The value of formalism: re-examining external costs and decision costs with multiple groups

Abstract

Several authors have examined the optimal k-majority rule using a variety of criteria. We formalize and extend the original argument laid out by Buchanan and Tullock (The calculus of consent: logical foundations of constitutional democracy, 1962) using a decision theoretic analysis from the perspective of an individual voter. Unlike previous formalizations, voters in our study are members of one or more groups. This allows us to examine cases wherein different voters have starkly different interests. Furthermore, voters in our study can err in their judgments of proposals allowing us to model potential irra- tionalities in the choice of an optimal k-majority rule. We consider both up or down votes on a single proposal as well as votes over a series of proposals. We find that the optimal k- majority rule depends on a number of parameters, most notably the number of rounds needed to create a proposal that will pass. Group membership has almost no affect. Fur- thermore, if two groups are at odds, then the external cost function can actually rise over some range of k; if voters err systematically in their judgment, more inclusive k-majority rules, such as unanimity rule, can fail to pass Pareto preferred proposals. Our results should help advance a classic work in Public Choice.