The Italian referendum: what can we get from game theory?

In Italy, the referendum represents the main form of direct democracy. At the national level, there exist 2 main forms of referendum: an abrogative referendum, in which the electorate is called to vote on whether they wish to abolish an existing law, and a constitutional referendum, which can be requested in some cases when a new constitutional law is approved by the Parliament. In the first case, the referendum has to meet a certain turnout requirement in order to be valid, namely, a participation quorum threshold has to be reached. The rationale for such a requirement is that, to change the status quo, a large proportion of citizens should care about the issue at stake and take part in the decision. In our work, we provide a game theoretic analysis of a voting rule with a participation quorum threshold. In particular, we focus on a binary dichotomous voting rule, in which the choices are vote “yes” and vote “no”, on a 3-option dichotomous voting rule, in which there is the additional choice to “stay at home”, and on a quaternary dichotomous voting rule, in which it is also possible to “abstain”. The possible outcomes are two, namely “approval” and “rejection”. We provide a graphical representation of these aforementioned voting rules, which allows for an easier analysis of these well-known voting scenarios, in particular focusing on the case of the Italian referendum. We analyze how the decisiveness (as a measure of agility), the blocking power (as a measure of inertia) of such voting situations, both at a collective and at an individual level, and the configurations in which a voter can become a swing voter are strongly impacted by the quorum threshold.