LINEAR VOTING RULES

How should a society choose between two social alternatives if participation in the decision process is voluntary and costly, and monetary transfers are not feasible? Assuming symmetric independent private values, we show that it is utilitarian-optimal to use a linear voting rule: votes get alternative-dependent weights, and a default obtains if the weighted sum of votes stays below some threshold. Any combination of weights and threshold can be optimal. A standard quorum rule can be optimal only when it yields the same outcome as a linear rule. A linear rule is called upper linear if the default is upset at every election result that meets the threshold exactly. We develop a perturbation method to characterize equilibria of voting rules in the case of small participation costs and show that leaving participation voluntary increases welfare for any two-sided upper linear rule that is optimal under compulsory participation.