Sen’s Liberal paradox describes a conflict between weak Pareto, minimal liberalism, and either transitivity or a best element over a domain of individual preferences. This paper examines variants of that paradox with varying amounts of unconditional preferences. We define a notion of unconditional preferences under which, in the absence of Pareto, there can be no cycles. We then define a stronger condition, that makes an individual’s preferences for her own private attributes independent of all other attributes. Under this assumption, there (...) can be no cycles with or without Pareto. We also show there exists a social decision function satisfying those conditions. We then determine the probability of a cycle assuming a much weaker independence condition that does not restrict the domain. This probability converges to one as the number of non-private attributes within the social states increases. Finally, we use simulations to determine the probability that liberalism and Pareto conflict with best elements, maximal elements, and transitivity separately. (shrink)

Florida International University, USA edwardj{at}fiu.edu ' + u + '@ ' + d + ' '/ /- -> 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. 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. Key Words: constitutional design • Pareto criterion • external costs • social contract. (shrink)