An expected utility analysis of k-majority rules

Buchanan and Tullok (The calculus of consent: logical foundations of constitutional democracy. University of Michigan Press, Ann Arbor, 1962) argued that the optimal k-majority rule should minimize the sum of external costs and decision costs. Dougherty et al. (Public Choice, 163(1–2):31–52, 2015) formalized their approach using various groups of voters. In this study, we analyze the optimal k-majority rule in terms of expected utility and compare our results to Dougherty et al. (2015), which focuses on costs alone. Specifically, we replace Buchanan and Tullock’s external cost function with an external utility function that accounts for both the benefits and costs of enacting proposals. We find that analyzing k-majority rules in terms of utility, rather than costs, affects the optima. Furthermore, in terms of utility, the optimal k-majority rule can vary depending on the group one expects to be in during a vote. With some interesting exceptions, individuals from groups that favor the proposal often find small k-majority rules optimal. Individuals from groups that oppose the proposal often find large k-majority rules optimal.