a mobilization model, found no substantial difference between participation and approval quorums, even if they find that parties may face the two kinds of mistakes that we describe above. Similarly, the two quorum rules are also difficult to compare when using ethical voting models like Feddersen and Sandroni (2006) or Coate and Conlin (2004), because in ethical voting models voters (if guided by the same type of group utilitarianism) are assumed to coordinate as if they were mobilized by a leader.

We define social preferences in terms of expected vote shares. We could have defined them in terms of actual vote shares. Given that we concentrate on equilibria in sufficiently large populations, the difference between the two, that is, the probability that the recommendation of social preferences based on expected shares differs from that based on actual shares, is negligible.

The quorums are expressed here in terms of absolute numbers of voters. They could not be expressed in fraction terms, as our population is potentially unbounded, due to our assumption of a Poisson distribution. This distribution, however, can be viewed as an approximation of a binomial distribution, where our n parameter corresponds to the expected number of interested citizens, pN, where N would be the actual size of the population and p the common probability of being called (by nature) to go to vote. With this interpretation, the quorums, in terms of a fraction of N, are pq and \(p\widehat{q}\).

See Venice Commission (2005) for the underpinnings of the various existing rules. One potential rationale for participation requirements, already mentioned in the introduction, is the fear that the majority of voters could be insufficiently informed and insufficiently motivated to express a preference for the status quo even though their total utility is negatively affected by the reform, which is instead pushed through by an informed and motivated minority.

It is the same stability requirement as in, for instance, Feddersen and Pesendorfer (1996). When there are three equilibria, this stability requirement rules out the intermediate equilibrium, as any perturbation of the expectations of other players’ behavior make all agents shift towards the strategies that form one of the other two equilibria (which, of course, are stable in that sense).

Given that we have an unbounded number of potential players, describing individual strategies and considering asymmetric equilibria would be complex. In the symmetric equilibrium that we select all agents of any given type use the same voting strategy.

Castanheira (2003) succeeds in characterizing the equilibria in a model with Poisson uncertainty and voting costs. A key step in his result is the explicit derivation of the probability of being pivotal. The same strategy cannot be applied here, because the modified Bessel function that Castanheira uses cannot be used to compute pivotal probabilities when the pivotal event depends on the sum of votes for two alternatives, as it is the case with quorums.