Optimal majority rule in referenda

Adopting the group turnout model of Herrera and Mattozzi, J. Eur. Econ. Assoc. 2010, 8, 838-871, we investigate direct democracy with supermajority rule and different preference intensities for two sides of a referendum: Reform versus status quo. Two parties spend money and effort to mobilize their voters. We characterize the set of pure strategy Nash equilibria. We investigate the optimal majority rule that maximizes voters' welfare. Using an example, we show that the relationship between the optimal majority rule and the preference intensity is not monotonic-the optimal majority rule is initially decreasing and then increasing in the preference intensity of the status quo side. We also show that when the preference intensity of the status quo side is higher, the easiness to mobilize voters on the status quo side is lower, or the payoff that the reform party receives is higher, the optimal majority rule is more likely to be supermajority.