The Integer Arithmetic of Legislative Dynamics

Every legislature may be defined by a finite integer partition of legislative seats between parties and a winning vote quota defining the minimum coalition size required to pass decisions. In this paper we explore the finite set of integer partitions of legislatures, categorizing all legislatures as one of five basic types. As legislatures approach the partition thresholds between these categories, they approach the thresholds for changing the set of winning coalitions. The criteria defining such partitions of legislatures thus define important thresholds for shifting the legislature from one bargaining environment to another. These different bargaining environments will generate different bargaining expectations for at least some of the set of parties, which in turn may create incentives for defections of legislators from one party to another, in order to shift the legislature from one bargaining environment to another and thereby create a gain in bargaining expectations for the switching legislator and the party to which she defects. The paper proceeds in five parts. First, we develop a theoretical categorization of legislatures into those likely a priori to have different bargaining environments. Second, we analyze the role of a particular type of party that is especially privileged under particular integer partitions of a legislature, which we call a k-dominant party. Third, we examine the category frequencies and partition characteristics in a “metadata” set that contains to universe of all logically possible integer partitions of seats between up to ten parties in a 100-seat legislature. Fourth, we examine similar patterns in real legislatures, linking the categories to outcomes such as control of the prime minister and single-party minority government. Finally, we explore the extent to which all of this helps us to explain party switching by individual legislators. ∗ Prepared for presentation at the Workshop on Party Switching, July 10-14, 2005, University of Virginia, Charlottesville, Va. With heartfelt thanks to Macartan Humphries for invaluable comments on an earlier draft. Integer arithmetic of legislative dynamics / 2