The Pareto Efficiency and Expected Costs of k-Majority Rules

Several authors have analyzed the optimal k -majority rule based on a variety of criteria. Buchanan and Tullock argued that, in constitutional settings, the criterion should be that all changes meet the Pareto criterion; otherwise the status quo should be preferred (we call this the BT criterion). They then asserted that unanimity rule would be the preferred voting rule in this setting. In parliamentary settings, they claimed that a near majority rule would be preferred because it minimizes the sum of decision costs and external costs. This article investigates both claims in an N -voter, two-alternative setting. We show the conditions under which unanimity rule is less likely to select BT preferred alternatives than other k -majority rules and prove that the difference in performance can be negligible when N is large and certain weak conditions are met. Furthermore, if we define external costs as the expected number of losers from a BT-inferior vote, then external costs become negligible for a range of supermajority rules. This implies that unanimity rule and a range of supermajority rules should be equally preferred when decision costs are added. Finally, we show that the external cost function can actually increase for certain populations. Many of the broader conclusions should also hold for multiple alternatives.