Counting votes in coupled decisions

Abstract

We consider scenarios with distributed decision processes, e.g., coupled majorities and personal union in parliament chambers, supranational decisions and supervisory boards. When computing the adoption rate for reaching a decision in these scenarios, multiple linear inequality restrictions in combinatorial countings are present. These rates cannot be computed in closed form. We introduce a general method for incorporating multiple inequality conditions in multiple majority decisions, which significantly reduces the number of involved summations and removes restrictions on the summation indices. Exact solutions are provided through (a) integral representations which can be evaluated numerically, and (b) unrestricted, contracted sums over discrete events. Further, we provide methods to reduce the number of necessary summations by splitting or recurring the original problem to easier sub-problems. For five dedicated scenarios, full results are given which indeed require a single unrestricted summation only.